On the sum of the cubes of generalized balancing numbers: The sum formula \(\sum\limits_{k=0}^{n}x^{k}W_{mk+j}^{3}\)

1. Introduction

Behera and Panda [1] defined balancing numbers \(n\) as solutions of the diophantine equation

\begin{equation*} 1+2+\cdots +(n-1)=(n+1)+(n+2)+\cdots +(n+r)\,, \end{equation*} for some natural number \(r\), called the balancer corresponding to \(n\). The \( n \)th balancing number is denoted by \(B_{n}\). Moreover, \(C_{n}=\sqrt{ 8B_{n}^{2}+1}\) is called the \(n\)th Lucas-balancing number (see [2]). In fact, \(B_{n}\) and \(C_{n}\) satisfy the second order linear recurrence relations \begin{equation*} B_{n}=6B_{n-1}-B_{n-2},\  \  B_{0}=0,B_{1}=1, \end{equation*} and \begin{equation*} C_{n}=6C_{n-1}-C_{n-2},\  \  C_{0}=1,C_{1}=3 \end{equation*} respectively. \((B_{n})_{n\geq 0}\) is the sequence \(A001109\) in the OEIS [3], whereas \((B_{n})_{n\geq 0}\) is the id-number \(A001541\) in OEIS. Balancing and Lucas-balancing sequences has been studied by many authors and more detail can be found in the extensive literature dedicated to these sequences, see for example, [1,2,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27].

A generalized balancing sequence \(\{W_{n}\}_{n\geq 0}=\{W_{n}(W_{0},W_{1})\}_{n\geq 0}\) is defined by the second-order recurrence relation

with the initial values \(W_{0}=c_{0},W_{1}=c_{1}\) not all being zero.

The sequence \(\{W_{n}\}_{n\geq 0}\) can be extended to negative subscripts by defining

\begin{equation*} W_{-n}=6W_{-(n-1)}-W_{-(n-2)}\,, \end{equation*} for \(n=1,2,3,....\) Therefore, recurrence (1) holds for all integer \(n.\)

The Binet formula of generalized balancing numbers can be written as

\begin{equation*} W_{n}=\dfrac{W_{1}-\beta W_{0}}{\alpha -\beta }\alpha ^{n}-\dfrac{ W_{1}-\alpha W_{0}}{\alpha -\beta }\beta ^{n}\,, \end{equation*} where \(\alpha \) and \(\beta \) are the roots of the quadratic equation \( x^{2}-6x+1=0.\) Moreover \begin{eqnarray*} \alpha &=&3+2\sqrt{2} \,,\\ \beta &=&3-2\sqrt{2}\,. \end{eqnarray*} Note that \begin{eqnarray*} \alpha +\beta &=&6, \\ \alpha \beta &=&1, \\ \alpha -\beta &=&4\sqrt{2}. \end{eqnarray*} Now we define three special cases of the sequence \(\{W_{n}\}\). Balancing sequence \(\{B_{n}\}_{n\geq 0}\), modified Lucas-balancing sequence \( \{H_{n}\}_{n\geq 0}\) and Lucas-balancing sequence \(\{C_{n}\}_{n\geq 0}\) are defined, respectively, by the second-order recurrence relations, The sequences \(\{B_{n}\}_{n\geq 0},\) \(\{H_{n}\}_{n\geq 0}\) and \( \{C_{n}\}_{n\geq 0}\) can be extended to negative subscripts by defining, \begin{eqnarray*} B_{-n} &=&6B_{-(n-1)}-B_{-(n-2)}, \\ H_{-n} &=&6H_{-(n-1)}-H_{-(n-2)}, \\ C_{-n} &=&6C_{-(n-1)}-C_{-(n-2)}, \end{eqnarray*} for \(n=1,2,3,...\) respectively. Therefore, recurrences (2)-(4) hold for all integer \(n.\) For more information on generalized balancing numbers, see Soykan [28].

2. The sum formula \(\sum\limits_{k=0}^{n}x^{k}W_{mk+j}^{3}\)

The following theorem presents sum formulas of generalized balancing numbers;

Theorem 1. Let \(x\) be a real (or complex) number. For all integers \(m\) and \(j,\) for generalized balancing numbers (the case \(r=6,s=-1\)), we have the following sum formulas:

• (a) If \((x^{2}-xH_{3m}+1)(x^{2}-xH_{m}+ 1)\neq 0\) then
  \begin{equation} \sum\limits_{k=0}^{n}x^{k}W_{mk+j}^{3}=\frac{\Psi _{1}}{ 32(x^{2}-xH_{3m}+1)(x^{2}-xH_{m}+ 1)} \,,\label{equa:tysgsamzn} \end{equation}

  (5)
  where

  \(\Psi _{1}=32 x^{n+1}(x^{2}-xH_{m}+1)W_{mn-m+j}^{3}+32 x^{n+1}(x-H_{3m})(x^{2}-xH_{m}+1) W_{mn+j}^{3}-32x(x^{2}-xH_{m}+1)W_{j-m}^{3}\)

  \(+32 (x^{2}-xH_{m}+1)W_{j}^{3}+3x^{n}x(x^{2}-xH_{3m}+1)(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1})W_{mn+m+j}\)

  \(+3x^{n}x(x^{2}-xH_{m}+1)(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1})W_{mn-m+j}- 3x^{n}x (x^{2}H_{m}-(xH_{m}-1)H_{3m})(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1}) W_{mn+j}\)

  \(-3x(x^{2}-xH_{3m}+1)(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1})W_{m+j}\)

  \(-3x(x^{2}-xH_{m}+1)(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1})W_{j-m}+3x(x^{2}H_{m}-H_{3m}(xH_{m}-1))(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1})W_{j}. \)

• (b) If \( (x^{2}-xH_{3m}+1)(x^{2}-xH_{m}+ 1)=u(x-a)(x-b)(x-c)(x-d)=0\) for some \(u,a,b,c,d\in \mathbb{C}\) with \(u\neq 0\) and \(a\neq b\neq c\neq d\), i.e., \(x=a\) or \(x=b\ \)or \(x=c\) or \(x=d,\) then \begin{equation*} \sum\limits_{k=0}^{n}x^{k}W_{mk+j}^{3}=\frac{\Psi _{2}}{\Lambda _{1}} \,,\end{equation*} where

