A note on binomial transform of the generalized fifth order Jacobsthal numbers

1. Introduction and preliminaries

In this paper, we introduce the binomial transform of the generalized fifth order Jacobsthal sequence and we investigate, in detail, two special cases which we call them the binomial transform of the fifth order Jacobsthal and fifth order Jacobsthal-Lucas sequences. We investigate their properties in the next sections. In this section, we present some properties of the generalized $(r,s,t,u,v)$ sequence (generalized Pentanacci) sequence.

The generalized $(r,s,t,u,v)$ sequence (the generalized Pentanacci sequence or 5-step Fibonacci sequence)

$$\{W_n{\}}_{n\ge 0}=\{W_n(W_0,W_1,W_2,W_3,W_4;r,s,t,u,v){\}}_{n\ge 0}$$ is defined by the fifth-order recurrence relations where the initial values $W_0,W_1,W_2,W_3,W_4$are arbitrary complex (or real) numbers and $r,s,t,u,v$ are real numbers. Pentanacci sequence has been studied by many authors and more detail can be found in the extensive literature dedicated to these sequences, see for example [1,2,3,4,5]. The sequence $\{W_n{\}}_{n\ge 0}$ can be extended to negative subscripts by defining $$W_{-n}=-\frac{u}{v}{W}_{-(n-1)}-\frac{t}{v}{W}_{-(n-2)}-\frac{s}{v}{W}_{-(n-3)}-\frac{r}{v}{W}_{-(n-4)}+\frac{1}{v}{W}_{-(n-5)}$$ for $n=1,2,3,\dots .$ Therefore, recurrence (1) holds for all integer $n.$

As $\{W_n\}$ is a fifth order recurrence sequence (difference equation), it’s characteristic equation is

whose roots are $\alpha ,\beta ,\gamma ,\delta ,\lambda .$ Note that we have the following identities: $$\begin{array}{rl}\alpha +\beta +\gamma +\delta +\lambda & =r,\\ \alpha \beta +\alpha \lambda +\alpha \gamma +\beta \lambda +\alpha \delta +\beta \gamma +\lambda \gamma +\beta \delta +\lambda \delta +\gamma \delta & =-s,\\ \alpha \beta \lambda +\alpha \beta \gamma +\alpha \lambda \gamma +\alpha \beta \delta +\alpha \lambda \delta +\beta \lambda \gamma +\alpha \gamma \delta +\beta \lambda \delta +\beta \gamma \delta +\lambda \gamma \delta & =t,\\ \alpha \beta \lambda \gamma +\alpha \beta \lambda \delta +\alpha \beta \gamma \delta +\alpha \lambda \gamma \delta +\beta \lambda \gamma \delta & =-u\\ \alpha \beta \gamma \delta \lambda & =v.\end{array}$$ Generalized Pentanacci numbers can be expressed, for all integers $n,$ using Binet’s formula.

Theorem 1. (Binet’s formula of generalized $(r,s,t,u,v)$ numbers (generalized Pentanacci numbers))

where $$\begin{array}{rl}{p}_1& =W_4{-(\beta +\gamma +\delta +\lambda )W}_3{+(\beta \lambda +\beta \gamma +\lambda \gamma +\beta \delta +\lambda \delta +\gamma \delta )W}_2{-(\beta \lambda \gamma +\beta \lambda \delta +\beta \gamma \delta +\lambda \gamma \delta )W}_1{+(\beta \lambda \gamma \delta )W}_0,\\ p_2& =W_4{-(\alpha +\gamma +\delta +\lambda )W}_3{+(\alpha \lambda +\alpha \gamma +\alpha \delta +\lambda \gamma +\lambda \delta +\gamma \delta )W}_2{-(\alpha \lambda \gamma +\alpha \lambda \delta +\alpha \gamma \delta +\lambda \gamma \delta )W}_1{+(\alpha \lambda \gamma \delta )W}_0,\\ p_3& =W_4{-(\alpha +\beta +\delta +\lambda )W}_3{+(\alpha \beta +\alpha \lambda +\beta \lambda +\alpha \delta +\beta \delta +\lambda \delta )W}_2{-(\alpha \beta \lambda +\alpha \beta \delta +\alpha \lambda \delta +\beta \lambda \delta )W}_1{+(\alpha \beta \lambda \delta )W}_0,\\ p_4& =W_4{-(\alpha +\beta +\gamma +\lambda )W}_3{+(\alpha \beta +\alpha \lambda +\alpha \gamma +\beta \lambda +\beta \gamma +\lambda \gamma )W}_2{-(\alpha \beta \lambda +\alpha \beta \gamma +\alpha \lambda \gamma +\beta \lambda \gamma )W}_1{+(\alpha \beta \lambda \gamma )W}_0,\\ p_5& =W_4{-(\alpha +\beta +\gamma +\delta )W}_3{+(\alpha \beta +\alpha \gamma +\alpha \delta +\beta \gamma +\beta \delta +\gamma \delta )W}_2{-(\alpha \beta \gamma +\alpha \beta \delta +\alpha \gamma \delta +\beta \gamma \delta )W}_1{+(\alpha \beta \gamma \delta )W}_0.\end{array}$$

Usually, it is customary to choose $r,s,t,u,v$ so that the Eq. (2) has at least one real (say $\alpha$) solutions. Eq. (3) can be written in the following form: $$W_n=A_1{\alpha }^n+A_2{\beta }^n+A_3{\gamma }^n+A_4{\delta }^n+A_5{\lambda }^n,$$ where $$\begin{array}{rl}{A}_1& =\frac{p_1}{(\alpha -\beta )(\alpha -\gamma )(\alpha -\delta )(\alpha -\lambda )},\\ A_2& =\frac{p_2}{(\beta -\alpha )(\beta -\gamma )(\beta -\delta )(\beta -\lambda )},\\ A_3& =\frac{p_3}{(\gamma -\alpha )(\gamma -\beta )(\gamma -\delta )(\gamma -\lambda )},\\ A_4& =\frac{p_4}{(\delta -\alpha )(\delta -\beta )(\delta -\gamma )(\delta -\lambda )},\\ A_5& =\frac{p_5}{(\lambda -\alpha )(\lambda -\beta )(\lambda -\gamma )(\lambda -\delta )}.\end{array}$$ Next, we give the ordinary generating function $\sum _{n=0}^{\mathrm{\infty }}{W}_n{x}^n$ of the sequence $W_n.$

Lemma 1. Suppose that $f_{W_n}(x)=\sum _{n=0}^{\mathrm{\infty }}{W}_n{x}^n$ is the ordinary generating function of the generalized $(r,s,t,u,v)$ sequence $\{W_n{\}}_{n\ge 0}.$ Then, $\sum _{n=0}^{\mathrm{\infty }}{W}_n{x}^n$ is given by

We next find Binet formula of generalized $(r,s,t,u,v)$ numbers $\{W_n\}$ by the use of generating function for $W_n.$

Theorem 2. (Binet’s formula of generalized $(r,s,t,u,v)$ numbers)

where $$\begin{array}{rl}{q}_1& =W_0{\alpha }^4{+(W}_1{-rW}_0{)\alpha }^3{+(W}_2{-rW}_1{-sW}_0{)\alpha }^2{+(W}_3{-rW}_2{-sW}_1{-tW}_0{)\alpha +(W}_4{-rW}_3{-sW}_2{-tW}_1{-vW}_0),\\ q_2& =W_0{\beta }^4{+(W}_1{-rW}_0{)\beta }^3{+(W}_2{-rW}_1{-sW}_0{)\beta }^2{+(W}_3{-rW}_2{-sW}_1{-tW}_0{)\beta +(W}_4{-rW}_3{-sW}_2{-tW}_1{-vW}_0),\\ q_3& =W_0{\gamma }^4{+(W}_1{-rW}_0{)\gamma }^3{+(W}_2{-rW}_1{-sW}_0{)\gamma }^2{+(W}_3{-rW}_2{-sW}_1{-tW}_0{)\gamma +(W}_4{-rW}_3{-sW}_2{-tW}_1{-vW}_0),\\ q_4& =W_0{\delta }^4{+(W}_1{-rW}_0{)\delta }^3{+(W}_2{-rW}_1{-sW}_0{)\delta }^2{+(W}_3{-rW}_2{-sW}_1{-tW}_0{)\delta +(W}_4{-rW}_3{-s3W}_2{-tW}_1{-vW}_0),\\ q_5& =W_0{\lambda }^4{+(W}_1{-rW}_0{)\lambda }^3{+(W}_2{-rW}_1{-sW}_0{)\lambda }^2{+(W}_3{-rW}_2{-sW}_1{-tW}_0{)\lambda +(W}_4{-rW}_3{-sW}_2{-tW}_1{-vW}_0).\end{array}$$

Matrix formulation of $W_n$ can be given as [6]: In fact, Kalman give the formula in the following form $$\left(\begin{array}{c}{W}_n\\ W_{n+1}\\ W_{n+2}\\ W_{n+3}\\ W_{n+4}\end{array}\right)={\left(\begin{array}{ccccc}0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\\ r& s& t& u& v\end{array}\right)}^n\left(\begin{array}{c}{W}_0\\ W_1\\ W_2\\ W_3\\ W_4\end{array}\right).$$ Next, we consider two special cases of the generalized $(r,s,t,u,v)$ sequence $\{W_n\}$ which we call them $(r,s,t,u,v)$ and Lucas $(r,s,t,u,v)$ sequences. $(r,s,t,u,v)$ sequence$\{G_n{\}}_{n\ge 0}$ and Lucas $(r,s,t,u,v)$ sequence $\{H_n{\}}_{n\ge 0}$ are defined, respectively, by the fifth-order recurrence relations The sequences $\{G_n{\}}_{n\ge 0}$ and $\{H_n{\}}_{n\ge 0}$ can be extended to negative subscripts by defining $$\begin{array}{rcl}{G}_{-n}& =& -\frac{u}{v}{G}_{-(n-1)}-\frac{t}{v}{G}_{-(n-2)}-\frac{s}{v}{G}_{-(n-3)}-\frac{r}{v}{G}_{-(n-4)}+\frac{1}{v}{G}_{-(n-5)},\\ H_{-n}& =& -\frac{u}{v}{H}_{-(n-1)}-\frac{t}{v}{H}_{-(n-2)}-\frac{s}{v}{H}_{-(n-3)}-\frac{r}{v}{H}_{-(n-4)}+\frac{1}{v}{H}_{-(n-5)},\end{array}$$ for $n=1,2,3,\dots$ respectively. Therefore, recurrences (7) and (8) hold for all integers $n.$

For more details on the generalized $(r,s,t,u,v)$ numbers, see [4].

Some special cases of $(r,s,t,u,v)$ sequence $\{G_n(0,1,r,r^2+s,r^3+2sr+t;r,s,t,u,v)\}$ and Lucas $(r,s,t,u,v)$ sequence $\{H_n(5,r,2s+r^2,r^3+3sr+3t,r^4+4r^{2}s+4tr+2s^2+4u;r,s,t,u,v)\}$ are as follows:

1. $G_n(0,1,1,2,4;1,1,1,1,1)=P_n,$ Pentanacci sequence,
2. $H_n(5,1,3,7,15;1,1,1,1,1)=Q_n,$ Pentanacci-Lucas sequence,
3. $G_n(0,1,2,5,13;2,1,1,1,1)=P_n,$ fifth-order Pell sequence,
4. $H_n(5,2,6,17,46;2,1,1,1,1)=Q_n,$ fifth-order Pell-Lucas sequence.

For all integers $n,$ $(r,s,t,u,v)$ and Lucas $(r,s,t,u,v)$ numbers (using initial conditions in (3) or (5)) can be expressed using Binet’s formulas as $$\begin{array}{rcl}{G}_n& =& \frac{{\alpha }^{n+3}}{(\alpha -\beta )(\alpha -\gamma )(\alpha -\delta )(\alpha -\lambda )}+\frac{{\beta }^{n+3}}{(\beta -\alpha )(\beta -\gamma )(\beta -\delta )(\beta -\lambda )}+\frac{{\gamma }^{n+3}}{(\gamma -\alpha )(\gamma -\beta )(\gamma -\delta )(\gamma -\lambda )}\\ & & +\frac{{\delta }^{n+3}}{(\delta -\alpha )(\delta -\beta )(\delta -\gamma )(\delta -\lambda )}+\frac{{\lambda }^{n+3}}{(\lambda -\alpha )(\lambda -\beta )(\lambda -\gamma )(\lambda -\delta )},\\ H_n& =& {\alpha }^n+{\beta }^n+{\gamma }^n+{\delta }^n+{\lambda }^n,\end{array}$$ respectively.

Lemma 1 gives the following results as particular examples (generating functions of $(r,s,t,u,v)$, Lucas $(r,s,t,u,v)$ and modified $(r,s,t,u,v)$ numbers).

Corollary 1. Generating functions of $(r,s,t,u,v)$, Lucas $(r,s,t,u,v)$ and modified $(r,s,t,u,v)$ numbers are $$\begin{array}{rcl}\sum _{n=0}^{\mathrm{\infty }}{G}_n{x}^n& =& \frac{x}{1-rx-s{x}^2-t{x}^3-u{x}^4-v{x}^5},\\ \sum _{n=0}^{\mathrm{\infty }}{H}_n{x}^n& =& \frac{5-4rx-3s{x}^2-2t{x}^3-u{x}^4}{1-rx-s{x}^2-t{x}^3-u{x}^4-v{x}^5},\end{array}$$ respectively.

The following theorem shows that the generalized Pentanacci sequence $W_n$ at negative indices can be expressed by the sequence itself at positive indices.

Theorem 3. For $n\in \mathbb{Z},$ for the generalized Pentanacci sequence (or generalized $(r,s,t,u,v)$-sequence or 5-step Fibonacci sequence) we have the following: $$\begin{array}{rcl}{W}_{-n}& =& \frac{1}{24}{v}^{-n}(W_0{H}_n^4-4W_n{H}_n^3+3W_0{H}_{2n}^2+12H_n^2{W}_{2n}-6W_0{H}_n^2{H}_{2n}-6W_0{H}_{4n}-8W_n{H}_{3n}-12H_{2n}{W}_{2n}\\ & & -24H_n{W}_{3n}+24W_{4n}+8W_0{H}_n{H}_{3n}+12W_n{H}_n{H}_{2n})\\ & =& v^{-n}(W_{4n}-H_n{W}_{3n}+\frac{1}{2}(H_n^2-H_{2n})W_{2n}-\frac{1}{6}(H_n^3+2H_{3n}-3H_{2n}{H}_n)W_n\\ & & +\frac{1}{24}(H_n^4+3H_{2n}^2-6H_n^2{H}_{2n}-6H_{4n}+8H_{3n}{H}_n)W_0).\end{array}$$

Proof. For the proof, see [5], Theorem 1.

Using Theorem 3, we have the following corollary, see [5], Corollary 4.

Corollary 2. For $n\in \mathbb{Z},$ we have $$H_{-n}=\frac{1}{24}{v}^{-n}(H_n^4+3H_{2n}^2-6H_n^2{H}_{2n}-6H_{4n}+8H_{3n}{H}_n).$$

Note that $G_{-n}$ and $H_{-n}$ can be given as follows by using $G_0=0$ and $H_0=5$ in Theorem 3: $$\begin{array}{rcl}{G}_{-n}& =& v^{-n}(G_{4n}-H_n{G}_{3n}+\frac{1}{2}(H_n^2-H_{2n})G_{2n}-\frac{1}{6}(H_n^3+2H_{3n}-3H_{2n}{H}_n)G_n),\\ H_{-n}& =& \frac{1}{24}{v}^{-n}(H_n^4+3H_{2n}^2-6H_n^2{H}_{2n}-6H_{4n}+8H_{3n}{H}_n),\end{array}$$ respectively.

Next, we consider the case $r=1,$ $s=1,t=1,u=1,v=2$ and in this case we write $V_n=W_n.$ A generalized fifth order Jacobsthal sequence $\{V_n{\}}_{n\ge 0}=\{V_n(V_0,V_1,V_2,V_3,V_4){\}}_{n\ge 0}$ is defined by the fifth order recurrence relations

with the initial values $V_0=c_0,V_1=c_1,V_2=c_2,$ $V_3=c_3,$ $V_4=c_4$ not all being zero.