  \(\Psi _{2}=32 x^{n}(x^{2}(n+3)-x(n+2)H_{m}+n+1) W_{mn-m+j}^{3}+32((n+4) x^{3}-(H_{m}+H_{3m})(n+3)x^{2}+(H_{m}H_{3m}+1)(n+2) x -(n+1)H_{3m}) x^{n}W_{mn+j}^{3}+32(-3x^{2}+2xH_{m}-1)W_{j-m}^{3}\)

  \(+32(2x-H_{m})W_{j}^{3}+3((n+3)x^{2}-x(n+2)H_{3m}+n+1)(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1})x^{n}W_{mn+m+j}+3((n+3)x^{2}-x(n+2)H_{m}+n+1)x^{n}(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1})W_{mn-m+j}+3(-(n+3)x^{2}H_{m}\)

  \(+x(n+2)H_{3m}H_{m}-(n+1)H_{3m})x^{n}(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1}) W_{mn+j}+3(-3x^{2}+2xH_{3m}-1)(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1}) W_{m+j}\)

  \(+3(-3x^{2}+2xH_{m}-1)(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1}) W_{j-m}\)

  \(+3(3x^{2}H_{m}-2xH_{m}H_{3m}+H_{3m}) (W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1})W_{j}\)

  and

  \(\Lambda _{1}=32(4x^{3}-3(H_{m}+H_{3m})x^{2}+2(2+H_{m}H_{3m})x-(H_{m}+H_{3m})).\)

• (c) If \( (x^{2}-xH_{3m}+1)(x^{2}-xH_{m}+ 1)=u(x-a)^{2}(x-b)(x-c)=0\) for some \(u,a,b,c\in \mathbb{C}\) with \(u\neq 0\) and \(a\neq b\neq c\), i.e., \(x=a\) or \(x=b\) or \(x=c,\) then if \(x=b\) or \(x=c\) then \begin{equation*} \sum\limits_{k=0}^{n}x^{k}W_{mk+j}^{3}=\frac{\Psi _{3}}{\Lambda _{2}} \,,\end{equation*} where

  \(\Psi _{3}=32 x^{n}(x^{2}(n+3)-x(n+2)H_{m}+n+1) W_{mn-m+j}^{3}+32((n+4) x^{3}-(H_{m}+H_{3m})(n+3)x^{2}+(H_{m}H_{3m}+1)(n+2) x -(n+1)H_{3m}) x^{n}W_{mn+j}^{3}+32(-3x^{2}+2xH_{m}-1)W_{j-m}^{3}\)

  \(+32(2x-H_{m})W_{j}^{3}+3((n+3)x^{2}-x(n+2)H_{3m}+n+1)(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1})x^{n}W_{mn+m+j}+3((n+3)x^{2}-x(n+2)H_{m}+n+1)x^{n}(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1})W_{mn-m+j}\)

  \(+3(-(n+3)x^{2}H_{m}+x(n+2)H_{3m}H_{m}-(n+1)H_{3m})x^{n}(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1}) W_{mn+j}+3(-3x^{2}+2xH_{3m}-1)(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1}) W_{m+j}\)

  \(+3(-3x^{2}+2xH_{m}-1)(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1}) W_{j-m}+3(3x^{2}H_{m}-2xH_{m}H_{3m}+H_{3m}) (W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1})W_{j}\,, \)

  and

  \(\Lambda _{2}=32(4x^{3}-3(H_{m}+H_{3m})x^{2}+2(2+H_{m}H_{3m})x-(H_{m}+H_{3m}))\), and if \(x=a\) then

  \begin{equation*} \sum\limits_{k=0}^{n}x^{k}W_{mk+j}^{3}=\frac{\Psi _{4}}{64 (6x^{2}-3x(H_{m}+H_{3m})+2+H_{m}H_{3m})} \,,\end{equation*} where

  \(\Psi _{4}=32 ((n+3)(n+2)x^{2}-x(n+2)(n+1)H_{m}+n(n+1)) x^{n-1}W_{mn-m+j}^{3}+32 ((n+4)(n+3) x^{3} -(n+3)(n+2)(H_{m}+H_{3m})x^{2} \)

  \(+x(n+2)(n+1)(H_{m}H_{3m}+1)-n(n+1)H_{3m})x^{n-1}W_{mn+j}^{3}+64(H_{m}-3x)W_{j-m}^{3}+64W_{j}^{3}\)

  \(+3((n+3)(n+2)x^{2}-x(n+2)(n+1)H_{3m}+n(n+1))(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1})x^{n-1} W_{mn+m+j}+3x^{n-1}((n+3)(n+2)x^{2}\)

  \(-x(n+2)(n+1)H_{m}+n(n+1))(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1})W_{mn-m+j}+3x^{n-1}(-x^{2}(n+3)(n+2)H_{m}\)

  \(+x(n+2)(n+1)H_{3m}H_{m}-n(n+1)H_{3m})(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1})W_{mn+j}+6(H_{3m}-3x)(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1})W_{m+j}\)

  \(+6(H_{m}-3x)(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1})W_{j-m}+6(3x-H_{3m})H_{m}(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1})W_{j}. \)

• (d) If \( (x^{2}-xH_{3m}+1)(x^{2}-xH_{m}+ 1)=u(x-a)^{3}(x-b)=0\) for some \( u,a,b\in \mathbb{C}\) with \(u\neq 0\) and \(a\neq b\), i.e., \(x=a\) or \(x=b,\) then if \(x=b\) then \begin{equation*} \sum\limits_{k=0}^{n}x^{k}W_{mk+j}^{3}=\frac{\Psi _{4}}{\Lambda _{3}} \,,\end{equation*} where

  \(\Psi _{5}=32 x^{n}(x^{2}(n+3)-x(n+2)H_{m}+n+1) W_{mn-m+j}^{3}+32((n+4) x^{3}-(H_{m}+H_{3m})(n+3)x^{2}+(H_{m}H_{3m}+1)(n+2) x -(n+1)H_{3m}) x^{n}W_{mn+j}^{3}+32(-3x^{2}+2xH_{m}-1)W_{j-m}^{3}\)

  \(+32(2x-H_{m})W_{j}^{3}+3((n+3)x^{2}-x(n+2)H_{3m}+n+1)(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1})x^{n}W_{mn+m+j}+3((n+3)x^{2}-x(n+2)H_{m}+n+1)x^{n}(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1})W_{mn-m+j}\)

  \(+3(-(n+3)x^{2}H_{m}+x(n+2)H_{3m}H_{m}-(n+1)H_{3m})x^{n}(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1}) W_{mn+j}+3(-3x^{2}+2xH_{3m}-1)(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1}) W_{m+j}\)

  \(+3(-3x^{2}+2xH_{m}-1)(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1}) W_{j-m}+3(3x^{2}H_{m}-2xH_{m}H_{3m}+H_{3m}) (W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1})W_{j}\,, \)

  and

  \(\Lambda _{3}=32(4x^{3}-3(H_{m}+H_{3m})x^{2}+2(2+H_{m}H_{3m})x-(H_{m}+H_{3m}))\,,\)

  and if \(x=a\) then \begin{equation*} \sum\limits_{k=0}^{n}x^{k}W_{mk+j}^{3}=\frac{\Psi _{6}}{192(4x-H_{m}-H_{3m})} \,,\end{equation*} where

  \(\Psi _{6}=32 (n+1)((n+3)(n+2)x^{2}-xn(n+2)H_{m}+n(n-1))x^{n-2}W_{mn-m+j}^{3}\)

  \(+32((n+3)(n+2)(n+4)x^{3} -(n+3)(n+2)(n+1)(H_{m}+H_{3m}) x^{2}+ n(n+2)(n+1)(H_{m}H_{3m}+1) x-n(n-1)(n+1)H_{3m}) x^{n-2}W_{mn+j}^{3}\)

  \(-192 W_{j-m}^{3}+3(n+1)((n+3)(n+2)x^{2}-xn(n+2)H_{3m}+n(n-1))(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1}) x^{n-2}W_{mn+m+j}\)

  \(+3(n+1)((n+3)(n+2)x^{2}-xn(n+2)H_{m}+n(n-1))(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1})x^{n-2}W_{mn-m+j}+3(n+1)(-x^{2}(n+3)(n+2)H_{m}\)

  \(+xn(n+2)H_{3m}H_{m}-n(n-1)H_{3m})(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1}) x^{n-2}W_{mn+j}-18(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1})W_{m+j}-18(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1})W_{j-m}+18H_{m}(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1})W_{j}. \)

• (e) If \( (x^{2}-xH_{3m}+1)(x^{2}-xH_{m}+ 1)=u(x-a)^{4}=0\) for some \(u,a\in \mathbb{C}\),\(u\neq 0\) i.e., \(x=a\) then \begin{equation*} \sum\limits_{k=0}^{n}x^{k}W_{mk+j}^{3}=\frac{\Psi _{7}}{768} \,,\end{equation*} where

  \(\Psi _{7}=32 n(n+1)((n+3)(n+2)x^{2}-x(n-1)(n+2)H_{m}+(n-1)(n-2)) x^{n-3}W_{mn-m+j}^{3}\)

  \(+32(n+1) ( x^{3}(n+4)(n+3)(n+2)-x^{2}n(n+3)(n+2)(H_{m}+H_{3m}) + xn(n-1)(n+2)(H_{m}H_{3m}+1) -n(n-1)(n-2)H_{3m})x^{n-3}W_{mn+j}^{3}\)

  \(+3n(n+1)(x^{2}(n+3)(n+2)-x(n+2)(n-1)H_{3m}+(n-1)(n-2))(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1}) x^{n-3}W_{mn+m+j}\)

  \(+3n(n+1)(x^{2}(n+3)(n+2)-x(n+2)(n-1)H_{m}+(n-1)(n-2)) (W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1})x^{n-3}W_{mn-m+j}\)

  \(+3n(n+1)( -x^{2}(n+3)(n+2)H_{m}+x(n+2)(n-1)H_{3m}H_{m}-(n-1)(n-2)H_{3m})(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1}) x^{n-3}W_{mn+j}. \)

Proof. Take \(r=6,s=-1\) and \(H_{n}=H_{n}\) in Soykan [29, Theorem 2.1.].

Note that (5) can be written in the following form: \begin{equation*} \sum\limits_{k=1}^{n}x^{k}W_{mk+j}^{2}=\frac{\Psi _{8}}{ 32(x^{2}-xH_{3m}+1)(x^{2}-xH_{m}+ 1)} \,,\end{equation*} where

\(\Psi _{8}=32 x^{n+1}(x^{2}-xH_{m}+1)W_{mn-m+j}^{3}+32 x^{n+1}(x-H_{3m})(x^{2}-xH_{m}+1) W_{mn+j}^{3}-32x(x^{2}-xH_{m}+1)W_{j-m}^{3}+32(H_{3m}-x)(x^{2}-xH_{m}+1) x W_{j}^{3}\)

\(+3x^{n}x(x^{2}-xH_{3m}+1)(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1}) W_{mn+m+j}+3x^{n}x(x^{2}-xH_{m}+1)(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1})W_{mn-m+j}- \)

\( 3x^{n}x (x^{2}H_{m}-(xH_{m}-1)H_{3m})(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1}) W_{mn+j}-3x(x^{2}-xH_{3m}+1)(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1})W_{m+j}\)

\(-3x(x^{2}-xH_{m}+1)(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1})W_{j-m}+3x(x^{2}H_{m}-H_{3m}(xH_{m}-1))(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1})W_{j}. \)

As special cases of \(m\) and \(j\) in the last Theorem, we obtain the following proposition;

Proposition 1. For generalized balancing numbers (the case \(r=6,s=-1\)), we have the following sum formulas for \(n\geq 0\):

• (a) \((m=1,\) \(j=0)\)

  If \( \left( x^{2}-6x+1\right) \left( x^{2}-198x+1\right) \neq 0,\) i.e., \(x\neq 99+70\sqrt{2},\) \(x\neq 99-70\sqrt{2},\) \(x\neq 3+2\sqrt{2},\) \( x\neq 3-2\sqrt{2},\) then

  \begin{equation*} \sum\limits_{k=0}^{n}x^{k}W_{k}^{3}=\frac{\Psi _{1}}{32\left( x^{2}-6x+1\right) \left( x^{2}-198x+1\right) } \,,\end{equation*} where

  \(\Psi _{1}=32x^{n+1}(x-198)(x^{2}-6x+1)W_{n}^{3}+32x^{n+1}(x^{2}-6x+1)W_{n-1}^{3}+3x^{n+1}(x^{2}-198x+1)(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1})W_{n+1}\)

  \(-18x^{n+1}(x^{2}-198x+33)(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1}) W_{n}+3x^{n+1}(x^{2}-6x+1)(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1})W_{n-1}\)