The sequence $\{V_n{\}}_{n\ge 0}$ can be extended to negative subscripts by defining

$$V_{-n}=-\frac{1}{2}{V}_{-(n-1)}-\frac{1}{2}{V}_{-(n-2)}-\frac{1}{2}{V}_{-(n-3)}-\frac{1}{2}{V}_{-(n-4)}+\frac{1}{2}{V}_{-(n-5)}$$ for $n=1,2,3,\dots .$ Therefore, recurrence (9) holds for all integer $n.$ For more information on the generalized fifth order Jacobsthal numbers, see [7].

The first few generalized fifth order Jacobsthal numbers with positive subscript and negative subscript are given in the Table 1

Table 1. A few generalized fifth order Jacobsthal numbers.

$n$	$V_n$	$V_{-n}$
$0$	$V_0$	$\dots$
$1$	$V_1$	$\frac{1}{2}{V}_4-\frac{1}{2}{V}_1-\frac{1}{2}{V}_2-\frac{1}{2}{V}_3-\frac{1}{2}{V}_0$
$2$	$V_2$	$\frac{3}{4}{V}_3-\frac{1}{4}{V}_1-\frac{1}{4}{V}_2-\frac{1}{4}{V}_0-\frac{1}{4}{V}_4$
$3$	$V_3$	$\frac{7}{8}{V}_2-\frac{1}{8}{V}_1-\frac{1}{8}{V}_0-\frac{1}{8}{V}_3-\frac{1}{8}{V}_4$
$4$	$V_4$	$\frac{15}{16}{V}_1-\frac{1}{16}{V}_0-\frac{1}{16}{V}_2-\frac{1}{16}{V}_3-\frac{1}{16}{V}_4$
$5$	$2V_0+V_1+V_2+V_3+V_4$	$\frac{31}{32}{V}_0-\frac{1}{32}{V}_1-\frac{1}{32}{V}_2-\frac{1}{32}{V}_3-\frac{1}{32}{V}_4$
$6$	$2V_0+3V_1+2V_2+2V_3+2V_4$	$\frac{31}{64}{V}_4-\frac{33}{64}{V}_1-\frac{33}{64}{V}_2-\frac{33}{64}{V}_3-\frac{33}{64}{V}_0$
$7$	$4V_0+4V_1+5V_2+4V_3+4V_4$	$\frac{95}{128}{V}_3-\frac{33}{128}{V}_1-\frac{33}{128}{V}_2-\frac{33}{128}{V}_0-\frac{33}{128}{V}_4$

Eq. (3) can be used to obtain Binet’s formula of generalized fifth order Jacobsthal numbers. Generalized fifth order Jacobsthal numbers can be expressed, for all integers $n,$ using Binet’s formula

$$\begin{array}{rcl}{V}_n& =& \frac{p_1{\alpha }^n}{(\alpha -\beta )(\alpha -\gamma )(\alpha -\delta )(\alpha -\lambda )}+\frac{p_2{\beta }^n}{(\beta -\alpha )(\beta -\gamma )(\beta -\delta )(\beta -\lambda )}+\frac{p_3{\gamma }^n}{(\gamma -\alpha )(\gamma -\beta )(\gamma -\delta )(\gamma -\lambda )}\\ & & +\frac{p_4{\delta }^n}{(\delta -\alpha )(\delta -\beta )(\delta -\gamma )(\delta -\lambda )}+\frac{p_5{\lambda }^n}{(\lambda -\alpha )(\lambda -\beta )(\lambda -\gamma )(\lambda -\delta )},\end{array}$$ where $$\begin{array}{rl}{p}_1& =V_4{-(\beta +\gamma +\delta +\lambda )V}_3{+(\beta \lambda +\beta \gamma +\lambda \gamma +\beta \delta +\lambda \delta +\gamma \delta )V}_2{-(\beta \lambda \gamma +\beta \lambda \delta +\beta \gamma \delta +\lambda \gamma \delta )V}_1{+(\beta \lambda \gamma \delta )V}_0,\\ p_2& =V_4{-(\alpha +\gamma +\delta +\lambda )V}_3{+(\alpha \lambda +\alpha \gamma +\alpha \delta +\lambda \gamma +\lambda \delta +\gamma \delta )V}_2{-(\alpha \lambda \gamma +\alpha \lambda \delta +\alpha \gamma \delta +\lambda \gamma \delta )V}_1{+(\alpha \lambda \gamma \delta )V}_0,\\ p_3& =V_4{-(\alpha +\beta +\delta +\lambda )V}_3{+(\alpha \beta +\alpha \lambda +\beta \lambda +\alpha \delta +\beta \delta +\lambda \delta )V}_2{-(\alpha \beta \lambda +\alpha \beta \delta +\alpha \lambda \delta +\beta \lambda \delta )V}_1{+(\alpha \beta \lambda \delta )V}_0,\\ p_4& =V_4{-(\alpha +\beta +\gamma +\lambda )V}_3{+(\alpha \beta +\alpha \lambda +\alpha \gamma +\beta \lambda +\beta \gamma +\lambda \gamma )V}_2{-(\alpha \beta \lambda +\alpha \beta \gamma +\alpha \lambda \gamma +\beta \lambda \gamma )V}_1{+(\alpha \beta \lambda \gamma )V}_0,\\ p_5& =V_4{-(\alpha +\beta +\gamma +\delta )V}_3{+(\alpha \beta +\alpha \gamma +\alpha \delta +\beta \gamma +\beta \delta +\gamma \delta )V}_2{-(\alpha \beta \gamma +\alpha \beta \delta +\alpha \gamma \delta +\beta \gamma \delta )V}_1{+(\alpha \beta \gamma \delta )V}_0.\end{array}$$ As $\{V_n\}$ is a fifth order recurrence sequence (difference equation), it’s characteristic equation is The roots $\alpha ,\beta ,\gamma ,\delta$ and $\lambda$ of Eq. (10) are given by: $$\begin{array}{rcl}\alpha & =& 2,\end{array}$$$$\begin{array}{rcl}\beta & =& \frac{1}{4}(\sqrt{5}-1)+\frac{\sqrt{2\sqrt{5}+10}}{4}i,\\ \gamma & =& \frac{1}{4}(\sqrt{5}-1)-\frac{\sqrt{2\sqrt{5}+10}}{4}i,\\ \delta & =& -\frac{1}{4}(\sqrt{5}+1)+\frac{\sqrt{-2\sqrt{5}+10}}{4}i,\\ \lambda & =& -\frac{1}{4}(\sqrt{5}+1)-\frac{\sqrt{-2\sqrt{5}+10}}{4}i.\end{array}$$ Note that we have the following identities: $$\begin{array}{rcl}\alpha +\beta +\gamma +\delta +\lambda & =& 1,\\ \alpha \beta +\alpha \lambda +\alpha \gamma +\beta \lambda +\alpha \delta +\beta \gamma +\lambda \gamma +\beta \delta +\lambda \delta +\gamma \delta & =& -1,\\ \alpha \beta \lambda +\alpha \beta \gamma +\alpha \lambda \gamma +\alpha \beta \delta +\alpha \lambda \delta +\beta \lambda \gamma +\alpha \gamma \delta +\beta \lambda \delta +\beta \gamma \delta +\lambda \gamma \delta & =& 1,\\ \alpha \beta \lambda \gamma +\alpha \beta \lambda \delta +\alpha \beta \gamma \delta +\alpha \lambda \gamma \delta +\beta \lambda \gamma \delta & =& -1,\\ \alpha \beta \gamma \delta \lambda & =& 2.\end{array}$$ Now we consider four special cases of the sequence $\{V_n\}$. Fifth-order Jacobsthal sequence $\{J_n{\}}_{n\ge 0}$, fifth order Jacobsthal-Lucas sequence $\{j_n{\}}_{n\ge 0}$, adjusted fifth order Jacobsthal sequence $\{S_n{\}}_{n\ge 0}$ and modified fifth order Jacobsthal-Lucas sequence $\{R_n{\}}_{n\ge 0}$ are defined, respectively, by the fifth order recurrence relations The sequences $\{J_n{\}}_{n\ge 0},$ $\{j_n{\}}_{n\ge 0}$, $\{S_n{\}}_{n\ge 0}$ and $\{R_n{\}}_{n\ge 0}$ can be extended to negative subscripts by defining $$\begin{array}{rcl}{J}_{-n}& =& -\frac{1}{2}{J}_{-(n-1)}-\frac{1}{2}{J}_{-(n-2)}-\frac{1}{2}{J}_{-(n-3)}-J_{-(n-4)}+\frac{1}{2}{J}_{-(n-5)},\\ j_{-n}& =& -\frac{1}{2}{j}_{-(n-1)}-\frac{1}{2}{j}_{-(n-2)}-\frac{1}{2}{j}_{-(n-3)}-\frac{1}{2}{j}_{-(n-4)}+\frac{1}{2}{j}_{-(n-5)},\\ S_{-n}& =& -\frac{1}{2}{S}_{-(n-1)}-\frac{1}{2}{S}_{-(n-2)}-\frac{1}{2}{S}_{-(n-3)}-\frac{1}{2}{S}_{-(n-4)}+\frac{1}{2}{S}_{-(n-5)},\\ R_{-n}& =& -\frac{1}{2}{R}_{-(n-1)}-\frac{1}{2}{R}_{-(n-2)}-\frac{1}{2}{R}_{-(n-3)}-\frac{1}{2}{R}_{-(n-4)}+\frac{1}{2}{R}_{-(n-5)},\end{array}$$ for $n=1,2,3,\dots$ respectively. Therefore, recurrences (11)-(14) hold for all integer $n.$

Next, we present the first few values of the fifth order Jacobsthal, fifth order Jacobsthal-Lucas, adjusted fifth order Jacobsthal and modified fifth order Jacobsthal-Lucas numbers with positive and negative subscripts in the following Table 2:

Table 2. The first few values of the special fifth order numbers with positive and negative subscripts .

$n$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$	$12$	$13$
$J_n$	$0$	$1$	$1$	$1$	$1$	$4$	$9$	$17$	$33$	$65$	$132$	$265$	$529$	$1057$
$J_{-n}$		$-1$	$0$	$\frac{1}{2}$	$\frac{3}{4}$	$-\frac{1}{8}$	$-\frac{17}{16}$	$-\frac{1}{32}$	$\frac{31}{64}$	$\frac{95}{128}$	$-\frac{33}{256}$	$-\frac{545}{512}$	$-\frac{33}{1024}$	$\frac{991}{2048}$
$j_n$	$2$	$1$	$5$	$10$	$20$	$40$	$77$	$157$	$314$	$628$	$1256$	$2509$	$5021$	$10042$
$j_{-n}$		$1$	$\frac{1}{2}$	$\frac{1}{4}$	$-\frac{11}{8}$	$\frac{13}{16}$	$\frac{13}{32}$	$\frac{13}{64}$	$\frac{13}{128}$	$-\frac{371}{256}$	$\frac{397}{512}$	$\frac{397}{1024}$	$\frac{397}{2048}$	$\frac{397}{4096}$
$S_n$	$0$	$1$	$1$	$2$	$4$	$8$	$17$	$33$	$66$	$132$	$264$	$529$	$1057$	$2114$
$S_{-n}$		$0$	$0$	$0$	$\frac{1}{2}$	$-\frac{1}{4}$	$-\frac{1}{8}$	$-\frac{1}{16}$	$-\frac{1}{32}$	$\frac{31}{64}$	$-\frac{33}{128}$	$-\frac{33}{256}$	$-\frac{33}{512}$	$-\frac{33}{1024}$
$R_n$	$5$	$1$	$3$	$7$	$15$	$36$	$63$	$127$	$255$	$511$	$1028$	$2047$	$4095$	$8191$
$R_{-n}$		$-\frac{1}{2}$	$-\frac{3}{4}$	$–\frac{7}{8}$	$-\frac{15}{16}$	$\frac{129}{32}$	$-\frac{63}{64}$	$-\frac{127}{128}$	$-\frac{255}{256}$	$-\frac{511}{512}$	$\frac{4097}{1024}$	$-\frac{2047}{2048}$	$–\frac{4095}{4096}$	$-\frac{8191}{8192}$

For all integers $n,$ Binet formulas of fifth order Jacobsthal, fifth order Jacobsthal-Lucas, adjusted fifth order Jacobsthal and modified fifth order Jacobsthal-Lucas numbers are

$$\begin{array}{rcl}{J}_n& =& \frac{({\alpha }^3-\alpha -2){\alpha }^n}{(\alpha -\beta )(\alpha -\gamma )(\alpha -\delta )(\alpha -\lambda )}+\frac{({\beta }^3-\beta -2){\beta }^n}{(\beta -\alpha )(\beta -\gamma )(\beta -\delta )(\beta -\lambda )}\\ & & +\frac{({\gamma }^3-\gamma -2){\gamma }^n}{(\gamma -\alpha )(\gamma -\beta )(\gamma -\delta )(\gamma -\lambda )}+\frac{({\delta }^3-\delta -2){\delta }^n}{(\delta -\alpha )(\delta -\beta )(\delta -\gamma )(\delta -\lambda )}\\ & & +\frac{({\lambda }^3-\lambda -2){\lambda }^n}{(\lambda -\alpha )(\lambda -\beta )(\lambda -\gamma )(\lambda -\delta )},\end{array}$$ $$\begin{array}{rcl}{j}_n& =& \frac{({\alpha }^4+4{\alpha }^3+4{\alpha }^2+4\alpha +4){\alpha }^{n-1}}{(\alpha -\beta )(\alpha -\gamma )(\alpha -\delta )(\alpha -\lambda )}+\frac{({\beta }^4+4{\beta }^3+4{\beta }^2+4\beta +4){\beta }^{n-1}}{(\beta -\alpha )(\beta -\gamma )(\beta -\delta )(\beta -\lambda )}\\ & & +\frac{({\gamma }^4+4{\gamma }^3+4{\gamma }^2+4\gamma +4){\gamma }^{n-1}}{(\gamma -\alpha )(\gamma -\beta )(\gamma -\delta )(\gamma -\lambda )}+\frac{({\delta }^4+4{\delta }^3+4{\delta }^2+4\delta +4){\delta }^{n-1}}{(\delta -\alpha )(\delta -\beta )(\delta -\gamma )(\delta -\lambda )}\\ & & +\frac{({\lambda }^4+4{\lambda }^3+4{\lambda }^2+4\lambda +4){\lambda }^{n-1}}{(\lambda -\alpha )(\lambda -\beta )(\lambda -\gamma )(\lambda -\delta )},\end{array}$$ $$\begin{array}{rcl}{S}_n& =& \frac{{\alpha }^{n+3}}{(\alpha -\beta )(\alpha -\gamma )(\alpha -\delta )(\alpha -\lambda )}+\frac{{\beta }^{n+3}}{(\beta -\alpha )(\beta -\gamma )(\beta -\delta )(\beta -\lambda )}\\ & & +\frac{{\gamma }^{n+3}}{(\gamma -\alpha )(\gamma -\beta )(\gamma -\delta )(\gamma -\lambda )}+\frac{{\delta }^{n+3}}{(\delta -\alpha )(\delta -\beta )(\delta -\gamma )(\delta -\lambda )}\\ & & +\frac{{\lambda }^{n+3}}{(\lambda -\alpha )(\lambda -\beta )(\lambda -\gamma )(\lambda -\delta )},\\ R_n& =& {\alpha }^n+{\beta }^n+{\gamma }^n+{\delta }^n+{\lambda }^n\end{array}$$ respectively.

Binet formulas of fifth order Jacobsthal, fifth order Jacobsthal-Lucas, adjusted fifth order Jacobsthal and modified fifth order Jacobsthal-Lucas numbers can be given in the following forms:

$$\begin{array}{rcl}{J}_n& =& \frac{4}{31}{\alpha }^n-\frac{1}{155}((6\sqrt{5}+5)+2\sqrt{2}\sqrt{\sqrt{5}+5}(6+\sqrt{5})i){\beta }^n\\ & & +\frac{1}{155}(-(6\sqrt{5}+5)+2\sqrt{2}\sqrt{\sqrt{5}+5}(6+\sqrt{5})i){\gamma }^n\\ & & +\frac{1}{155}((6\sqrt{5}-5)+2\sqrt{2}\sqrt{5-\sqrt{5}}(\sqrt{5}-6)i){\delta }^n\\ & & +\frac{1}{155}((6\sqrt{5}-5)+2\sqrt{2}\sqrt{5-\sqrt{5}}(-\sqrt{5}+6)i){\lambda }^n,\end{array}$$ $$\begin{array}{rcl}{j}_n& =& \frac{38}{31}{\alpha }^n+\frac{1}{1240}(12(20-7\sqrt{5})+\sqrt{\sqrt{5}+5}(111\sqrt{2}+3\sqrt{10})i){\beta }^n\\ & & +\frac{1}{1240}(12(20-7\sqrt{5})-\sqrt{\sqrt{5}+5}(111\sqrt{2}+3\sqrt{10})i){\gamma }^n\\ & & +\frac{1}{1240}(12(20+7\sqrt{5})+\sqrt{5-\sqrt{5}}(111\sqrt{2}-3\sqrt{10})i){\delta }^n\\ & & +\frac{1}{1240}(12(20+7\sqrt{5})+\sqrt{5-\sqrt{5}}(-111\sqrt{2}+3\sqrt{10})i){\lambda }^n,\end{array}$$ $$\begin{array}{rcl}{S}_n& =& \frac{8}{31}{\alpha }^n+\frac{1}{1240}(4(-20+7\sqrt{5})-\sqrt{2}\sqrt{\sqrt{5}+5}(37i+i\sqrt{5})){\beta }^n\\ & & +\frac{1}{1240}(4(-20+7\sqrt{5})+\sqrt{2}\sqrt{\sqrt{5}+5}(37i+i\sqrt{5})){\gamma }^n\end{array}$$$$\begin{array}{rcl}& & +\frac{1}{1240}(-4(20+7\sqrt{5})+\sqrt{2}\sqrt{\sqrt{5}+5}(21-19\sqrt{5})i){\delta }^n\\ & & +\frac{1}{1240}(-4(20+7\sqrt{5})+\sqrt{2}\sqrt{\sqrt{5}+5}(-21+19\sqrt{5})i){\lambda }^n,\\ R_n& =& {\alpha }^n+{\beta }^n+{\gamma }^n+{\delta }^n+{\lambda }^n.\end{array}$$ Next, we give the ordinary generating function $\sum _{n=0}^{\mathrm{\infty }}{V}_n{x}^n$ of the sequence $V_n.$

Lemma 2. Suppose that $f_{V_n}(x)=\sum _{n=0}^{\mathrm{\infty }}{V}_n{x}^n$ is the ordinary generating function of the generalized fifth order Jacobsthal sequence $\{V_n{\}}_{n\ge 0}.$ Then, $\sum _{n=0}^{\mathrm{\infty }}{V}_n{x}^n$ is given by $$\sum _{n=0}^{\mathrm{\infty }}{V}_n{x}^n=\frac{V_0{+(V}_1{-V}_0{)x+(V}_2{-V}_1{-V}_0{)x}^2{+(V}_3{-V}_2{-V}_1{-V}_0{)x}^3{+(V}_4{-V}_3{-V}_2{-V}_1{-V}_0{)x}^4}{1-x-x^2-x^3-x^4-2x^5}.$$

The previous Lemma gives the following results as particular examples: generating function of the fifth order Jacobsthal, fifth order Jacobsthal-Lucas, adjusted fifth order Jacobsthal and modified fifth order Jacobsthal-Lucas are $$\begin{array}{rcl}{f}_{J_n}(x)& =& \sum _{n=0}^{\mathrm{\infty }}{J}_n{x}^n=\frac{x-x^3-2x^4}{1-x-x^2-x^3-x^4-2x^5},\\ f_{j_n}(x)& =& \sum _{n=0}^{\mathrm{\infty }}{j}_n{x}^n=\frac{2-x+2x^2+2x^3+2x^4}{1-x-x^2-x^3-x^4-2x^5},\\ f_{S_n}(x)& =& \sum _{n=0}^{\mathrm{\infty }}{S}_n{x}^n=\frac{x}{1-x-x^2-x^3-x^4-2x^5},\\ f_{R_n}(x)& =& \sum _{n=0}^{\mathrm{\infty }}{R}_n{x}^n=\frac{5-4x-3x^2-2x^3-x^4}{1-x-x^2-x^3-x^4-2x^5},\end{array}$$ respectively.

2. Binomial transform of the generalized fifth order Jacobsthal sequence $V_n$

In [8], p. 137, Knuth introduced the idea of the binomial transform. Given a sequence of numbers $(a_n)$, its binomial transform $({\hat{a}}_n)$ may be defined by the rule $${\hat{a}}_n=\sum _{i=0}^n(\genfrac{}{}{0ex}{}{n}{i})a_i,\text{ with inversion }{a}_n=\sum _{i=0}^n(\genfrac{}{}{0ex}{}{n}{i})(-1{)}^{n-i}{\hat{a}}_i,$$ or, in the symmetric version $${\hat{a}}_n=\sum _{i=0}^n(\genfrac{}{}{0ex}{}{n}{i})(-1{)}^{i+1}{a}_i,\text{ withinversion }{a}_n=\sum _{i=0}^n(\genfrac{}{}{0ex}{}{n}{i})(-1{)}^{i+1}{\hat{a}}_i.$$ For more information on binomial transform, see, for example, [9,10,11,12] and references therein. For recent works on binomial transform of well-known sequences, see for example, [13,14,15,16,17,18,19,20,21,22,23,24,25].

In this section, we define the binomial transform of the generalized fifth order Jacobsthal sequence $V_n$ and as special cases the binomial transform of the fifth order Jacobsthal, fifth order Jacobsthal-Lucas, adjusted fifth order Jacobsthal and modified fifth order Jacobsthal-Lucas sequences will be introduced.

Definition 1. The binomial transform of the generalized fifth order Jacobsthal sequence $V_n$ is defined by $$b_n={\hat{V}}_n=\sum _{i=0}^n(\genfrac{}{}{0ex}{}{n}{i})V_i.$$

The few terms of $b_n$ are $$\begin{array}{rcl}{b}_0& =& \sum _{i=0}^0(\genfrac{}{}{0ex}{}{0}{i})V_i=V_0,\\ b_1& =& \sum _{i=0}^1(\genfrac{}{}{0ex}{}{1}{i})V_i=V_0+V_1,\\ b_2& =& \sum _{i=0}^2(\genfrac{}{}{0ex}{}{2}{i})V_i=V_0+2V_1+V_2,\\ b_3& =& \sum _{i=0}^3(\genfrac{}{}{0ex}{}{3}{i})V_i=V_0+3V_1+3V_2+V_3,\\ b_4& =& \sum _{i=0}^4(\genfrac{}{}{0ex}{}{4}{i})V_i=V_0+4V_1+6V_2+4V_3+V_4.\end{array}$$ Translated to matrix language, $b_n$ has the nice (lower-triangular matrix) form $$\left(\begin{array}{c}{b}_0\\ b_1\\ b_2\\ b_3\\ b_4\\ ⋮\end{array}\right)=\left(\begin{array}{cccccc}1& 0& 0& 0& 0& \cdots \\ 1& 1& 0& 0& 0& \cdots \\ 1& 2& 1& 0& 0& \cdots \\ 1& 3& 3& 1& 0& \cdots \\ 1& 4& 6& 4& 1& \cdots \\ ⋮& ⋮& ⋮& ⋮& ⋮& \ddots \end{array}\right)\left(\begin{array}{c}{V}_0\\ V_1\\ V_2\\ V_3\\ V_4\\ ⋮\end{array}\right).$$ As special cases of $b_n={\hat{V}}_n$, the binomial transforms of the fifth order Jacobsthal and fifth order Jacobsthal-Lucas sequences are defined as follows: The binomial transform of the fifth order Jacobsthal sequence $J_n$ is $${\hat{J}}_n=\sum _{i=0}^n(\genfrac{}{}{0ex}{}{n}{i})J_i,$$ and the binomial transform of the fifth order Jacobsthal-Lucas sequence $j_n$ is $${\hat{j}}_n=\sum _{i=0}^n(\genfrac{}{}{0ex}{}{n}{i})j_i,$$ The binomial transform of the adjusted fifth order Jacobsthal sequence $S_n$ is $${\hat{S}}_n=\sum _{i=0}^n(\genfrac{}{}{0ex}{}{n}{i})S_i,$$ and the binomial transform of the modified fifth order Jacobsthal-Lucas sequence $R_n$ is $${\hat{R}}_n=\sum _{i=0}^n(\genfrac{}{}{0ex}{}{n}{i})R_i.$$

Lemma 3. For $n\ge 0,$ the binomial transform of the generalized fifth order Jacobsthal sequence $V_n$ satisfies the following relation: $$b_{n+1}=\sum _{i=0}^n(\genfrac{}{}{0ex}{}{n}{i})(V_i+V_{i+1}).$$

Proof. We use the following well-known identity: $$(\genfrac{}{}{0ex}{}{n+1}{i})=(\genfrac{}{}{0ex}{}{n}{i})+(\genfrac{}{}{0ex}{}{n}{i-1}).$$ Note also that $$(\genfrac{}{}{0ex}{}{n+1}{0})=(\genfrac{}{}{0ex}{}{n}{0})=1\text{ and }(\genfrac{}{}{0ex}{}{n}{n+1})=0.$$ Then $$\begin{array}{rcl}{b}_{n+1}& =& V_0+\sum _{i=1}^{n+1}(\genfrac{}{}{0ex}{}{n+1}{i})V_i\\ & =& V_0+\sum _{i=1}^{n+1}(\genfrac{}{}{0ex}{}{n}{i})V_i+\sum _{i=1}^{n+1}(\genfrac{}{}{0ex}{}{n}{i-1})V_i\\ & =& V_0+\sum _{i=1}^n(\genfrac{}{}{0ex}{}{n}{i})V_i+\sum _{i=0}^n(\genfrac{}{}{0ex}{}{n}{i})V_{i+1}\\ & =& \sum _{i=0}^n(\genfrac{}{}{0ex}{}{n}{i})V_i+\sum _{i=0}^n(\genfrac{}{}{0ex}{}{n}{i})V_{i+1}\\ & =& \sum _{i=0}^n(\genfrac{}{}{0ex}{}{n}{i})(V_i+V_{i+1}).\end{array}$$ This completes the proof.

Remark 1. From the Lemma 3, we see that $$b_{n+1}=b_n+\sum _{i=0}^n(\genfrac{}{}{0ex}{}{n}{i})V_{i+1}.$$

The following theorem gives recurrent relations of the binomial transform of the generalized fifth order Jacobsthal sequence.

Theorem 4. For $n\ge 0,$ the binomial transform of the generalized fifth order Jacobsthal sequence $V_n$ satisfies the following recurrence relation:

Proof. To show (15), writing $$b_{n+5}=r_1\times b_{n+4}+s_1\times b_{n+3}+t_1\times b_{n+2}+u_1\times b_{n+1}+v_1\times b_n$$ and taking the values $n=0,1,2,3,4$ and then solving the system of equations $$\begin{array}{rcl}{b}_5& =& r_1\times b_4+s_1\times b_3+t_1\times b_2+u_1\times b_1+v_1\times b_0\\ b_6& =& r_1\times b_5+s_1\times b_4+t_1\times b_3+u_1\times b_2+v_1\times b_1\\ b_7& =& r_1\times b_6+s_1\times b_5+t_1\times b_4+u_1\times b_3+v_1\times b_2\\ b_8& =& r_1\times b_7+s_1\times b_6+t_1\times b_5+u_1\times b_4+v_1\times b_3\\ b_9& =& r_1\times b_8+s_1\times b_7+t_1\times b_6+u_1\times b_5+v_1\times b_4\end{array}$$ we find that $r_1=6,s_1=-13,t_1=14,u_1=-7,v_1=3.$

The sequence $\{b_n{\}}_{n\ge 0}$ can be extended to negative subscripts by defining $$b_{-n}=\frac{7}{3}{b}_{-(n-1)}-\frac{14}{3}{b}_{-(n-2)}+\frac{13}{3}{b}_{-(n-3)}-2b_{-(n-4)}+\frac{1}{3}{b}_{-(n-5)}$$ for $n=1,2,3,\dots$. Therefore, recurrence (15) holds for all integer $n.$

Note that the recurence relation (15) is independent from initial values. So,

$$\begin{array}{rcl}{\hat{J}}_{n+5}& =& 6{\hat{J}}_{n+4}-13{\hat{J}}_{n+3}+14{\hat{J}}_{n+2}-7{\hat{J}}_{n+1}+3{\hat{J}}_n,\\ {\hat{j}}_{n+5}& =& 6{\hat{j}}_{n+4}-13{\hat{j}}_{n+3}+14{\hat{j}}_{n+2}-7{\hat{j}}_{n+1}+3{\hat{j}}_n,\\ {\hat{S}}_{n+5}& =& 6{\hat{S}}_{n+4}-13{\hat{S}}_{n+3}+14{\hat{S}}_{n+2}-7{\hat{S}}_{n+1}+3{\hat{S}}_n,\end{array}$$$$\begin{array}{rcl}{\hat{R}}_{n+5}& =& 6{\hat{R}}_{n+4}-13{\hat{R}}_{n+3}+14{\hat{R}}_{n+2}-7{\hat{R}}_{n+1}+3{\hat{R}}_n.\end{array}$$

The first few terms of the binomial transform of the generalized fifth order Jacobsthal sequence with positive subscript and negative subscript are given in the following Table 3.

Table 3. A few binomial transform (terms) of the generalized fifth order Jacobsthal sequence.

$n$	$b_n$	$b_{-n}$
$0$	$V_0$	$V_0$
$1$	$V_0+V_1$	$\frac{1}{3}(V_0-2V_1+V_2-2V_3+V_4)$
$2$	$V_0+2V_1+V_2$	$-\frac{1}{9}(11V_0+2V_1+2V_2+11V_3-7V_4)$
$3$	$V_0+3V_1+3V_2+V_3$	$-\frac{1}{27}(47V_0-34V_1+47V_2-7V_3-7V_4)$
$4$	$V_0+4V_1+6V_2+4V_3+V_4$	$\frac{1}{81}(115V_0+115V_1-128V_2+277V_3-128V_4)$
$5$	$3V_0+6V_1+11V_2+11V_3+6V_4$	$\frac{1}{243}(1411V_0-533V_1+682V_2+682V_3-533V_4)$
$6$	$15V_0+15V_1+23V_2+28V_3+23V_4$	$\frac{1}{729}(1411V_0-4421V_1+5056V_2-4421V_3+1411V_4)$
$7$	$61V_0+53V_1+61V_2+74V_3+74V_4$	$-\frac{1}{2187}(29207V_0+776V_1+776V_2+29207V_3-16720V_4)$

The first few terms of the binomial transform numbers of the fifth order Jacobsthal, fifth order Jacobsthal-Lucas, adjusted fifth order Jacobsthal and modified fifth order Jacobsthal-Lucas sequences with positive subscript and negative subscript are given in the following Table 4.

Table 4. A few binomial transform (terms).