  \(-32(-x(x^{2}+12x+1)W_{1}^{3}+(216x^{3}-1189x^{2}+204x-1)W_{0}^{3}+18x^{2}(x+6)W_{1}^{2}W_{0}-18x^{2}(6x+1)W_{0}^{2}W_{1})\,, \)

  and

  if \( \left( x^{2}-6x+1\right) \left( x^{2}-198x+1\right) =0,\) i.e., \(x=99+70\sqrt{2}\) or \(x=99-70\sqrt{2}\) or \(x=3+2\sqrt{2}\) or \(x=3-2 \sqrt{2}\) then

  \begin{equation*} \sum\limits_{k=0}^{n}x^{k}W_{k}^{3}=\frac{\Psi _{2}}{128\left( x^{3}-153x^{2}+595x-51\right) } \,,\end{equation*} where

  \(\Psi _{2}=32x^{n}(n(x-198)(x^{2}-6x+1)+4x^{3}-612x^{2}+2378x-198)W_{n}^{3}+32x^{n}(n(x^{2}-6x+1)+3x^{2}-12x+1)W_{n-1}^{3} \)

  \(+3x^{n}(n(x^{2}-198x+1)+3x^{2}-396x+1)(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1})W_{n+1}-18x^{n}(n(x^{2}-198x+33)+3x^{2}-396x+33)(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1})W_{n} \)

  \(+3x^{n}(n(x^{2}-6x+1)+3x^{2}-12x+1)(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1})W_{n-1}+32((3x^{2}+24x+1)W_{1}^{3}-2(324x^{2}-1189x+102)W_{0}^{3}-54x(x+4)W_{1}^{2}W_{0}+36x(9x+1)W_{0}^{2}W_{1}). \)

• (b) \((m=2,\) \(j=0)\)

  If \( \left( x^{2}-34x+1\right) \left( x^{2}-39202x+1\right) \neq 0,\) i.e., \(x\neq 19601+13860\sqrt{2},\) \(x\neq 19601-13860\sqrt{2},\) \(x\neq 17+12 \sqrt{2},\) \(x\neq 17-12\sqrt{2},\) then

  \begin{equation*} \sum\limits_{k=0}^{n}x^{k}W_{2k}^{3}=\frac{\Psi _{1}}{3 2\left( x^{2}-34x+1\right) \left( x^{2}-39202x+1\right) } \,,\end{equation*} where

  \(\Psi _{1}=32x^{n+1}(x-39202)(x^{2}-34x+1)W_{2n}^{3}+32x^{n+1}(x^{2}-34x+1)W_{2n-2}^{3}\)

  \(+3x^{n+1}(x^{2}-39202x+1)(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1})W_{2n+2}-102x^{n+1}(x^{2}-39202x+1153)(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1}) W_{2n}\)

  \(+3x^{n+1}(x^{2}-34x+1)(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1})W_{2n-2}+32(216x(x^{2}+68x+1)W_{1}^{3}-(42875x^{3}-1329231x^{2}\)

  \(+39237x-1)W_{0}^{3}-108x(35x^{2}+1224x+1)W_{1}^{2}W_{0}+18x(1225x^{2}+2414x+1)W_{0}^{2}W_{1})\,, \)

  and

  if \( \left( x^{2}-34x+1\right) \left( x^{2}-39202x+1\right) =0,\) i.e., \(x=19601+13860\sqrt{2}\) or \(x=19601-13860\sqrt{2}\) or \( x=17+12 \sqrt{2}\) or \(x=17-12\sqrt{2}\) then

  \begin{equation*} \sum\limits_{k=0}^{n}x^{k}W_{2k}^{3}=\frac{\Psi _{2}}{128\left( x^{3}-29427x^{2}+666435x-9809\right) } \,,\end{equation*} where

  \(\Psi _{2}=32x^{n}(n(x-39202)(x^{2}-34x+1)+2(2x^{3}-58854x^{2}+1332869x-19601))W_{2n}^{3}+32x^{n}(n(x^{2}-34x+1)+3x^{2}-68x+1)W_{2n-2}^{3} \)

  \(+3x^{n}(n(x^{2}-39202x+1)+3x^{2}-78404x+1)(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1})W_{2n+2}-102x^{n}(n(x^{2}-39202x+1153)\)

  \(+3x^{2}-78404x+1153)(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1})W_{2n} +3x^{n}(n(x^{2}-34x+1)\)

  \(+3x^{2}-68x+1)(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1})W_{2n-2}+96(72(3x^{2}+136x+1)W_{1}^{3}-(42875x^{2}-886154x+13079)W_{0}^{3}\)

  \(-36(105x^{2}+2448x+1)W_{1}^{2}W_{0}+6(3675x^{2}+4828x+1)W_{0}^{2}W_{1}). \)

• (c) \((m=2,\) \(j=1)\)

  If \( \left( x^{2}-34x+1\right) \left( x^{2}-39202x+1\right) \neq 0,\) i.e., \(x\neq 19601+13860\sqrt{2},\) \(x\neq 19601-13860\sqrt{2},\) \(x\neq 17+12 \sqrt{2},\) \(x\neq 17-12\sqrt{2},\) then

  \begin{equation*} \sum\limits_{k=0}^{n}x^{k}W_{2k+1}^{3}=\frac{\Psi _{1}}{32 \left( x^{2}-34x+1\right) \left( x^{2}-39202x+1\right) } \,,\end{equation*} where

  \(\Psi _{1}=32x^{n+1}(x-39202)(x^{2}-34x+1)W_{2n+1}^{3}+32x^{n+1}(x^{2}-34x+1)W_{2n-1}^{3}\)

  \(+3x^{n+1}(x^{2}-39202x+1)(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1})W_{2n+3}\)

  \(-102x^{n+1}\left( (x^{2}-39202x+1153)\right) (W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1})W_{2n+1}\)

  \(+3x^{n+1}(x^{2}-34x+1) (W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1})W_{2n-1}+32((x+1)(x^{2}+3638x+1)W_{1}^{3}\)

  \(-216x(x^{2}+68x+1)W_{0}^{3}-18x(x^{2}+2414x+1225)W_{0}W_{1}^{2}+108x(x^{2}+1224x+35)W_{0}^{2}W_{1}) \,,\)

  and

  if \( \left( x^{2}-34x+1\right) \left( x^{2}-39202x+1\right) =0,\) i.e., \(x=19601+13860\sqrt{2}\) or \(x=19601-13860\sqrt{2}\) or \( x=17+12 \sqrt{2}\) or \(x=17-12\sqrt{2}\) then

  \begin{equation*} \sum\limits_{k=0}^{n}x^{k}W_{2k+1}^{3}=\frac{\Psi _{2}}{128\left( x^{3}-29427x^{2}+666435x-9809\right) } \,,\end{equation*} where