$n$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$
${\hat{J}}_n$	$0$	$1$	$3$	$7$	$15$	$34$	$89$	$262$	$807$	$2489$	$7590$	$22914$
${\hat{J}}_{-n}$		$-\frac{2}{3}$	$-\frac{8}{9}$	$\frac{1}{27}$	$\frac{136}{81}$	$\frac{298}{243}$	$-\frac{2375}{729}$	$-\frac{14039}{2187}$	$\frac{14392}{6561}$	$\frac{375247}{19683}$	$\frac{788590}{59049}$	$-\frac{6474437}{177147}$
${\hat{j}}_n$	$2$	$3$	$9$	$30$	$96$	$297$	$900$	$2700$	$8076$	$24165$	$72393$	$217077$
${\hat{j}}_{-n}$		$\frac{5}{3}$	$-\frac{4}{9}$	$-\frac{85}{27}$	$-\frac{85}{81}$	$\frac{1859}{243}$	$\frac{7691}{729}$	$-\frac{20740}{2187}$	$-\frac{243814}{6561}$	$-\frac{243814}{19683}$	$\frac{5011547}{59049}$	$\frac{20777630}{177147}$
${\hat{S}}_n$	$0$	$1$	$3$	$8$	$22$	$63$	$186$	$558$	$1682$	$5067$	$15235$	$45739$
${\hat{S}}_{-n}$		$-\frac{1}{3}$	$\frac{2}{9}$	$\frac{29}{27}$	$\frac{29}{81}$	$-\frac{619}{243}$	$-\frac{2563}{729}$	$\frac{6914}{2187}$	$\frac{81272}{6561}$	$\frac{81272}{19683}$	$-\frac{1670515}{59049}$	$-\frac{6925876}{177147}$
${\hat{R}}_n$	$5$	$6$	$10$	$24$	$70$	$221$	$700$	$2169$	$6590$	$19806$	$59295$	$177469$
${\hat{R}}_{-n}$		$\frac{7}{3}$	$-\frac{35}{9}$	$-\frac{188}{27}$	$\frac{325}{81}$	$\frac{5347}{243}$	$\frac{8020}{729}$	$-\frac{102788}{2187}$	$-\frac{498635}{6561}$	$\frac{925102}{19683}$	$\frac{14526055}{59049}$	$\frac{21789082}{177147}$

Eq. (3) can be used to obtain Binet’s formula of the binomial transform of generalized fifth order Jacobsthal numbers. Binet’s formula of the binomial transform of generalized fifth order Jacobsthal numbers can be given as

where $$\begin{array}{rcl}{C}_1& =& b_4{-({\theta }_2+{\theta }_3+{\theta }_4+{\theta }_5)b}_3{+({\theta }_2{\theta }_5+{\theta }_2{\theta }_3+{\theta }_5{\theta }_3+{\theta }_2{\theta }_4+{\theta }_5{\theta }_4+{\theta }_3{\theta }_4)b}_2\\ & & {-({\theta }_2{\theta }_5{\theta }_3+{\theta }_2{\theta }_5{\theta }_4+{\theta }_2{\theta }_3{\theta }_4+{\theta }_5{\theta }_3{\theta }_4)b}_1{+({\theta }_2{\theta }_5{\theta }_3{\theta }_4)b}_0,\\ C_2& =& b_4{-({\theta }_1+{\theta }_3+{\theta }_4+{\theta }_5)b}_3{+({\theta }_1{\theta }_5+{\theta }_1{\theta }_3+{\theta }_1{\theta }_4+{\theta }_5{\theta }_3+{\theta }_5{\theta }_4+{\theta }_3{\theta }_4)b}_2\\ & & {-({\theta }_1{\theta }_5{\theta }_3+{\theta }_1{\theta }_5{\theta }_4+{\theta }_1{\theta }_3{\theta }_4+{\theta }_5{\theta }_3{\theta }_4)b}_1{+({\theta }_1{\theta }_5{\theta }_3{\theta }_4)b}_0,\end{array}$$$$\begin{array}{rcl}{C}_3& =& b_4{-({\theta }_1+{\theta }_2+{\theta }_4+{\theta }_5)b}_3{+({\theta }_1{\theta }_2+{\theta }_1{\theta }_5+{\theta }_2{\theta }_5+{\theta }_1{\theta }_4+{\theta }_2{\theta }_4+{\theta }_5{\theta }_4)b}_2\\ & & {-({\theta }_1{\theta }_2{\theta }_5+{\theta }_1{\theta }_2{\theta }_4+{\theta }_1{\theta }_5{\theta }_4+{\theta }_2{\theta }_5{\theta }_4)b}_1{+({\theta }_1{\theta }_2{\theta }_5{\theta }_4)b}_0,\\ C_4& =& b_4{-({\theta }_1+{\theta }_2+{\theta }_3+{\theta }_5)b}_3{+({\theta }_1{\theta }_2+{\theta }_1{\theta }_5+{\theta }_1{\theta }_3+{\theta }_2{\theta }_5+{\theta }_2{\theta }_3+{\theta }_5{\theta }_3)b}_2\\ & & {-({\theta }_1{\theta }_2{\theta }_5+{\theta }_1{\theta }_2{\theta }_3+{\theta }_1{\theta }_5{\theta }_3+{\theta }_2{\theta }_5{\theta }_3)b}_1{+({\theta }_1{\theta }_2{\theta }_5{\theta }_3)b}_0,\\ C_5& =& b_4{-({\theta }_1+{\theta }_2+{\theta }_3+{\theta }_4)b}_3{+({\theta }_1{\theta }_2+{\theta }_1{\theta }_3+{\theta }_1{\theta }_4+{\theta }_2{\theta }_3+{\theta }_2{\theta }_4+{\theta }_3{\theta }_4)b}_2\\ & & {-({\theta }_1{\theta }_2{\theta }_3+{\theta }_1{\theta }_2{\theta }_4+{\theta }_1{\theta }_3{\theta }_4+{\theta }_2{\theta }_3{\theta }_4)b}_1{+({\theta }_1{\theta }_2{\theta }_3{\theta }_4)b}_0.\end{array}$$ Here, ${\theta }_1,{\theta }_2,{\theta }_3,{\theta }_4$ and ${\theta }_5$ are the roots of the equation $$x^5-6x^4+13x^3-14x^2+7x-3=(x-3)(x^4-3x^3+4x^2-2x+1)=0.$$ Moreover, the approximate value of ${\theta }_1,{\theta }_2,{\theta }_3,{\theta }_4$ and ${\theta }_5$ are given by $$\begin{array}{rcl}{\theta }_1& =& 3,\\ {\theta }_2& =& 1.30901699437495+0.951056516295154i,\\ {\theta }_3& =& 1.30901699437495-0.951056516295154i,\\ {\theta }_4& =& 0.190983005625053+0.587785252292473i,\\ {\theta }_5& =& 0.190983005625053-0.587785252292473i.\end{array}$$ Note that $$\begin{array}{rcl}{\theta }_1+{\theta }_2+{\theta }_3+{\theta }_4+{\theta }_5& =& 6,\\ {\theta }_1{\theta }_2+{\theta }_1{\theta }_5+{\theta }_1{\theta }_3+{\theta }_2{\theta }_5+{\theta }_1{\theta }_4+{\theta }_2{\theta }_3+{\theta }_5{\theta }_3+{\theta }_2{\theta }_4+{\theta }_5{\theta }_4+{\theta }_3{\theta }_4& =& 13,\\ {\theta }_1{\theta }_2{\theta }_5+{\theta }_1{\theta }_2{\theta }_3+{\theta }_1{\theta }_5{\theta }_3+{\theta }_1{\theta }_2{\theta }_4+{\theta }_1{\theta }_5{\theta }_4+{\theta }_2{\theta }_5{\theta }_3+{\theta }_1{\theta }_3{\theta }_4+{\theta }_2{\theta }_5{\theta }_4+{\theta }_2{\theta }_3{\theta }_4+{\theta }_5{\theta }_3{\theta }_4& =& 14,\\ {\theta }_1{\theta }_2{\theta }_5{\theta }_3+{\theta }_1{\theta }_2{\theta }_5{\theta }_4+{\theta }_1{\theta }_2{\theta }_3{\theta }_4+{\theta }_1{\theta }_5{\theta }_3{\theta }_4+{\theta }_2{\theta }_5{\theta }_3{\theta }_4& =& 7,\\ {\theta }_1{\theta }_2{\theta }_3{\theta }_4{\theta }_5& =& 3.\end{array}$$ For all integers $n,$ (Binet’s formulas of) binomial transforms of fifth order Jacobsthal, fifth order Jacobsthal-Lucas, adjusted fifth order Jacobsthal and modified fifth order Jacobsthal-Lucas numbers (using initial conditions in (16)) can be expressed using Binet’s formulas as $$\begin{array}{rcl}{\hat{J}}_n& =& \frac{({\theta }_1^3-3{\theta }_1^2+2{\theta }_1-2){\theta }_1^n}{({\theta }_1-{\theta }_2)({\theta }_1-{\theta }_3)({\theta }_1-{\theta }_4)({\theta }_1-{\theta }_5)}+\frac{({\theta }_2^3-3{\theta }_2^2+2{\theta }_2-2){\theta }_2^n}{({\theta }_2-{\theta }_1)({\theta }_2-{\theta }_3)({\theta }_2-{\theta }_4)({\theta }_2-{\theta }_5)}\\ & & +\frac{({\theta }_3^3-3{\theta }_3^2+2{\theta }_3-2){\theta }_3^n}{({\theta }_3-{\theta }_1)({\theta }_3-{\theta }_2)({\theta }_3-{\theta }_4)({\theta }_3-{\theta }_5)}+\frac{({\theta }_4^3-3{\theta }_4^2+2{\theta }_4-2){\theta }_4^n}{({\theta }_4-{\theta }_1)({\theta }_4-{\theta }_2)({\theta }_4-{\theta }_3)({\theta }_4-{\theta }_5)}\\ & & +\frac{({\theta }_5^3-3{\theta }_5^2+2{\theta }_5-2){\theta }_5^n}{({\theta }_5-{\theta }_1)({\theta }_5-{\theta }_2)({\theta }_5-{\theta }_3)({\theta }_5-{\theta }_4)},\end{array}$$ $$\begin{array}{rcl}{\hat{j}}_n& =& \frac{(2{\theta }_1^4-9{\theta }_1^3+17{\theta }_1^2-13{\theta }_1+5){\theta }_1^n}{({\theta }_1-{\theta }_2)({\theta }_1-{\theta }_3)({\theta }_1-{\theta }_4)({\theta }_1-{\theta }_5)}+\frac{(2{\theta }_2^4-9{\theta }_2^3+17{\theta }_2^2-13{\theta }_2+5){\theta }_2^n}{({\theta }_2-{\theta }_1)({\theta }_2-{\theta }_3)({\theta }_2-{\theta }_4)({\theta }_2-{\theta }_5)}\\ & & +\frac{(2{\theta }_3^4-9{\theta }_3^3+17{\theta }_3^2-13{\theta }_3+5){\theta }_3^n}{({\theta }_3-{\theta }_1)({\theta }_3-{\theta }_2)({\theta }_3-{\theta }_4)({\theta }_3-{\theta }_5)}+\frac{(2{\theta }_4^4-9{\theta }_4^3+17{\theta }_4^2-13{\theta }_4+5){\theta }_4^n}{({\theta }_4-{\theta }_1)({\theta }_4-{\theta }_2)({\theta }_4-{\theta }_3)({\theta }_4-{\theta }_5)}\\ & & +\frac{(2{\theta }_5^4-9{\theta }_5^3+17{\theta }_5^2-13{\theta }_5+5){\theta }_5^n}{({\theta }_5-{\theta }_1)({\theta }_5-{\theta }_2)({\theta }_5-{\theta }_3)({\theta }_5-{\theta }_4)},\end{array}$$ $$\begin{array}{rcl}{\hat{S}}_n& =& \frac{({\theta }_1-1{)}^3{\theta }_1^n}{({\theta }_1-{\theta }_2)({\theta }_1-{\theta }_3)({\theta }_1-{\theta }_4)({\theta }_1-{\theta }_5)}+\frac{({\theta }_2-1{)}^3{\theta }_2^n}{({\theta }_2-{\theta }_1)({\theta }_2-{\theta }_3)({\theta }_2-{\theta }_4)({\theta }_2-{\theta }_5)}\\ & & +\frac{({\theta }_3-1{)}^3{\theta }_3^n}{({\theta }_3-{\theta }_1)({\theta }_3-{\theta }_2)({\theta }_3-{\theta }_4)({\theta }_3-{\theta }_5)}+\frac{({\theta }_4-1{)}^3{\theta }_4^n}{({\theta }_4-{\theta }_1)({\theta }_4-{\theta }_2)({\theta }_4-{\theta }_3)({\theta }_4-{\theta }_5)}\end{array}$$$$\begin{array}{rcl}& & +\frac{({\theta }_5-1{)}^3{\theta }_5^n}{({\theta }_5-{\theta }_1)({\theta }_5-{\theta }_2)({\theta }_5-{\theta }_3)({\theta }_5-{\theta }_4)},\\ {\hat{R}}_n& =& {\theta }_1^n+{\theta }_2^n+{\theta }_3^n+{\theta }_4^n+{\theta }_5^n,\end{array}$$ respectively.

3. Generating functions and obtaining Binet formula of binomial transform from generating function

The generating function of the binomial transform of the generalized fifth order Jacobsthal sequence $V_n$ is a power series centered at the origin whose coefficients are the binomial transform of the generalized fifth order Jacobsthal sequence.

Next, we give the ordinary generating function $f_{b_n}(x)=\sum _{n=0}^{\mathrm{\infty }}{b}_n{x}^n$ of the sequence $b_n.$

Lemma 4. Suppose that $f_{b_n}(x)=\sum _{n=0}^{\mathrm{\infty }}{b}_n{x}^n$ is the ordinary generating function of the binomial transform of the generalized fifth order Jacobsthal sequence $\{V_n{\}}_{n\ge 0}.$ Then, $f_{b_n}(x)$ is given by

Proof. Using Lemma 1, we obtain $$\begin{array}{rcl}{f}_{b_n}(x)& =& \frac{b_0{+(b}_1{-6b}_0{)x+(b}_2{-6b}_1+13b_0{)x}^2{+(b}_3{-6b}_2+13b_1{-14b}_0{)x}^3{+(b}_4{-6b}_3+13b_2{-14b}_1+7b_0{)x}^4}{1-6x+13x^2-14x^3+7x^4-3x^5}\\ & =& \frac{V_0{+(V_1-5V_0)x+(8V_0-4V_1+V_2)x}^2{+(4V_1-6V_0-3V_2+V_3)x}^3{+(V_0-2V_1+V_2-2V_3+V_4)x}^4}{1-6x+13x^2-14x^3+7x^4-3x^5},\end{array}$$ where $$\begin{array}{rcl}{b}_0& =& V_0,\\ b_1& =& V_0+V_1,\\ b_2& =& V_0+2V_1+V_2,\\ b_3& =& V_0+3V_1+3V_2+V_3,\\ b_4& =& V_0+4V_1+6V_2+4V_3+V_4.\end{array}$$

Note that P. Barry shows in [26] that if $A(x)$ is the generating function of the sequence $\{a_n\},$ then $$S(x)=\frac{1}{1-x}A\left(\frac{x}{1-x}\right)$$ is the generating function of the sequence $\{b_n\}$ with $b_n=\sum _{i=0}^n(\genfrac{}{}{0ex}{}{n}{i})a_i.$ In our case, since $$A(x)=\frac{V_0{+(V}_1{-V}_0{)x+(V}_2{-V}_1{-V}_0{)x}^2{+(V}_3{-V}_2{-V}_1{-V}_0{)x}^3{+(V}_4{-V}_3{-V}_2{-V}_1{-V}_0{)x}^4}{1-x-x^2-x^3-x^4-2x^5},\text{ see Lemma 2,}$$ we obtain $$\begin{array}{rcl}S(x)& =& \frac{1}{1-x}A(\frac{x}{1-x})\\ & =& \frac{V_0{+(V_1-5V_0)x+(8V_0-4V_1+V_2)x}^2{+(4V_1-6V_0-3V_2+V_3)x}^3{+(V_0-2V_1+V_2-2V_3+V_4)x}^4}{1-6x+13x^2-14x^3+7x^4-3x^5}.\end{array}$$ The Lemma 4 gives the following results as particular examples.

Corollary 3. Generating functions of the binomial transform of the fifth order Jacobsthal, fifth order Jacobsthal-Lucas, adjusted fifth order Jacobsthal and modified fifth order Jacobsthal-Lucas numbers are $$\begin{array}{rcl}\sum _{n=0}^{\mathrm{\infty }}{\hat{J}}_n{x}^n& =& \frac{x-3x^2+2x^3-2x^4}{1-6x+13x^2-14x^3+7x^4-3x^5},\\ \sum _{n=0}^{\mathrm{\infty }}{\hat{j}}_n{x}^n& =& \frac{2-9x+17x^2-13x^3+5x^4}{1-6x+13x^2-14x^3+7x^4-3x^5},\\ \sum _{n=0}^{\mathrm{\infty }}{\hat{S}}_n{x}^n& =& \frac{x-3x^2+3x^3-x^4}{1-6x+13x^2-14x^3+7x^4-3x^5},\\ \sum _{n=0}^{\mathrm{\infty }}{\hat{R}}_n{x}^n& =& \frac{5-24x+39x^2-28x^3+7x^4}{1-6x+13x^2-14x^3+7x^4-3x^5}.\end{array}$$ respectively.