  \(\Psi _{2}=32x^{n}(n(x-39202)(x^{2}-34x+1)+2(2x^{3}-58854x^{2}+1332869x-19601))W_{2n+1}^{3}\)

  \(+32x^{n}(n(x^{2}-34x+1)+3x^{2}-68x+1)W_{2n-1}^{3}+3x^{n}(n(x^{2}-39\,202x+1)\)

  \(+3x^{2}-78404x+1)(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1})W_{2n+3}-102x^{n}(n(x^{2}-39202x+1153)\)

  \(+3x^{2}-78404x+1153)(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1}) W_{2n+1}+3x^{n}(n(x^{2}-34x+1)\)

  \(+3x^{2}-68x+1)(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1})W_{2n-1}+96((x^{2}+2426x+1213)W_{1}^{3}-72(3x^{2}+136x+1)W_{0}^{3}\)

  \(-6(3x^{2}+4828x+1225)W_{1}^{2}W_{0}+36(3x^{2}+2448x+35)W_{0}^{2}W_{1}). \)

• (d) \((m=-1,\) \(j=0)\)

  If \( \left( x^{2}-6x+1\right) \left( x^{2}-198x+1\right) \neq 0,\) i.e., \(x\neq 99+70\sqrt{2},\) \(x\neq 99-70\sqrt{2},\) \(x\neq 3+2\sqrt{2},\) \( x\neq 3-2\sqrt{2},\) then

  \begin{equation*} \sum\limits_{k=0}^{n}x^{k}W_{-k}^{3}=\frac{\Psi _{1}}{32 \left( x^{2}-6x+1\right) \left( x^{2}-198x+1\right) } \,,\end{equation*} where

  \(\Psi _{1}=32x^{n+1}(x^{2}-6x+1)W_{-n+1}^{3}+32x^{n+1}(x-198)(x^{2}-6x+1)W_{-n}^{3}\)

  \(+3x^{n+1}(x^{2}-6x+1)(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1})W_{-n+1}-18x^{n+1}(x^{2}-198x+33)(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1}) W_{-n}\)

  \(+3x^{n+1}(x^{2}-198x+1)(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1}) W_{-n-1}\)

  \(+32(-x(x^{2}+12x+1)W_{1}^{3}+(x^{2}+12x+1)W_{0}^{3}+18x(6x+1)W_{1}^{2}W_{0}-18x(x+6)W_{0}^{2}W_{1}) \,,\)

  and

  if \( \left( x^{2}-6x+1\right) \left( x^{2}-198x+1\right) =0,\) i.e., \(x=99+70\sqrt{2}\) or \(x=99-70\sqrt{2}\) or \(x=3+2\sqrt{2}\) or \(x=3-2 \sqrt{2}\) then

  \begin{equation*} \sum\limits_{k=0}^{n}x^{k}W_{-k}^{3}=\frac{\Psi _{2}}{128\left( x^{3}-153x^{2}+595x-51\right) } \,,\end{equation*} where

  \(\Psi _{2}=32x^{n}(n(x^{2}-6x+1)+3x^{2}-12x+1)W_{-n+1}^{3}+32x^{n}(n(x-198)(x^{2}-6x+1)\)

  \(+2(2x^{3}-306x^{2}+1189x-99))W_{-n}^{3}+3x^{n}(n(x^{2}-6x+1)+3x^{2}-12x+1)(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1})W_{-n+1} \)

  \(-18x^{n}(n(x^{2}-198x+33)+3x^{2}-396x+33)(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1})W_{-n}+3x^{n}(n(x^{2}-198x+1)\)

  \(+3x^{2}-396x+1)(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1})W_{-n-1}+32(-(3x^{2}+24x+1)W_{1}^{3}\)

  \(+2(x+6)W_{0}^{3}+18(12x+1)W_{1}^{2}W_{0}-36(x+3)W_{0}^{2}W_{1}). \)

• (e) \((m=-2,\) \(j=0)\)

  If \( \left( x^{2}-34x+1\right) \left( x^{2}-39202x+1\right) \neq 0,\) i.e., \(x\neq 19601+13860\sqrt{2},\) \(x\neq 19601-13860\sqrt{2},\) \(x\neq 17+12 \sqrt{2},\) \(x\neq 17-12\sqrt{2},\) then

  \begin{equation*} \sum\limits_{k=0}^{n}x^{k}W_{-2k}^{3}=\frac{\Psi _{1}}{32 \left( x^{2}-34x+1\right) \left( x^{2}-39202x+1\right) } \,,\end{equation*} where

  \(\Psi _{1}=32x^{n+1}(x^{2}-34x+1)W_{-2n+2}^{3}+32x^{n+1}(x-39202)(x^{2}-34x+1)W_{-2n}^{3}\)

  \(+3x^{n+1}(x^{2}-34x+1)(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1})W_{-2n+2}-102x^{n+1}(x^{2}-39202x+1153)(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1}) W_{-2n}\)

  \(+3x^{n+1}(x^{2}-39202x+1)(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1}) W_{-2n-2}+32(-216x(x^{2}+68x+1)W_{1}^{3}+(x+1)(x^{2}+3638x+1)W_{0}^{3}\)

  \(+108x(x^{2}+1224x+35)W_{1}^{2}W_{0}-18x(x^{2}+2414x+1225)W_{0}^{2}W_{1}) \,,\)

  and

  if \( \left( x^{2}-34x+1\right) \left( x^{2}-39202x+1\right) =0,\) i.e., \(x=19601+13860\sqrt{2}\) or \(x=19601-13860\sqrt{2}\) or \( x=17+12 \sqrt{2}\) or \(x=17-12\sqrt{2}\) then

  \begin{equation*} \sum\limits_{k=0}^{n}x^{k}W_{-2k}^{3}=\frac{\Psi _{2}}{128\left( x^{3}-29427x^{2}+666435x-9809\right) } \,,\end{equation*} where

  \(\Psi _{2}=32x^{n}(n(x^{2}-34x+1)+3x^{2}-68x+1)W_{-2n+2}^{3}+32x^{n}(n(x-39202)(x^{2}-34x+1)+2(2x^{3}-58854x^{2}+1332869x-19601))W_{-2n}^{3}\)

  \(+3x^{n}(n(x^{2}-34x+1)+3x^{2}-68x+1)(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1})W_{-2n+2}-102x^{n}(n(x^{2}-39202x+1153)\)