4. Simson formulas

There is a well-known Simson Identity (formula) for Fibonacci sequence $\{F_n\}$, namely, $$F_{n+1}{F}_{n-1}-F_n^2=(-1{)}^n,$$ which was derived first by R. Simson in 1753 and it is now called as Cassini Identity (formula) as well. This can be written in the form $$\left|\begin{array}{cc}{F}_{n+1}& F_n\\ F_n& F_{n-1}\end{array}\right|=(-1{)}^n.$$ The following theorem gives generalization of this result to the generalized Pentanacci sequence $\{W_n\}.$

Theorem 5.(Simson formula of generalized Pentanacci numbers) For all integers $n,$ we have

Proof. Eq. (18) is given in [27], Theorem 3.1.

Taking $\{W_n\}=\{b_n\}$ in the above theorem and considering $b_{n+5}=6b_{n+4}-13b_{n+3}+14b_{n+2}-7b_{n+1}+3b_n,$ $r=6,s=-13,t=14,u=-7,v=3,$ we have the following proposition.

Proposition 1. For all integers $n,$ Simson formula of binomial transforms of generalized fifth order Jacobsthal numbers is given as $$\left|\begin{array}{ccccc}{b}_{n+4}& b_{n+3}& b_{n+2}& b_{n+1}& b_n\\ b_{n+3}& b_{n+2}& b_{n+1}& b_n& b_{n-1}\\ b_{n+2}& b_{n+1}& b_n& b_{n-1}& b_{n-2}\\ b_{n+1}& b_n& b_{n-1}& b_{n-2}& b_{n-3}\\ b_n& b_{n-1}& b_{n-2}& b_{n-3}& b_{n-4}\end{array}\right|=3^n\left|\begin{array}{ccccc}{b}_4& b_3& b_2& b_1& b_0\\ b_3& b_2& b_1& b_0& b_{-1}\\ b_2& b_1& b_0& b_{-1}& b_{-2}\\ b_1& b_0& b_{-1}& b_{-2}& b_{-3}\\ b_0& b_{-1}& b_{-2}& b_{-3}& b_{-4}\end{array}\right|.$$

The Proposition 1 gives the following results as particular examples.

Corollary 4. For all integers $n,$ Simson formula of binomial transforms of the fifth order Jacobsthal, fifth order Jacobsthal-Lucas, adjusted fifth order Jacobsthal and modified fifth order Jacobsthal-Lucas numbers are given as $$\begin{array}{rcl}\left|\begin{array}{ccccc}{\hat{J}}_{n+4}& {\hat{J}}_{n+3}& {\hat{J}}_{n+2}& {\hat{J}}_{n+1}& {\hat{J}}_n\\ {\hat{J}}_{n+3}& {\hat{J}}_{n+2}& {\hat{J}}_{n+1}& {\hat{J}}_n& {\hat{J}}_{n-1}\\ {\hat{J}}_{n+2}& {\hat{J}}_{n+1}& {\hat{J}}_n& {\hat{J}}_{n-1}& {\hat{J}}_{n-2}\\ {\hat{J}}_{n+1}& {\hat{J}}_n& {\hat{J}}_{n-1}& {\hat{J}}_{n-2}& {\hat{J}}_{n-3}\\ {\hat{J}}_n& {\hat{J}}_{n-1}& {\hat{J}}_{n-2}& {\hat{J}}_{n-3}& {\hat{J}}_{n-4}\end{array}\right|& =& 44\times 3^{n-4},\\ \left|\begin{array}{ccccc}{\hat{j}}_{n+4}& {\hat{j}}_{n+3}& {\hat{j}}_{n+2}& {\hat{j}}_{n+1}& {\hat{j}}_n\\ {\hat{j}}_{n+3}& {\hat{j}}_{n+2}& {\hat{j}}_{n+1}& {\hat{j}}_n& {\hat{j}}_{n-1}\\ {\hat{j}}_{n+2}& {\hat{j}}_{n+1}& {\hat{j}}_n& {\hat{j}}_{n-1}& {\hat{j}}_{n-2}\\ {\hat{j}}_{n+1}& {\hat{j}}_n& {\hat{j}}_{n-1}& {\hat{j}}_{n-2}& {\hat{j}}_{n-3}\\ {\hat{j}}_n& {\hat{j}}_{n-1}& {\hat{j}}_{n-2}& {\hat{j}}_{n-3}& {\hat{j}}_{n-4}\end{array}\right|& =& 38\times 3^n,\\ \left|\begin{array}{ccccc}{\hat{S}}_{n+4}& {\hat{S}}_{n+3}& {\hat{S}}_{n+2}& {\hat{S}}_{n+1}& {\hat{S}}_n\\ {\hat{S}}_{n+3}& {\hat{S}}_{n+2}& {\hat{S}}_{n+1}& {\hat{S}}_n& {\hat{S}}_{n-1}\\ {\hat{S}}_{n+2}& {\hat{S}}_{n+1}& {\hat{S}}_n& {\hat{S}}_{n-1}& {\hat{S}}_{n-2}\\ {\hat{S}}_{n+1}& {\hat{S}}_n& {\hat{S}}_{n-1}& {\hat{S}}_{n-2}& {\hat{S}}_{n-3}\\ {\hat{S}}_n& {\hat{S}}_{n-1}& {\hat{S}}_{n-2}& {\hat{S}}_{n-3}& {\hat{S}}_{n-4}\end{array}\right|& =& 8\times 3^{n-4},\\ \left|\begin{array}{ccccc}{\hat{R}}_{n+4}& {\hat{R}}_{n+3}& {\hat{R}}_{n+2}& {\hat{R}}_{n+1}& {\hat{R}}_n\\ {\hat{R}}_{n+3}& {\hat{R}}_{n+2}& {\hat{R}}_{n+1}& {\hat{R}}_n& {\hat{R}}_{n-1}\\ {\hat{R}}_{n+2}& {\hat{R}}_{n+1}& {\hat{R}}_n& {\hat{R}}_{n-1}& {\hat{R}}_{n-2}\\ {\hat{R}}_{n+1}& {\hat{R}}_n& {\hat{R}}_{n-1}& {\hat{R}}_{n-2}& {\hat{R}}_{n-3}\\ {\hat{R}}_n& {\hat{R}}_{n-1}& {\hat{R}}_{n-2}& {\hat{R}}_{n-3}& {\hat{R}}_{n-4}\end{array}\right|& =& 120125\times 3^{n-4},\end{array}$$ respectively.

5. Some identities

In this section, we obtain some identities of binomial transforms of fifth order Jacobsthal, fifth order Jacobsthal-Lucas, adjusted fifth order Jacobsthal and modified fifth order Jacobsthal-Lucas numbers. First, we present a few basic relations between $\{{\hat{J}}_n\}$ and $\{{\hat{j}}_n\}$.

Lemma 5. The following equalities are true: $$\begin{array}{rcl}171{\hat{J}}_n& =& -22{\hat{j}}_{n+6}+123{\hat{j}}_{n+5}-259{\hat{j}}_{n+4}+329{\hat{j}}_{n+3}-193{\hat{j}}_{n+2},\\ 57{\hat{J}}_n& =& -3{\hat{j}}_{n+5}+9{\hat{j}}_{n+4}+7{\hat{j}}_{n+3}-13{\hat{j}}_{n+2}-22{\hat{j}}_{n+1},\\ 57{\hat{J}}_n& =& -9{\hat{j}}_{n+4}+46{\hat{j}}_{n+3}-55{\hat{j}}_{n+2}-{\hat{j}}_{n+1}-9{\hat{j}}_n,\\ 57{\hat{J}}_n& =& -8{\hat{j}}_{n+3}+62{\hat{j}}_{n+2}-127{\hat{j}}_{n+1}+54{\hat{j}}_n-27{\hat{j}}_{n-1},\\ 57{\hat{J}}_n& =& 14{\hat{j}}_{n+2}-23{\hat{j}}_{n+1}-58{\hat{j}}_n+29{\hat{j}}_{n-1}-24{\hat{j}}_{n-2},\end{array}$$ and $$\begin{array}{rcl}198{\hat{j}}_n& =& 5{\hat{J}}_{n+6}-285{\hat{J}}_{n+5}+1532{\hat{J}}_{n+4}-2764{\hat{J}}_{n+3}+2003{\hat{J}}_{n+2},\\ 66{\hat{j}}_n& =& -85{\hat{J}}_{n+5}+489{\hat{J}}_{n+4}-898{\hat{J}}_{n+3}+656{\hat{J}}_{n+2}+5{\hat{J}}_{n+1},\\ 22{\hat{j}}_n& =& -7{\hat{J}}_{n+4}+69{\hat{J}}_{n+3}-178{\hat{J}}_{n+2}+200{\hat{J}}_{n+1}-85{\hat{J}}_n,\\ 22{\hat{j}}_n& =& 27{\hat{J}}_{n+3}-87{\hat{J}}_{n+2}+102{\hat{J}}_{n+1}-36{\hat{J}}_n-21{\hat{J}}_{n-1},\\ 22{\hat{j}}_n& =& 75{\hat{J}}_{n+2}-249{\hat{J}}_{n+1}+342{\hat{J}}_n-210{\hat{J}}_{n-1}+81{\hat{J}}_{n-2}.\end{array}$$

Proof. Writing $${\hat{J}}_n=a\times {\hat{j}}_{n+6}+b\times {\hat{j}}_{n+5}+c\times {\hat{j}}_{n+4}+d\times {\hat{j}}_{n+3}+e\times {\hat{j}}_{n+2}$$ and solving the system of equations $$\begin{array}{rcl}{\hat{J}}_0& =& a\times {\hat{j}}_6+b\times {\hat{j}}_5+c\times {\hat{j}}_4+d\times {\hat{j}}_3+e\times {\hat{j}}_2\\ {\hat{J}}_1& =& a\times {\hat{j}}_7+b\times {\hat{j}}_6+c\times {\hat{j}}_5+d\times {\hat{j}}_4+e\times {\hat{j}}_3\\ {\hat{J}}_2& =& a\times {\hat{j}}_8+b\times {\hat{j}}_7+c\times {\hat{j}}_6+d\times {\hat{j}}_5+e\times {\hat{j}}_4\\ {\hat{J}}_3& =& a\times {\hat{j}}_9+b\times {\hat{j}}_8+c\times {\hat{j}}_7+d\times {\hat{j}}_6+e\times {\hat{j}}_5\\ {\hat{J}}_4& =& a\times {\hat{j}}_{10}+b\times {\hat{j}}_9+c\times {\hat{j}}_8+d\times {\hat{j}}_7+e\times {\hat{j}}_6\end{array}$$ we find that $a=-\frac{22}{171},b=\frac{41}{57},c=-\frac{259}{171},d=\frac{329}{171},e=-\frac{193}{171}.$ The other equalities can be proved similarly.

Now, we give a few basic relations between $\{{\hat{J}}_n\}$ and $\{{\hat{S}}_n\}$.

Lemma 6. The following equalities are true: $$\begin{array}{rcl}18{\hat{J}}_n& =& 7{\hat{S}}_{n+6}-39{\hat{S}}_{n+5}+82{\hat{S}}_{n+4}-104{\hat{S}}_{n+3}+61{\hat{S}}_{n+2},\\ 6{\hat{J}}_n& =& {\hat{S}}_{n+5}-3{\hat{S}}_{n+4}-2{\hat{S}}_{n+3}+4{\hat{S}}_{n+2}+7{\hat{S}}_{n+1},\\ 2{\hat{J}}_n& =& {\hat{S}}_{n+4}-5{\hat{S}}_{n+3}+6{\hat{S}}_{n+2}+{\hat{S}}_n,\\ 2{\hat{J}}_n& =& {\hat{S}}_{n+3}-7{\hat{S}}_{n+2}+14{\hat{S}}_{n+1}-6{\hat{S}}_n+3{\hat{S}}_{n-1},\\ 2{\hat{J}}_n& =& -{\hat{S}}_{n+2}+{\hat{S}}_{n+1}+8{\hat{S}}_n-4{\hat{S}}_{n-1}+3{\hat{S}}_{n-2},\end{array}$$ and $$\begin{array}{rcl}99{\hat{S}}_n& =& {\hat{J}}_{n+6}+42{\hat{J}}_{n+5}-248{\hat{J}}_{n+4}+457{\hat{J}}_{n+3}-332{\hat{J}}_{n+2},\\ 33{\hat{S}}_n& =& 16{\hat{J}}_{n+5}-87{\hat{J}}_{n+4}+157{\hat{J}}_{n+3}-113{\hat{J}}_{n+2}+{\hat{J}}_{n+1},\\ 11{\hat{S}}_n& =& 3{\hat{J}}_{n+4}-17{\hat{J}}_{n+3}+37{\hat{J}}_{n+2}-37{\hat{J}}_{n+1}+16{\hat{J}}_n,\\ 11{\hat{S}}_n& =& {\hat{J}}_{n+3}-2{\hat{J}}_{n+2}+5{\hat{J}}_{n+1}-5{\hat{J}}_n+9{\hat{J}}_{n-1},\\ 11{\hat{S}}_n& =& 4{\hat{J}}_{n+2}-8{\hat{J}}_{n+1}+9{\hat{J}}_n+2{\hat{J}}_{n-1}+3{\hat{J}}_{n-2}.\end{array}$$

Next, we present a few basic relations between $\{{\hat{J}}_n\}$ and $\{{\hat{R}}_n\}$.

Lemma 7. The following equalities are true: $$\begin{array}{rcl}43245{\hat{J}}_n& =& 164{\hat{R}}_{n+6}+1182{\hat{R}}_{n+5}-7993{\hat{R}}_{n+4}+15017{\hat{R}}_{n+3}-17692{\hat{R}}_{n+2},\\ 14415{\hat{J}}_n& =& 722{\hat{R}}_{n+5}-3375{\hat{R}}_{n+4}+5771{\hat{R}}_{n+3}-6280{\hat{R}}_{n+2}+164{\hat{R}}_{n+1},\\ 4805{\hat{J}}_n& =& 319{\hat{R}}_{n+4}-1205{\hat{R}}_{n+3}+1276{\hat{R}}_{n+2}-1630{\hat{R}}_{n+1}+722{\hat{R}}_n,\\ 4805{\hat{J}}_n& =& 709{\hat{R}}_{n+3}-2871{\hat{R}}_{n+2}+2836{\hat{R}}_{n+1}-1511{\hat{R}}_n+957{\hat{R}}_{n-1},\\ 4805{\hat{J}}_n& =& 1383{\hat{R}}_{n+2}-6381{\hat{R}}_{n+1}+8415{\hat{R}}_n-4006{\hat{R}}_{n-1}+2127{\hat{R}}_{n-2},\end{array}$$ and $$\begin{array}{rcl}396{\hat{R}}_n& =& 1163{\hat{J}}_{n+6}-7881{\hat{J}}_{n+5}+19664{\hat{J}}_{n+4}-22810{\hat{J}}_{n+3}+10379{\hat{J}}_{n+2},\\ 132{\hat{R}}_n& =& -301{\hat{J}}_{n+5}+1515{\hat{J}}_{n+4}-2176{\hat{J}}_{n+3}+746{\hat{J}}_{n+2}+1163{\hat{J}}_{n+1},\\ 44{\hat{R}}_n& =& -97{\hat{J}}_{n+4}+579{\hat{J}}_{n+3}-1156{\hat{J}}_{n+2}+1090{\hat{J}}_{n+1}-301{\hat{J}}_n,\\ 44{\hat{R}}_n& =& -3{\hat{J}}_{n+3}+105{\hat{J}}_{n+2}-268{\hat{J}}_{n+1}+378{\hat{J}}_n-291{\hat{J}}_{n-1},\\ 44{\hat{R}}_n& =& 87{\hat{J}}_{n+2}-229{\hat{J}}_{n+1}+336{\hat{J}}_n-270{\hat{J}}_{n-1}-9{\hat{J}}_{n-2}.\end{array}$$

Now, we give a few basic relations between $\{{\hat{j}}_n\}$ and $\{{\hat{S}}_n\}$.