  \(+3x^{2}-78404x+1153)(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1}) W_{-2n}+3x^{n}(n(x^{2}-39202x+1)\)

  \(+3x^{2}-78404x+1)(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1})W_{-2n-2}+96(-72(3x^{2}+136x+1)W_{1}^{3}+(x^{2}+2426x+1213)W_{0}^{3}\)

  \(+36(3x^{2}+2448x+35)W_{1}^{2}W_{0}-6(3x^{2}+4828x+1225)W_{0}^{2}W_{1}). \)

• (f) \((m=-2,\) \(j=1)\)

  If \( \left( x^{2}-34x+1\right) \left( x^{2}-39202x+1\right) \neq 0,\) i.e., \(x\neq 19601+13860\sqrt{2},\) \(x\neq 19601-13860\sqrt{2},\) \(x\neq 17+12 \sqrt{2},\) \(x\neq 17-12\sqrt{2},\) then

  \begin{equation*} \sum\limits_{k=0}^{n}x^{k}W_{-2k+1}^{3}=\frac{\Psi _{1}}{32 \left( x^{2}-34x+1\right) \left( x^{2}-39202x+1\right) } \,,\end{equation*} where

  \(\Psi _{1}=32x^{n+1}(x^{2}-34x+1)W_{-2n+3}^{3}+32x^{n+1}(x-39202)(x^{2}-34x+1)W_{-2n+1}^{3}\)

  \(+3x^{n+1}(x^{2}-34x+1)\left( W_{0}^{2}+W_{1}^{2}-6W_{0}W_{1}\right) W_{-2n+3}\)

  \(-102x^{n+1}(x^{2}-39202x+1153)(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1}) W_{-2n+1}\)

  \(+3x^{n+1}(x^{2}-39202x+1)(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1})W_{-2n-1}+32(-(42875x^{3}-1329231x^{2}\)

  \(+39237x-1)W_{1}^{3}+216x(x^{2}+68x+1)W_{0}^{3}+18x(1225x^{2}+2414x+1)W_{0}W_{1}^{2}-108x(35x^{2}+1224x+1)W_{0}^{2}W_{1}) \,,\)

  and

  if \( \left( x^{2}-34x+1\right) \left( x^{2}-39202x+1\right) =0,\) i.e., \(x=19601+13860\sqrt{2}\) or \(x=19601-13860\sqrt{2}\) or \( x=17+12 \sqrt{2}\) or \(x=17-12\sqrt{2}\) then

  \begin{equation*} \sum\limits_{k=0}^{n}x^{k}W_{-2k+1}^{3}=\frac{\Psi _{2}}{128\left( x^{3}-29427x^{2}+666435x-9809\right) } \,,\end{equation*} where

  \(\Psi _{2}=32x^{n}(n(x^{2}-34x+1)+3x^{2}-68x+1)W_{-2n+3}^{3}+32x^{n}(n(x-39202)(x^{2}-34x+1)\)

  \(+4x^{3}-117708x^{2}+2665738x-39202)W_{-2n+1}^{3}+3x^{n}(n(x^{2}-34x+1)+3x^{2}-68x+1)(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1})W_{-2n+3} \)

  \(-102x^{n}(n(x^{2}-39202x+1153)+3x^{2}-78404x+1153)(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1})W_{-2n+1}+3x^{n}(n(x^{2}-39202x+1)\)

  \(+3x^{2}-78404x+1)(W_{1}^{2}+W_{0}^{2}-6W_{0}W_{1})W_{-2n-1}+96(-(42875x^{2}-886154x+13079)W_{1}^{3} \)

  \(+72(3x^{2}+136x+1)W_{0}^{3}+6(3675x^{2}+4828x+1)W_{1}^{2}W_{0}-36(105x^{2}+2448x+1)W_{0}^{2}W_{1}) . \)

From the above proposition, we have the following corollary which gives sum formulas of balancing numbers (take \(W_{n}=B_{n}\) with \(B_{0}=0,B_{1}=1\));

Corollary 2. For \(n\geq 0,\) balancing numbers have the following properties:

• (a) \((m=1,\) \(j=0)\)

  If \( \left( x^{2}-6x+1\right) \left( x^{2}-198x+1\right) \neq 0,\) i.e., \(x\neq 99+70\sqrt{2},\) \(x\neq 99-70\sqrt{2},\) \(x\neq 3+2\sqrt{2},\) \( x\neq 3-2\sqrt{2},\) then

  \begin{equation*} \sum\limits_{k=0}^{n}x^{k}B_{k}^{3}=\frac{\Psi _{1}}{32\left( x^{2}-6x+1\right) \left( x^{2}-198x+1\right) } \,,\end{equation*} where

  \(\Psi _{1}=32x^{n+1}(x-198)(x^{2}-6x+1)B_{n}^{3}+32x^{n+1}(x^{2}-6x+1)B_{n-1}^{3}\)

  \(+3x^{n+1}(x^{2}-198x+1)B_{n+1}-18x^{n+1}(x^{2}-198x+33)B_{n}+3x^{n+1}(x^{2}-6x+1)B_{n-1}+32x(x^{2}+12x+1)\,, \)

  and

  if \( \left( x^{2}-6x+1\right) \left( x^{2}-198x+1\right) =0,\) i.e., \(x=99+70\sqrt{2}\) or \(x=99-70\sqrt{2}\) or \(x=3+2\sqrt{2}\) or \(x=3-2 \sqrt{2}\) then

  \begin{equation*} \sum\limits_{k=0}^{n}x^{k}B_{k}^{3}=\frac{\Psi _{2}}{128\left( x^{3}-153x^{2}+595x-51\right) } \,,\end{equation*} where

  \(\Psi _{2}=32x^{n}(n(x-198)(x^{2}-6x+1)+4x^{3}-612x^{2}+2378x-198)B_{n}^{3}+32x^{n}(n(x^{2}-6x+1)\)

  \(+3x^{2}-12x+1)B_{n-1}^{3}+3x^{n}(n(x^{2}-198x+1)+3x^{2}-396x+1)B_{n+1}-18x^{n}(n(x^{2}-198x+33)\)

  \(+3x^{2}-396x+33)B_{n} +3x^{n}(n(x^{2}-6x+1)+3x^{2}-12x+1)B_{n-1}+32(3x^{2}+24x+1).\)

• (b) \((m=2,\) \(j=0)\)