Lemma 8. The following equalities are true: $$\begin{array}{rcl}36{\hat{j}}_n& =& -83{\hat{S}}_{n+6}+465{\hat{S}}_{n+5}-872{\hat{S}}_{n+4}+706{\hat{S}}_{n+3}-83{\hat{S}}_{n+2},\end{array}$$$$\begin{array}{rcl}12{\hat{j}}_n& =& -11{\hat{S}}_{n+5}+69{\hat{S}}_{n+4}-152{\hat{S}}_{n+3}+166{\hat{S}}_{n+2}-83{\hat{S}}_{n+1},\\ 4{\hat{j}}_n& =& {\hat{S}}_{n+4}-3{\hat{S}}_{n+3}+4{\hat{S}}_{n+2}-2{\hat{S}}_{n+1}-11{\hat{S}}_n,\\ 4{\hat{j}}_n& =& 3{\hat{S}}_{n+3}-9{\hat{S}}_{n+2}+12{\hat{S}}_{n+1}-18{\hat{S}}_n+3{\hat{S}}_{n-1},\\ 4{\hat{j}}_n& =& 9{\hat{S}}_{n+2}-27{\hat{S}}_{n+1}+24{\hat{S}}_n-18{\hat{S}}_{n-1}+9{\hat{S}}_{n-2},\end{array}$$ and $$\begin{array}{rcl}171{\hat{S}}_n& =& -44{\hat{j}}_{n+6}+246{\hat{j}}_{n+5}-461{\hat{j}}_{n+4}+373{\hat{j}}_{n+3}-44{\hat{j}}_{n+2},\\ 57{\hat{S}}_n& =& -6{\hat{j}}_{n+5}+37{\hat{j}}_{n+4}-81{\hat{j}}_{n+3}+88{\hat{j}}_{n+2}-44{\hat{j}}_{n+1},\\ 57{\hat{S}}_n& =& {\hat{j}}_{n+4}-3{\hat{j}}_{n+3}+4{\hat{j}}_{n+2}-2{\hat{j}}_{n+1}-18{\hat{j}}_n,\\ 57{\hat{S}}_n& =& 3{\hat{j}}_{n+3}-9{\hat{j}}_{n+2}+12{\hat{j}}_{n+1}-25{\hat{j}}_n+3{\hat{j}}_{n-1},\\ 57{\hat{S}}_n& =& 9{\hat{j}}_{n+2}-27{\hat{j}}_{n+1}+17{\hat{j}}_n-18{\hat{j}}_{n-1}+9{\hat{j}}_{n-2}.\end{array}$$

Next, we present a few basic relations between $\{{\hat{j}}_n\}$ and $\{{\hat{R}}_n\}$.

Lemma 9. The following equalities are true: $$\begin{array}{rcl}43245{\hat{j}}_n& =& 11881{\hat{R}}_{n+6}-71634{\hat{R}}_{n+5}+151312{\hat{R}}_{n+4}-141779{\hat{R}}_{n+3}+41176{\hat{R}}_{n+2},\\ 14415{\hat{j}}_n& =& -116{\hat{R}}_{n+5}-1047{\hat{R}}_{n+4}+8185{\hat{R}}_{n+3}-13997{\hat{R}}_{n+2}+11881{\hat{R}}_{n+1},\\ 4805{\hat{j}}_n& =& -581{\hat{R}}_{n+4}+3231{\hat{R}}_{n+3}-5207{\hat{R}}_{n+2}+4231{\hat{R}}_{n+1}-116{\hat{R}}_n,\\ 4805{\hat{j}}_n& =& -255{\hat{R}}_{n+3}+2346{\hat{R}}_{n+2}-3903{\hat{R}}_{n+1}+3951{\hat{R}}_n-1743{\hat{R}}_{n-1},\\ 4805{\hat{j}}_n& =& 816{\hat{R}}_{n+2}-588{\hat{R}}_{n+1}+381{\hat{R}}_n+42{\hat{R}}_{n-1}-765{\hat{R}}_{n-2},\end{array}$$ and $$\begin{array}{rcl}342{\hat{R}}_n& =& 457{\hat{j}}_{n+6}-2397{\hat{j}}_{n+5}+3994{\hat{j}}_{n+4}-2396{\hat{j}}_{n+3}-1025{\hat{j}}_{n+2},\\ 114{\hat{R}}_n& =& 115{\hat{j}}_{n+5}-649{\hat{j}}_{n+4}+1334{\hat{j}}_{n+3}-1408{\hat{j}}_{n+2}+457{\hat{j}}_{n+1},\\ 114{\hat{R}}_n& =& 41{\hat{j}}_{n+4}-161{\hat{j}}_{n+3}+202{\hat{j}}_{n+2}-348{\hat{j}}_{n+1}+345{\hat{j}}_n,\\ 114{\hat{R}}_n& =& 85{\hat{j}}_{n+3}-331{\hat{j}}_{n+2}+226{\hat{j}}_{n+1}+58{\hat{j}}_n+123{\hat{j}}_{n-1},\\ 114{\hat{R}}_n& =& 179{\hat{j}}_{n+2}-879{\hat{j}}_{n+1}+1248{\hat{j}}_n-472{\hat{j}}_{n-1}+255{\hat{j}}_{n-2}.\end{array}$$

Now, we give a few basic relations between $\{{\hat{S}}_n\}$ and $\{{\hat{R}}_n\}$.

Lemma 10. The following equalities are true: $$\begin{array}{rcl}43245{\hat{S}}_n& =& -3857{\hat{R}}_{n+6}+23568{\hat{R}}_{n+5}-50024{\hat{R}}_{n+4}+47053{\hat{R}}_{n+3}-13622{\hat{R}}_{n+2},\\ 14415{\hat{S}}_n& =& 142{\hat{R}}_{n+5}+39{\hat{R}}_{n+4}-2315{\hat{R}}_{n+3}+4459{\hat{R}}_{n+2}-3857{\hat{R}}_{n+1},\\ 4805{\hat{S}}_n& =& 297{\hat{R}}_{n+4}-1387{\hat{R}}_{n+3}+2149{\hat{R}}_{n+2}-1617{\hat{R}}_{n+1}+142{\hat{R}}_n,\\ 4805{\hat{S}}_n& =& 395{\hat{R}}_{n+3}-1712{\hat{R}}_{n+2}+2541{\hat{R}}_{n+1}-1937{\hat{R}}_n+891{\hat{R}}_{n-1},\\ 4805{\hat{S}}_n& =& 658{\hat{R}}_{n+2}-2594{\hat{R}}_{n+1}+3593{\hat{R}}_n-1874{\hat{R}}_{n-1}+1185{\hat{R}}_{n-2},\end{array}$$ and $$\begin{array}{rcl}72{\hat{R}}_n& =& -287{\hat{S}}_{n+6}+1509{\hat{S}}_{n+5}-2516{\hat{S}}_{n+4}+1510{\hat{S}}_{n+3}+649{\hat{S}}_{n+2},\\ 24{\hat{R}}_n& =& -71{\hat{S}}_{n+5}+405{\hat{S}}_{n+4}-836{\hat{S}}_{n+3}+886{\hat{S}}_{n+2}-287{\hat{S}}_{n+1},\\ 8{\hat{R}}_n& =& -7{\hat{S}}_{n+4}+29{\hat{S}}_{n+3}-36{\hat{S}}_{n+2}+70{\hat{S}}_{n+1}-71{\hat{S}}_n,\\ 8{\hat{R}}_n& =& -13{\hat{S}}_{n+3}+55{\hat{S}}_{n+2}-28{\hat{S}}_{n+1}-22{\hat{S}}_n-21{\hat{S}}_{n-1},\\ 8{\hat{R}}_n& =& -23{\hat{S}}_{n+2}+141{\hat{S}}_{n+1}-204{\hat{S}}_n+70{\hat{S}}_{n-1}-39{\hat{S}}_{n-2}.\end{array}$$

6. On the recurrence properties of binomial transform of the generalized fifth order Jacobsthal sequence

Taking $r_1=6,s_1=-13,t_1=14,u_1=-7,v_1=3$ and $H_n={\hat{R}}_n$ in Theorem 3, we obtain the following Proposition.

Proposition 2. For $n\in \mathbb{Z},$ binomial Transform of the generalized fifth order Jacobsthal sequence have the following identity: $$\begin{array}{rcl}{b}_{-n}& =& \frac{1}{24}{3}^{-n}(b_0{\hat{R}}_n^4-4b_n{\hat{R}}_n^3+3b_0{\hat{R}}_{2n}^2+12{\hat{R}}_n^2{b}_{2n}-6b_0{\hat{R}}_n^2{\hat{R}}_{2n}-6b_0{\hat{R}}_{4n}-8b_n{\hat{R}}_{3n}-12{\hat{R}}_{2n}{b}_{2n}-24{\hat{R}}_n{b}_{3n}+24b_{4n}\\ & & +8b_0{\hat{R}}_n{\hat{R}}_{3n}+12b_n{\hat{R}}_n{\hat{R}}_{2n})\\ & =& 3^{-n}(b_{4n}-{\hat{R}}_n{b}_{3n}+\frac{1}{2}({\hat{R}}_n^2-{\hat{R}}_{2n})b_{2n}-\frac{1}{6}({\hat{R}}_n^3+2{\hat{R}}_{3n}-3{\hat{R}}_{2n}{\hat{R}}_n)b_n\\ & & +\frac{1}{24}({\hat{R}}_n^4+3{\hat{R}}_{2n}^2-6{\hat{R}}_n^2{\hat{R}}_{2n}-6{\hat{R}}_{4n}+8{\hat{R}}_{3n}{\hat{R}}_n)b_0).\end{array}$$

Using Proposition 2 (and Corollary 2), we obtain the following corollary which gives the connection between the special cases of binomial transform of generalized fifth order Jacobsthal sequence at the positive index and the negative index: for binomial transform of fifth order Jacobsthal, fifth order Jacobsthal-Lucas numbers: take $b_n={\hat{J}}_n$ with ${\hat{J}}_0=0,{\hat{J}}_1=1,{\hat{J}}_2=3,{\hat{J}}_3=7,{\hat{J}}_4=15,$ take $b_n={\hat{j}}_n$ with ${\hat{j}}_0=2,{\hat{j}}_1=3,{\hat{j}}_2=9,{\hat{j}}_3=30,{\hat{j}}_4=96,$ take $b_n={\hat{S}}_n$ with ${\hat{S}}_0=0,{\hat{S}}_1=1,{\hat{S}}_2=3,{\hat{S}}_3=8,{\hat{S}}_4=22,$ take $b_n={\hat{R}}_n$ with ${\hat{R}}_0=5,{\hat{R}}_1=6,{\hat{R}}_2=10,{\hat{R}}_3=24,{\hat{R}}_4=70,$ respectively. Note that in this case we have $H_n={\hat{R}}_n$. Note also that $G_n\ne {\hat{S}}_n$.

Corollary 5. For $n\in \mathbb{Z},$ we have the following recurrence relations:

• (a) Recurrence relations of binomial transforms of fifth order Jacobsthal numbers (take $b_n={\hat{J}}_n$ in Proposition 2): $${\hat{J}}_{-n}=3^{-n}({\hat{J}}_{4n}-{\hat{R}}_n{\hat{J}}_{3n}+\frac{1}{2}({\hat{R}}_n^2-{\hat{R}}_{2n}){\hat{J}}_{2n}-\frac{1}{6}({\hat{R}}_n^3+2{\hat{R}}_{3n}-3{\hat{R}}_{2n}{\hat{R}}_n){\hat{J}}_n).$$
• (b) Recurrence relations of binomial transforms of fifth order Jacobsthal-Lucas numbers (take $b_n={\hat{j}}_n$ in Proposition 2): $${\hat{j}}_{-n}=3^{-n}({\hat{j}}_{4n}-{\hat{R}}_n{\hat{j}}_{3n}+\frac{1}{2}({\hat{R}}_n^2-{\hat{R}}_{2n}){\hat{j}}_{2n}-\frac{1}{6}({\hat{R}}_n^3+2{\hat{R}}_{3n}-3{\hat{R}}_{2n}{\hat{R}}_n){\hat{j}}_n+\frac{1}{12}({\hat{R}}_n^4+3{\hat{R}}_{2n}^2-6{\hat{R}}_n^2{\hat{R}}_{2n}-6{\hat{R}}_{4n}+8{\hat{R}}_{3n}{\hat{R}}_n)).$$
• (c) Recurrence relations of binomial transforms of adjusted fifth order Jacobsthal numbers (take $b_n={\hat{S}}_n$ in Proposition 2): $${\hat{S}}_{-n}=3^{-n}({\hat{S}}_{4n}-{\hat{R}}_n{\hat{S}}_{3n}+\frac{1}{2}({\hat{R}}_n^2-{\hat{R}}_{2n}){\hat{S}}_{2n}-\frac{1}{6}({\hat{R}}_n^3+2{\hat{R}}_{3n}-3{\hat{R}}_{2n}{\hat{R}}_n){\hat{S}}_n).$$
• (d) Recurrence relations of binomial transforms of modified fifth order Jacobsthal-Lucas numbers (take $b_n={\hat{R}}_n$ in Proposition 2 or take $H_n={\hat{R}}_n$ in Corollary 2): $${\hat{R}}_{-n}=\frac{1}{24}{3}^{-n}({\hat{R}}_n^4+3{\hat{R}}_{2n}^2-6{\hat{R}}_n^2{\hat{R}}_{2n}-6{\hat{R}}_{4n}+8{\hat{R}}_{3n}{\hat{R}}_n).$$

7. Sum formulas

7.1. Sums of terms with positive subscripts

The following proposition presents some formulas of binomial transform of generalized fifth order Jacobsthal numbers with positive subscripts.

Proposition 3. If $r=6,s=-13,t=14,u=-7,v=3,$ then for $n\ge 0$ we have the following formulas:

• (a) $\sum _{k=0}^n{b}_k=\frac{1}{2}(b_{n+5}-5b_{n+4}+8b_{n+3}-6b_{n+2}+b_{n+1}-b_4+5b_3-8b_2+6b_1-b_0).$
• (b) $\sum _{k=0}^n{b}_{2k}=\frac{1}{88}(21b_{2n+2}-103b_{2n+1}+244b_{2n}-98b_{2n-1}+69b_{2n-2}-21b_4+103b_3-156b_2+98b_1+19b_0).$
• (c) $\sum _{k=0}^n{b}_{2k+1}=\frac{1}{88}(23b_{2n+2}-29b_{2n+1}+196b_{2n}-78b_{2n-1}+63b_{2n-2}-23b_4+117b_3-196b_2+166b_1-63b_0).$

Proof. Take $r=6,s=-13,t=14,u=-7,v=3,$ in Theorem 2.1 in [28].

From the last proposition, we have the following corollary which gives sum formulas of binomial transform of fifth order Jacobsthal numbers (take $b_n={\hat{J}}_n$ with ${\hat{J}}_0=0,{\hat{J}}_1=1,{\hat{J}}_2=3,{\hat{J}}_3=7,{\hat{J}}_4=15$).

Corollary 6. For $n\ge 0$ we have the following formulas:

• (a) $\sum _{k=0}^n{\hat{J}}_k=\frac{1}{2}({\hat{J}}_{n+5}-5{\hat{J}}_{n+4}+8{\hat{J}}_{n+3}-6{\hat{J}}_{n+2}+{\hat{J}}_{n+1}+2).$
• (b) $\sum _{k=0}^n{\hat{J}}_{2k}=\frac{1}{88}(21{\hat{J}}_{2n+2}-103{\hat{J}}_{2n+1}+244{\hat{J}}_{2n}-98{\hat{J}}_{2n-1}+69{\hat{J}}_{2n-2}+36).$
• (c) $\sum _{k=0}^n{\hat{J}}_{2k+1}=\frac{1}{88}(23{\hat{J}}_{2n+2}-29{\hat{J}}_{2n+1}+196{\hat{J}}_{2n}-78{\hat{J}}_{2n-1}+63{\hat{J}}_{2n-2}+52).$

Taking $b_n={\hat{j}}_n$ with ${\hat{j}}_0=2,{\hat{j}}_1=3,{\hat{j}}_2=9,{\hat{j}}_3=30,{\hat{j}}_4=96$ in the last proposition, we have the following corollary which presents sum formulas of binomial transform of fifth order Jacobsthal-Lucas numbers.

Corollary 7. For $n\ge 0$ we have the following formulas:

• (a) $\sum _{k=0}^n{\hat{j}}_k=\frac{1}{2}({\hat{j}}_{n+5}-5{\hat{j}}_{n+4}+8{\hat{j}}_{n+3}-6{\hat{j}}_{n+2}+{\hat{j}}_{n+1}-2).$
• (b) $\sum _{k=0}^n{\hat{j}}_{2k}=\frac{1}{88}(21{\hat{j}}_{2n+2}-103{\hat{j}}_{2n+1}+244{\hat{j}}_{2n}-98{\hat{j}}_{2n-1}+69{\hat{j}}_{2n-2}+2).$
• (c) $\sum _{k=0}^n{\hat{j}}_{2k+1}=\frac{1}{88}(23{\hat{j}}_{2n+2}-29{\hat{j}}_{2n+1}+196{\hat{j}}_{2n}-78{\hat{j}}_{2n-1}+63{\hat{j}}_{2n-2}-90).$

From the last proposition, we have the following corollary which gives sum formulas of binomial transform of adjusted fifth order Jacobsthal numbers (take $b_n={\hat{S}}_n$ with ${\hat{S}}_0=0,{\hat{S}}_1=1,{\hat{S}}_2=3,{\hat{S}}_3=8,{\hat{S}}_4=22$).

Corollary 8. For $n\ge 0$ we have the following formulas:

• (a) $\sum _{k=0}^n{\hat{S}}_k=\frac{1}{2}({\hat{S}}_{n+5}-5{\hat{S}}_{n+4}+8{\hat{S}}_{n+3}-6{\hat{S}}_{n+2}+{\hat{S}}_{n+1}).$
• (b) $\sum _{k=0}^n{\hat{S}}_{2k}=\frac{1}{88}(21{\hat{S}}_{2n+2}-103{\hat{S}}_{2n+1}+244{\hat{S}}_{2n}-98{\hat{S}}_{2n-1}+69{\hat{S}}_{2n-2}-8).$
• (c) $\sum _{k=0}^n{\hat{S}}_{2k+1}=\frac{1}{88}(23{\hat{S}}_{2n+2}-29{\hat{S}}_{2n+1}+196{\hat{S}}_{2n}-78{\hat{S}}_{2n-1}+63{\hat{S}}_{2n-2}+8).$

Taking $b_n={\hat{R}}_n$ with ${\hat{R}}_0=5,{\hat{R}}_1=6,{\hat{R}}_2=10,{\hat{R}}_3=24,{\hat{R}}_4=70$ in the last proposition, we have the following corollary which presents sum formulas of binomial transform of modified fifth order Jacobsthal-Lucas numbers.

Corollary 9. For $n\ge 0$ we have the following formulas:

• (a) $\sum _{k=0}^n{\hat{R}}_k=\frac{1}{2}({\hat{R}}_{n+5}-5{\hat{R}}_{n+4}+8{\hat{R}}_{n+3}-6{\hat{R}}_{n+2}+{\hat{R}}_{n+1}+1).$
• (b) $\sum _{k=0}^n{\hat{R}}_{2k}=\frac{1}{88}(21{\hat{R}}_{2n+2}-103{\hat{R}}_{2n+1}+244{\hat{R}}_{2n}-98{\hat{R}}_{2n-1}+69{\hat{R}}_{2n-2}+125).$
• (c) $\sum _{k=0}^n{\hat{R}}_{2k+1}=\frac{1}{88}(23{\hat{R}}_{2n+2}-29{\hat{R}}_{2n+1}+196{\hat{R}}_{2n}-78{\hat{R}}_{2n-1}+63{\hat{R}}_{2n-2}-81).$

7.2. Sums of terms with negative subscripts

The following proposition presents some formulas of binomial transform of generalized fifth order Jacobsthal numbers with negative subscripts.

Proposition 4. If $r=6,s=-13,t=14,u=-7,v=3,$ then for $n\ge 1$ we have the following formulas:

• (a) $\sum _{k=1}^n{b}_{-k}=\frac{1}{2}(-b_{-n+4}+5b_{-n+3}-8b_{-n+2}+6b_{-n+1}-b_{-n}+b_4-5b_3+8b_2-6b_1+b_0).$
• (b) $\sum _{k=1}^n{b}_{-2k}=\frac{1}{88}(-23b_{-2n+3}+117b_{-2n+2}-196b_{-2n+1}+166b_{-2n}-63b_{-2n-1}+21b_4-103b_3+156b_2-98b_1-19b_0).$
• (c) $\sum _{k=1}^n{b}_{-2k+1}=\frac{1}{88}(-21b_{-2n+3}+103b_{-2n+2}-156b_{-2n+1}+98b_{-2n}-69b_{-2n-1}+23b_4-117b_3+196b_2-166b_1+63b_0).$

Proof. Take $r=6,s=-13,t=14,u=-7,v=3,$ in Theorem 3.1 in [28].

From the last proposition, we have the following corollary which gives sum formulas of binomial transform of fifth order Jacobsthal numbers (take $b_n={\hat{J}}_n$ with ${\hat{J}}_0=0,{\hat{J}}_1=1,{\hat{J}}_2=3,{\hat{J}}_3=7,{\hat{J}}_4=15).$

Corollary 10. For $n\ge 1,$ binomial transform of fifth order Jacobsthal numbers have the following properties.

• (a) $\sum _{k=1}^n{\hat{J}}_{-k}=\frac{1}{2}(-{\hat{J}}_{-n+4}+5{\hat{J}}_{-n+3}-8{\hat{J}}_{-n+2}+6{\hat{J}}_{-n+1}-{\hat{J}}_{-n}-2).$
• (b) $\sum _{k=1}^n{\hat{J}}_{-2k}=\frac{1}{88}(-23{\hat{J}}_{-2n+3}+117{\hat{J}}_{-2n+2}-196{\hat{J}}_{-2n+1}+166{\hat{J}}_{-2n}-63{\hat{J}}_{-2n-1}-36).$
• (c) $\sum _{k=1}^n{\hat{J}}_{-2k+1}=\frac{1}{88}(-21{\hat{J}}_{-2n+3}+103{\hat{J}}_{-2n+2}-156{\hat{J}}_{-2n+1}+98{\hat{J}}_{-2n}-69{\hat{J}}_{-2n-1}-52).$

Taking $b_n={\hat{j}}_n$ with ${\hat{j}}_0=2,{\hat{j}}_1=3,{\hat{j}}_2=9,{\hat{j}}_3=30,{\hat{j}}_4=96$ in the last proposition, we have the following corollary which presents sum formulas of binomial transform of fifth order Jacobsthal-Lucas numbers.

Corollary 11. For $n\ge 1,$ binomial transform of fifth order Jacobsthal-Lucas numbers have the following properties.

• (a) $\sum _{k=1}^n{\hat{j}}_{-k}=\frac{1}{2}(-{\hat{j}}_{-n+4}+5{\hat{j}}_{-n+3}-8{\hat{j}}_{-n+2}+6{\hat{j}}_{-n+1}-{\hat{j}}_{-n}+2).$
• (b) $\sum _{k=1}^n{\hat{j}}_{-2k}=\frac{1}{88}(-23{\hat{j}}_{-2n+3}+117{\hat{j}}_{-2n+2}-196{\hat{j}}_{-2n+1}+166{\hat{j}}_{-2n}-63{\hat{j}}_{-2n-1}-2).$
• (c) $\sum _{k=1}^n{\hat{j}}_{-2k+1}=\frac{1}{88}(-21{\hat{j}}_{-2n+3}+103{\hat{j}}_{-2n+2}-156{\hat{j}}_{-2n+1}+98{\hat{j}}_{-2n}-69{\hat{j}}_{-2n-1}+90).$

From the last proposition, we have the following corollary which gives sum formulas of binomial transform of adjusted fifth order Jacobsthal numbers (take $b_n={\hat{S}}_n$ with ${\hat{S}}_0=0,{\hat{S}}_1=1,{\hat{S}}_2=3,{\hat{S}}_3=8,{\hat{S}}_4=22$).

Corollary 12. For $n\ge 1,$ binomial transform of adjusted fifth order Jacobsthal numbers have the following properties.

• (a) $\sum _{k=1}^n{\hat{S}}_{-k}=\frac{1}{2}(-{\hat{S}}_{-n+4}+5{\hat{S}}_{-n+3}-8{\hat{S}}_{-n+2}+6{\hat{S}}_{-n+1}-{\hat{S}}_{-n}).$
• (b) $\sum _{k=1}^n{\hat{S}}_{-2k}=\frac{1}{88}(-23{\hat{S}}_{-2n+3}+117{\hat{S}}_{-2n+2}-196{\hat{S}}_{-2n+1}+166{\hat{S}}_{-2n}-63{\hat{S}}_{-2n-1}+8).$
• (c) $\sum _{k=1}^n{\hat{S}}_{-2k+1}=\frac{1}{88}(-21{\hat{S}}_{-2n+3}+103{\hat{S}}_{-2n+2}-156{\hat{S}}_{-2n+1}+98{\hat{S}}_{-2n}-69{\hat{S}}_{-2n-1}-8).$

Taking $b_n={\hat{R}}_n$ with ${\hat{R}}_0=5,{\hat{R}}_1=6,{\hat{R}}_2=10,{\hat{R}}_3=24,{\hat{R}}_4=70$ in the last proposition, we have the following corollary which presents sum formulas of binomial transform of modified fifth order Jacobsthal-Lucas numbers.

Corollary 13. For $n\ge 1,$ binomial transform of modified fifth order Jacobsthal-Lucas numbers have the following properties.

• (a) $\sum _{k=1}^n{\hat{R}}_{-k}=\frac{1}{2}(-{\hat{R}}_{-n+4}+5{\hat{R}}_{-n+3}-8{\hat{R}}_{-n+2}+6{\hat{R}}_{-n+1}-{\hat{R}}_{-n}-1).$
• (b) $\sum _{k=1}^n{\hat{R}}_{-2k}=\frac{1}{88}(-23{\hat{R}}_{-2n+3}+117{\hat{R}}_{-2n+2}-196{\hat{R}}_{-2n+1}+166{\hat{R}}_{-2n}-63{\hat{R}}_{-2n-1}-125).$
• (c) $\sum _{k=1}^n{\hat{R}}_{-2k+1}=\frac{1}{88}(-21{\hat{R}}_{-2n+3}+103{\hat{R}}_{-2n+2}-156{\hat{R}}_{-2n+1}+98{\hat{R}}_{-2n}-69{\hat{R}}_{-2n-1}+81).$