  If \( \left( x^{2}-34x+1\right) \left( x^{2}-39202x+1\right) \neq 0,\) i.e., \(x\neq 19601+13860\sqrt{2},\) \(x\neq 19601-13860\sqrt{2},\) \(x\neq 17+12 \sqrt{2},\) \(x\neq 17-12\sqrt{2},\) then

  \begin{equation*} \sum\limits_{k=0}^{n}x^{k}B_{2k}^{3}=\frac{\Psi _{1}}{3 2\left( x^{2}-34x+1\right) \left( x^{2}-39202x+1\right) } \,,\end{equation*} where

  \(\Psi _{1}=32x^{n+1}(x-39202)(x^{2}-34x+1)B_{2n}^{3}+32x^{n+1}(x^{2}-34x+1)B_{2n-2}^{3}+3x^{n+1}(x^{2}-39202x+1)B_{2n+2}\)

  \(-102x^{n+1}(x^{2}-39202x+1153)B_{2n}+3x^{n+1}(x^{2}-34x+1)B_{2n-2}+6912x\left( x^{2}+68x+1\right)\,, \)

  and

  if \( \left( x^{2}-34x+1\right) \left( x^{2}-39202x+1\right) =0,\) i.e., \(x=19601+13860\sqrt{2}\) or \(x=19601-13860\sqrt{2}\) or \( x=17+12 \sqrt{2}\) or \(x=17-12\sqrt{2}\) then

  \begin{equation*} \sum\limits_{k=0}^{n}x^{k}B_{2k}^{3}=\frac{\Psi _{2}}{128\left( x^{3}-29427x^{2}+666435x-9809\right) } \,,\end{equation*} where

  \(\Psi _{2}=32x^{n}(n(x-39202)(x^{2}-34x+1)+2(2x^{3}-58854x^{2}+1332869x-19601))B_{2n}^{3}+32x^{n}(n(x^{2}-34x+1)\)

  \(+3x^{2}-68x+1)B_{2n-2}^{3}+3x^{n}(n(x^{2}-39202x+1)+3x^{2}-78404x+1)B_{2n+2} -102x^{n}(n(x^{2}-39202x+1153)\)

  \(+3x^{2}-78404x+1153)B_{2n}+3x^{n}(n(x^{2}-34x+1)+3x^{2}-68x+1)B_{2n-2}+6912(3x^{2}+136x+1). \)

• (c) \((m=2,\) \(j=1)\)

  If \( \left( x^{2}-34x+1\right) \left( x^{2}-39202x+1\right) \neq 0,\) i.e., \(x\neq 19601+13860\sqrt{2},\) \(x\neq 19601-13860\sqrt{2},\) \(x\neq 17+12 \sqrt{2},\) \(x\neq 17-12\sqrt{2},\) then

  \begin{equation*} \sum\limits_{k=0}^{n}x^{k}B_{2k+1}^{3}=\frac{\Psi _{1}}{32 \left( x^{2}-34x+1\right) \left( x^{2}-39202x+1\right) } \,,\end{equation*} where

  \(\Psi _{1}=32x^{n+1}(x-39202)(x^{2}-34x+1)B_{2n+1}^{3}+32x^{n+1}(x^{2}-34x+1)B_{2n-1}^{3}+3x^{n+1}(x^{2}-39202x+1)B_{2n+3}\)

  \(-102x^{n+1}\left( (x^{2}-39202x+1153)\right) B_{2n+1}+3x^{n+1}(x^{2}-34x+1) B_{2n-1}+32(x+1)(x^{2}+3638x+1) \,,\)

  and

  if \( \left( x^{2}-34x+1\right) \left( x^{2}-39202x+1\right) =0,\) i.e., \(x=19601+13860\sqrt{2}\) or \(x=19601-13860\sqrt{2}\) or \( x=17+12 \sqrt{2}\) or \(x=17-12\sqrt{2}\) then

  \begin{equation*} \sum\limits_{k=0}^{n}x^{k}B_{2k+1}^{3}=\frac{\Psi _{2}}{128\left( x^{3}-29427x^{2}+666435x-9809\right) } \,,\end{equation*} where

  \(\Psi _{2}=32x^{n}(n(x-39202)(x^{2}-34x+1)+2(2x^{3}-58854x^{2}+1332869x-19601))B_{2n+1}^{3}\)

  \(+32x^{n}(n(x^{2}-34x+1)+3x^{2}-68x+1)B_{2n-1}^{3}+3x^{n}(n(x^{2}-39\,202x+1)+3x^{2}-78404x+1)B_{2n+3}\)

  \(-102x^{n}(n(x^{2}-39202x+1153)+3x^{2}-78404x+1153)B_{2n+1}+3x^{n}(n(x^{2}-34x+1)+3x^{2}-68x+1)B_{2n-1}+96(x^{2}+2426x+1213). \)

• (d) \((m=-1,\) \(j=0)\)

  If \( \left( x^{2}-6x+1\right) \left( x^{2}-198x+1\right) \neq 0,\) i.e., \(x\neq 99+70\sqrt{2},\) \(x\neq 99-70\sqrt{2},\) \(x\neq 3+2\sqrt{2},\) \( x\neq 3-2\sqrt{2},\) then

  \begin{equation*} \sum\limits_{k=0}^{n}x^{k}B_{-k}^{3}=\frac{\Psi _{1}}{32 \left( x^{2}-6x+1\right) \left( x^{2}-198x+1\right) } \,,\end{equation*} where

  \(\Psi _{1}=32x^{n+1}(x^{2}-6x+1)B_{-n+1}^{3}+32x^{n+1}(x-198)(x^{2}-6x+1)B_{-n}^{3}+3x^{n+1}(x^{2}-6x+1)B_{-n+1}-18x^{n+1}(x^{2}\)

  \(-198x+33)B_{-n}+3x^{n+1}(x^{2}-198x+1)B_{-n-1}-32x(x^{2}+12x+1) \,,\)

  and

  if \( \left( x^{2}-6x+1\right) \left( x^{2}-198x+1\right) =0,\) i.e., \(x=99+70\sqrt{2}\) or \(x=99-70\sqrt{2}\) or \(x=3+2\sqrt{2}\) or \(x=3-2 \sqrt{2}\) then

  \begin{equation*} \sum\limits_{k=0}^{n}x^{k}B_{-k}^{3}=\frac{\Psi _{2}}{128\left( x^{3}-153x^{2}+595x-51\right) } \,,\end{equation*} where