8. Matrices related with binomial transform of generalized fifth order Jacobsthal numbers

We define the square matrix $A$ of order $5$ as: $$A=\left(\begin{array}{ccccc}6& -13& 14& -7& 3\\ 1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\end{array}\right),$$ such that $detA=3.$ From (1) we have and from (6) (or using (19) and induction) we have $$\left(\begin{array}{c}{b}_{n+4}\\ b_{n+3}\\ b_{n+2}\\ b_{n+1}\\ b_n\end{array}\right)={\left(\begin{array}{ccccc}6& -13& 14& -7& 3\\ 1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\end{array}\right)}^n\left(\begin{array}{c}{b}_4\\ b_3\\ b_2\\ b_1\\ b_0\end{array}\right).$$ If we take $b_n={\hat{S}}_n$ in (19) we have We also, for $n\ge 0,$ define $$B_n=\left(\begin{array}{ccccc}\sum _{k=0}^{n+1}\sum _{l=k}^{n+1}\sum _{p=l}^{n+1}{\hat{S}}_k& E_1& E_6& E_{11}& 3\sum _{k=0}^n\sum _{l=k}^n\sum _{p=l}^n{\hat{S}}_k\\ \sum _{k=0}^n\sum _{l=k}^n\sum _{p=l}^n{\hat{S}}_k& E_2& E_7& E_{12}& 3\sum _{k=0}^{n-1}\sum _{l=k}^{n-1}\sum _{p=l}^{n-1}{\hat{S}}_k\\ \sum _{k=0}^{n-1}\sum _{l=k}^{n-1}\sum _{p=l}^{n-1}{\hat{S}}_k& E_3& E_8& E_{13}& 3\sum _{k=0}^{n-2}\sum _{l=k}^{n-2}\sum _{p=l}^{n-2}{\hat{S}}_k\\ \sum _{k=0}^{n-2}\sum _{l=k}^{n-2}\sum _{p=l}^{n-2}{\hat{S}}_k& E_4& E_9& E_{14}& 3\sum _{k=0}^{n-3}\sum _{l=k}^{n-3}\sum _{p=l}^{n-3}{\hat{S}}_k\\ \sum _{k=0}^{n-3}\sum _{l=k}^{n-3}\sum _{p=l}^{n-3}{\hat{S}}_k& E_5& E_{10}& E_{15}& 3\sum _{k=0}^{n-4}\sum _{l=k}^{n-4}\sum _{p=l}^{n-4}{\hat{S}}_k\end{array}\right),$$ and $$C_n=\left(\begin{array}{c}\begin{array}{ccccc}{b}_{n+1}& -13b_n+14b_{n-1}-7b_{n-2}+3b_{n-3}& 14b_n-7b_{n-1}+3b_{n-2}& -7b_n+3b_{n-1}& 3b_n\\ b_n& -13b_{n-1}+14b_{n-2}-7b_{n-3}+3b_{n-4}& 14b_{n-1}-7b_{n-2}+3b_{n-3}& -7b_{n-1}+3b_{n-2}& 3b_{n-1}\\ b_{n-1}& -13b_{n-2}+14b_{n-3}-7b_{n-4}+3b_{n-5}& 14b_{n-2}-7b_{n-3}+3b_{n-4}& -7b_{n-2}+3b_{n-3}& 3b_{n-2}\\ b_{n-2}& -13b_{n-3}+14b_{n-4}-7b_{n-5}+3b_{n-6}& 14b_{n-3}-7b_{n-4}+3b_{n-5}& -7b_{n-3}+3b_{n-4}& 3b_{n-3}\\ b_{n-3}& -13b_{n-4}+14b_{n-5}-7b_{n-6}+3b_{n-7}& 14b_{n-4}-7b_{n-5}+3b_{n-6}& -7b_{n-4}+3b_{n-5}& 3b_{n-4}\end{array}\end{array}\right),$$ where $$\left\{\begin{array}{c}{E}_1\\ E_2\\ E_3\\ E_4\\ E_5\\ E_6\\ E_7\\ E_8\\ E_9\\ E_{10}\\ E_{11}\\ E_{12}\\ E_{13}\\ E_{14}\\ E_{15}\end{array}\right\}=\left\{\begin{array}{c}-13\sum _{k=0}^n\sum _{l=k}^n\sum _{p=l}^n{\hat{S}}_k+14\sum _{k=0}^{n-1}\sum _{l=k}^{n-1}\sum _{p=l}^{n-1}{\hat{S}}_k-7\sum _{k=0}^{n-2}\sum _{l=k}^{n-2}\sum _{p=l}^{n-2}{\hat{S}}_k+3\sum _{k=0}^{n-3}\sum _{l=k}^{n-3}\sum _{p=l}^{n-3}{\hat{S}}_k\\ -13\sum _{k=0}^{n-1}\sum _{l=k}^{n-1}\sum _{p=l}^{n-1}{\hat{S}}_k+14\sum _{k=0}^{n-2}\sum _{l=k}^{n-2}\sum _{p=l}^{n-2}{\hat{S}}_k-7\sum _{k=0}^{n-3}\sum _{l=k}^{n-3}\sum _{p=l}^{n-3}{\hat{S}}_k+3\sum _{k=0}^{n-4}\sum _{l=k}^{n-4}\sum _{p=l}^{n-4}{\hat{S}}_k\\ -13\sum _{k=0}^{n-2}\sum _{l=k}^{n-2}\sum _{p=l}^{n-2}{\hat{S}}_k+14\sum _{k=0}^{n-3}\sum _{l=k}^{n-3}\sum _{p=l}^{n-3}{\hat{S}}_k-7\sum _{k=0}^{n-4}\sum _{l=k}^{n-4}\sum _{p=l}^{n-4}{\hat{S}}_k+3\sum _{k=0}^{n-5}\sum _{l=k}^{n-5}\sum _{p=l}^{n-5}{\hat{S}}_k\\ -13\sum _{k=0}^{n-3}\sum _{l=k}^{n-3}\sum _{p=l}^{n-3}{\hat{S}}_k+14\sum _{k=0}^{n-4}\sum _{l=k}^{n-4}\sum _{p=l}^{n-4}{\hat{S}}_k-7\sum _{k=0}^{n-5}\sum _{l=k}^{n-5}\sum _{p=l}^{n-5}{\hat{S}}_k+3\sum _{k=0}^{n-6}\sum _{l=k}^{n-6}\sum _{p=l}^{n-6}{\hat{S}}_k\\ -13\sum _{k=0}^{n-4}\sum _{l=k}^{n-4}\sum _{p=l}^{n-4}{\hat{S}}_k+14\sum _{k=0}^{n-5}\sum _{l=k}^{n-5}\sum _{p=l}^{n-5}{\hat{S}}_k-7\sum _{k=0}^{n-6}\sum _{l=k}^{n-6}\sum _{p=l}^{n-6}{\hat{S}}_k+3\sum _{k=0}^{n-7}\sum _{l=k}^{n-7}\sum _{p=l}^{n-7}{\hat{S}}_k\\ 14\sum _{k=0}^n\sum _{l=k}^n\sum _{p=l}^n{\hat{S}}_k-7\sum _{k=0}^{n-1}\sum _{l=k}^{n-1}\sum _{p=l}^{n-1}{\hat{S}}_k+3\sum _{k=0}^{n-2}\sum _{l=k}^{n-2}\sum _{p=l}^{n-2}{\hat{S}}_k\\ 14\sum _{k=0}^{n-1}\sum _{l=k}^{n-1}\sum _{p=l}^{n-1}{\hat{S}}_k-7\sum _{k=0}^{n-2}\sum _{l=k}^{n-2}\sum _{p=l}^{n-2}{\hat{S}}_k+3\sum _{k=0}^{n-3}\sum _{l=k}^{n-3}\sum _{p=l}^{n-3}{\hat{S}}_k\\ 14\sum _{k=0}^{n-2}\sum _{l=k}^{n-2}\sum _{p=l}^{n-2}{\hat{S}}_k-7\sum _{k=0}^{n-3}\sum _{l=k}^{n-3}\sum _{p=l}^{n-3}{\hat{S}}_k+3\sum _{k=0}^{n-4}\sum _{l=k}^{n-4}\sum _{p=l}^{n-4}{\hat{S}}_k\\ 14\sum _{k=0}^{n-3}\sum _{l=k}^{n-3}\sum _{p=l}^{n-3}{\hat{S}}_k-7\sum _{k=0}^{n-4}\sum _{l=k}^{n-4}\sum _{p=l}^{n-4}{\hat{S}}_k+3\sum _{k=0}^{n-5}\sum _{l=k}^{n-5}\sum _{p=l}^{n-5}{\hat{S}}_k\\ 14\sum _{k=0}^{n-4}\sum _{l=k}^{n-4}\sum _{p=l}^{n-4}{\hat{S}}_k-7\sum _{k=0}^{n-5}\sum _{l=k}^{n-5}\sum _{p=l}^{n-5}{\hat{S}}_k+3\sum _{k=0}^{n-6}\sum _{l=k}^{n-6}\sum _{p=l}^{n-6}{\hat{S}}_k\\ -7\sum _{k=0}^n\sum _{l=k}^n\sum _{p=l}^n{\hat{S}}_k+3\sum _{k=0}^{n-1}\sum _{l=k}^{n-1}\sum _{p=l}^{n-1}{\hat{S}}_k\\ -7\sum _{k=0}^{n-1}\sum _{l=k}^{n-1}\sum _{p=l}^{n-1}{\hat{S}}_k+3\sum _{k=0}^{n-2}\sum _{l=k}^{n-2}\sum _{p=l}^{n-2}{\hat{S}}_k\\ -7\sum _{k=0}^{n-2}\sum _{l=k}^{n-2}\sum _{p=l}^{n-2}{\hat{S}}_k+3\sum _{k=0}^{n-3}\sum _{l=k}^{n-3}\sum _{p=l}^{n-3}{\hat{S}}_k\\ -7\sum _{k=0}^{n-3}\sum _{l=k}^{n-3}\sum _{p=l}^{n-3}{\hat{S}}_k+3\sum _{k=0}^{n-4}\sum _{l=k}^{n-4}\sum _{p=l}^{n-4}{\hat{S}}_k\\ -7\sum _{k=0}^{n-4}\sum _{l=k}^{n-4}\sum _{p=l}^{n-4}{\hat{S}}_k+3\sum _{k=0}^{n-5}\sum _{l=k}^{n-5}\sum _{p=l}^{n-5}{\hat{S}}_k\end{array}\right\}.$$ By convention, we assume that $$\begin{array}{r}\sum _{k=0}^0\sum _{l=k}^0\sum _{p=l}^0{\hat{S}}_k=0,\sum _{k=0}^{-1}\sum _{l=k}^{-1}\sum _{p=l}^{-1}{\hat{S}}_k=0,\sum _{k=0}^{-2}\sum _{l=k}^{-2}\sum _{p=l}^{-2}{\hat{S}}_k=0,\sum _{k=0}^{-3}\sum _{l=k}^{-3}\sum _{p=l}^{-3}{\hat{S}}_k=0,\\ \sum _{k=0}^{-4}\sum _{l=k}^{-4}\sum _{p=l}^{-4}{\hat{S}}_k=\frac{1}{3},\sum _{k=0}^{-5}\sum _{l=k}^{-5}\sum _{p=l}^{-5}{\hat{S}}_k=\frac{7}{9},\sum _{k=0}^{-6}\sum _{l=k}^{-6}\sum _{p=l}^{-6}{\hat{S}}_k=\frac{7}{27},\sum _{k=0}^{-7}\sum _{l=k}^{-7}\sum _{p=l}^{-7}{\hat{S}}_k=-\frac{128}{81}.\end{array}$$

Theorem 6. For all integers $m,n\ge 0,$ we have

• (a) $B_n=A^n.$
• (b) $C_1{A}^n=A^n{C}_1.$
• (c) $C_{n+m}=C_n{B}_m=B_m{C}_n.$

Proof.

• (a) Proof can be done by mathematical induction on $n.$
• (b) After matrix multiplication, (b) follows.
• (c) We have $C_n=A{C}_{n-1}.$ From the last equation, using induction, we obtain $C_n=A^{n-1}{C}_1.$ Now $$C_{n+m}=A^{n+m-1}{C}_1=A^{n-1}{A}^m{C}_1=A^{n-1}{C}_1{A}^m=C_n{B}_m$$ and similarly $$C_{n+m}=B_m{C}_n.$$

Theorem 7. For $m,n\ge 0,$ we have $$\begin{array}{rcl}{b}_{n+m}& =& b_n\sum _{k=0}^{m+1}\sum _{l=k}^{m+1}\sum _{p=l}^{m+1}{\hat{S}}_k\\ & & +b_{n-1}(-13\sum _{k=0}^m\sum _{l=k}^m\sum _{p=l}^m{\hat{S}}_k+14\sum _{k=0}^{m-1}\sum _{l=k}^{m-1}\sum _{p=l}^{m-1}{\hat{S}}_k-7\sum _{k=0}^{m-2}\sum _{l=k}^{m-2}\sum _{p=l}^{m-2}{\hat{S}}_k+3\sum _{k=0}^{m-3}\sum _{l=k}^{m-3}\sum _{p=l}^{m-3}{\hat{S}}_k)\\ & & +b_{n-2}(14\sum _{k=0}^m\sum _{l=k}^m\sum _{p=l}^m{\hat{S}}_k-7\sum _{k=0}^{m-1}\sum _{l=k}^{m-1}\sum _{p=l}^{m-1}{\hat{S}}_k+3\sum _{k=0}^{m-2}\sum _{l=k}^{m-2}\sum _{p=l}^{m-2}{\hat{S}}_k)\\ & & +b_{n-3}(-7\sum _{k=0}^m\sum _{l=k}^m\sum _{p=l}^m{\hat{S}}_k+3\sum _{k=0}^{m-1}\sum _{l=k}^{m-1}\sum _{p=l}^{m-1}{\hat{S}}_k)+3b_{n-4}\sum _{k=0}^m\sum _{l=k}^m\sum _{p=l}^m{\hat{S}}_k.\end{array}$$

Proof. From the equation $C_{n+m}=C_n{B}_m=B_m{C}_n,$ we see that an element of $C_{n+m}$ is the product of row $C_n$ and a column $B_m.$ From the last equation, we say that an element of $C_{n+m}$ is the product of a row $C_n$ and column $B_m.$ We just compare the linear combination of the 2nd row and 1st column entries of the matrices $C_{n+m}$ and $C_n{B}_m$. This completes the proof.

Corollary 14. For $m,n\ge 0,$ we have $$\begin{array}{rcl}{\hat{J}}_{n+m}& =& {\hat{J}}_n\sum _{k=0}^{m+1}\sum _{l=k}^{m+1}\sum _{p=l}^{m+1}{\hat{S}}_k+{\hat{J}}_{n-1}(-13\sum _{k=0}^m\sum _{l=k}^m\sum _{p=l}^m{\hat{S}}_k+14\sum _{k=0}^{m-1}\sum _{l=k}^{m-1}\sum _{p=l}^{m-1}{\hat{S}}_k-7\sum _{k=0}^{m-2}\sum _{l=k}^{m-2}\sum _{p=l}^{m-2}{\hat{S}}_k+3\sum _{k=0}^{m-3}\sum _{l=k}^{m-3}\sum _{p=l}^{m-3}{\hat{S}}_k)\\ & & +{\hat{J}}_{n-2}(14\sum _{k=0}^m\sum _{l=k}^m\sum _{p=l}^m{\hat{S}}_k-7\sum _{k=0}^{m-1}\sum _{l=k}^{m-1}\sum _{p=l}^{m-1}{\hat{S}}_k+3\sum _{k=0}^{m-2}\sum _{l=k}^{m-2}\sum _{p=l}^{m-2}{\hat{S}}_k)\\ & & +{\hat{J}}_{n-3}(-7\sum _{k=0}^m\sum _{l=k}^m\sum _{p=l}^m{\hat{S}}_k+3\sum _{k=0}^{m-1}\sum _{l=k}^{m-1}\sum _{p=l}^{m-1}{\hat{S}}_k)+3{\hat{J}}_{n-4}\sum _{k=0}^m\sum _{l=k}^m\sum _{p=l}^m{\hat{S}}_k,\end{array}$$ $$\begin{array}{rcl}{\hat{j}}_{n+m}& =& {\hat{j}}_n\sum _{k=0}^{m+1}\sum _{l=k}^{m+1}\sum _{p=l}^{m+1}{\hat{S}}_k\\ & & +{\hat{j}}_{n-1}(-13\sum _{k=0}^m\sum _{l=k}^m\sum _{p=l}^m{\hat{S}}_k+14\sum _{k=0}^{m-1}\sum _{l=k}^{m-1}\sum _{p=l}^{m-1}{\hat{S}}_k-7\sum _{k=0}^{m-2}\sum _{l=k}^{m-2}\sum _{p=l}^{m-2}{\hat{S}}_k+3\sum _{k=0}^{m-3}\sum _{l=k}^{m-3}\sum _{p=l}^{m-3}{\hat{S}}_k)\\ & & +{\hat{j}}_{n-2}(14\sum _{k=0}^m\sum _{l=k}^m\sum _{p=l}^m{\hat{S}}_k-7\sum _{k=0}^{m-1}\sum _{l=k}^{m-1}\sum _{p=l}^{m-1}{\hat{S}}_k+3\sum _{k=0}^{m-2}\sum _{l=k}^{m-2}\sum _{p=l}^{m-2}{\hat{S}}_k)\\ & & +{\hat{j}}_{n-3}(-7\sum _{k=0}^m\sum _{l=k}^m\sum _{p=l}^m{\hat{S}}_k+3\sum _{k=0}^{m-1}\sum _{l=k}^{m-1}\sum _{p=l}^{m-1}{\hat{S}}_k)+3{\hat{j}}_{n-4}\sum _{k=0}^m\sum _{l=k}^m\sum _{p=l}^m{\hat{S}}_k,\end{array}$$ $$\begin{array}{rcl}{\hat{S}}_{n+m}& =& {\hat{S}}_n\sum _{k=0}^{m+1}\sum _{l=k}^{m+1}\sum _{p=l}^{m+1}{\hat{S}}_k\\ & & +{\hat{S}}_{n-1}(-13\sum _{k=0}^m\sum _{l=k}^m\sum _{p=l}^m{\hat{S}}_k+14\sum _{k=0}^{m-1}\sum _{l=k}^{m-1}\sum _{p=l}^{m-1}{\hat{S}}_k-7\sum _{k=0}^{m-2}\sum _{l=k}^{m-2}\sum _{p=l}^{m-2}{\hat{S}}_k+3\sum _{k=0}^{m-3}\sum _{l=k}^{m-3}\sum _{p=l}^{m-3}{\hat{S}}_k)\\ & & +{\hat{S}}_{n-2}(14\sum _{k=0}^m\sum _{l=k}^m\sum _{p=l}^m{\hat{S}}_k-7\sum _{k=0}^{m-1}\sum _{l=k}^{m-1}\sum _{p=l}^{m-1}{\hat{S}}_k+3\sum _{k=0}^{m-2}\sum _{l=k}^{m-2}\sum _{p=l}^{m-2}{\hat{S}}_k)\\ & & +{\hat{S}}_{n-3}(-7\sum _{k=0}^m\sum _{l=k}^m\sum _{p=l}^m{\hat{S}}_k+3\sum _{k=0}^{m-1}\sum _{l=k}^{m-1}\sum _{p=l}^{m-1}{\hat{S}}_k)+3{\hat{S}}_{n-4}\sum _{k=0}^m\sum _{l=k}^m\sum _{p=l}^m{\hat{S}}_k,\end{array}$$ and $$\begin{array}{rcl}{\hat{R}}_{n+m}& =& {\hat{R}}_n\sum _{k=0}^{m+1}\sum _{l=k}^{m+1}\sum _{p=l}^{m+1}{\hat{S}}_k\\ & & +{\hat{R}}_{n-1}(-13\sum _{k=0}^m\sum _{l=k}^m\sum _{p=l}^m{\hat{S}}_k+14\sum _{k=0}^{m-1}\sum _{l=k}^{m-1}\sum _{p=l}^{m-1}{\hat{S}}_k-7\sum _{k=0}^{m-2}\sum _{l=k}^{m-2}\sum _{p=l}^{m-2}{\hat{S}}_k+3\sum _{k=0}^{m-3}\sum _{l=k}^{m-3}\sum _{p=l}^{m-3}{\hat{S}}_k)\\ & & +{\hat{R}}_{n-2}(14\sum _{k=0}^m\sum _{l=k}^m\sum _{p=l}^m{\hat{S}}_k-7\sum _{k=0}^{m-1}\sum _{l=k}^{m-1}\sum _{p=l}^{m-1}{\hat{S}}_k+3\sum _{k=0}^{m-2}\sum _{l=k}^{m-2}\sum _{p=l}^{m-2}{\hat{S}}_k)\\ & & +{\hat{R}}_{n-3}(-7\sum _{k=0}^m\sum _{l=k}^m\sum _{p=l}^m{\hat{S}}_k+3\sum _{k=0}^{m-1}\sum _{l=k}^{m-1}\sum _{p=l}^{m-1}{\hat{S}}_k)+3{\hat{R}}_{n-4}\sum _{k=0}^m\sum _{l=k}^m\sum _{p=l}^m{\hat{S}}_k.\end{array}$$

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflicts of Interest

The authors declare no conflict of interest.