  \(\Psi _{2}=32x^{n}(n(x^{2}-6x+1)+3x^{2}-12x+1)B_{-n+1}^{3}+32x^{n}(n(x-198)(x^{2}-6x+1)+2(2x^{3}-306x^{2}+1189x-99))B_{-n}^{3}+3x^{n}(n(x^{2}-6x+1)\)

  \(+3x^{2}-12x+1)B_{-n+1}-18x^{n}(n(x^{2}-198x+33)+3x^{2}-396x+33)B_{-n} +3x^{n}(n(x^{2}-198x+1)\)

  \(+3x^{2}-396x+1)B_{-n-1}-32(3x^{2}+24x+1).\)

• (e) \((m=-2,\) \(j=0)\)

  If \( \left( x^{2}-34x+1\right) \left( x^{2}-39202x+1\right) \neq 0,\) i.e., \(x\neq 19601+13860\sqrt{2},\) \(x\neq 19601-13860\sqrt{2},\) \(x\neq 17+12 \sqrt{2},\) \(x\neq 17-12\sqrt{2},\) then

  \begin{equation*} \sum\limits_{k=0}^{n}x^{k}B_{-2k}^{3}=\frac{\Psi _{1}}{32 \left( x^{2}-34x+1\right) \left( x^{2}-39202x+1\right) } \,,\end{equation*} where

  \(\Psi _{1}=32x^{n+1}(x^{2}-34x+1)B_{-2n+2}^{3}+32x^{n+1}(x-39202)(x^{2}-34x+1)B_{-2n}^{3}+3x^{n+1}(x^{2}-34x+1)B_{-2n+2}\)

  \(-102x^{n+1}(x^{2}-39202x+1153)B_{-2n}+3x^{n+1}(x^{2}-39202x+1)B_{-2n-2}-6912x(x^{2}+68x+1) \,,\)

  and

  if \( \left( x^{2}-34x+1\right) \left( x^{2}-39202x+1\right) =0,\) i.e., \(x=19601+13860\sqrt{2}\) or \(x=19601-13860\sqrt{2}\) or \( x=17+12 \sqrt{2}\) or \(x=17-12\sqrt{2}\) then

  \begin{equation*} \sum\limits_{k=0}^{n}x^{k}B_{-2k}^{3}=\frac{\Psi _{2}}{128\left( x^{3}-29427x^{2}+666435x-9809\right) } \,,\end{equation*} where

  \(\Psi _{2}=32x^{n}(n(x^{2}-34x+1)+3x^{2}-68x+1)B_{-2n+2}^{3}+32x^{n}(n(x-39202)(x^{2}-34x+1)\)

  \(+2(2x^{3}-58854x^{2}+1332869x-19601))B_{-2n}^{3}+3x^{n}(n(x^{2}-34x+1)+3x^{2}-68x+1)B_{-2n+2}-102x^{n}(n(x^{2}-39202x+1153)\)

  \(+3x^{2}-78404x+1153)B_{-2n}+3x^{n}(n(x^{2}-39202x+1)+3x^{2}-78404x+1)B_{-2n-2}-6912(3x^{2}+136x+1). \)

• (f) \((m=-2,\) \(j=1)\)

  If \( \left( x^{2}-34x+1\right) \left( x^{2}-39202x+1\right) \neq 0,\) i.e., \(x\neq 19601+13860\sqrt{2},\) \(x\neq 19601-13860\sqrt{2},\) \(x\neq 17+12 \sqrt{2},\) \(x\neq 17-12\sqrt{2},\) then

  \begin{equation*} \sum\limits_{k=0}^{n}x^{k}B_{-2k+1}^{3}=\frac{\Psi _{1}}{32 \left( x^{2}-34x+1\right) \left( x^{2}-39202x+1\right) } \,,\end{equation*} where

  \(\Psi _{1}=32x^{n+1}(x^{2}-34x+1)B_{-2n+3}^{3}+32x^{n+1}(x-39202)(x^{2}-34x+1)B_{-2n+1}^{3}\)

  \(+3x^{n+1}(x^{2}-34x+1)\left( B_{0}^{2}+B_{1}^{2}-6B_{0}B_{1}\right) B_{-2n+3}-102x^{n+1}(x^{2}-39202x+1153)B_{-2n+1}\)

  \(+3x^{n+1}(x^{2}-39202x+1)B_{-2n-1}-32(42875x^{3}-1329231x^{2}+39237x-1) \,,\)

  and

  if \( \left( x^{2}-34x+1\right) \left( x^{2}-39202x+1\right) =0,\) i.e., \(x=19601+13860\sqrt{2}\) or \(x=19601-13860\sqrt{2}\) or \( x=17+12 \sqrt{2}\) or \(x=17-12\sqrt{2}\) then

  \begin{equation*} \sum\limits_{k=0}^{n}x^{k}B_{-2k+1}^{3}=\frac{\Psi _{2}}{128\left( x^{3}-29427x^{2}+666435x-9809\right) } \,,\end{equation*} where

  \(\Psi _{2}=32x^{n}(n(x^{2}-34x+1)+3x^{2}-68x+1)B_{-2n+3}^{3}+32x^{n}(n(x-39202)(x^{2}-34x+1)+4x^{3}-117708x^{2}+2665738x-39202)B_{-2n+1}^{3}\)

  \(+3x^{n}(n(x^{2}-34x+1)+3x^{2}-68x+1)B_{-2n+3} -102x^{n}(n(x^{2}-39202x+1153)+3x^{2}-78404x+1153)B_{-2n+1}+3x^{n}(n(x^{2}-39202x+1)\)

  \(+3x^{2}-78404x+1)B_{-2n-1}-96(42875x^{2}-886154x+13079). \)

Taking \(W_{n}=H_{n}\) with \(H_{0}=2,H_{1}=6\) in the last proposition, we have the following corollary which presents sum formulas of modified Lucas-balancing numbers;

Corollary 3. For \(n\geq 0,\) modified Lucas-balancing numbers have the following properties:

• (a) \((m=1,\) \(j=0)\)

  If \( \left( x^{2}-6x+1\right) \left( x^{2}-198x+1\right) \neq 0,\) i.e., \(x\neq 99+70\sqrt{2},\) \(x\neq 99-70\sqrt{2},\) \(x\neq 3+2\sqrt{2},\) \( x\neq 3-2\sqrt{2},\) then

  \begin{equation*} \sum\limits_{k=0}^{n}x^{k}H_{k}^{3}=\frac{\Psi _{1}}{\left( x^{2}-6x+1\right) \left( x^{2}-198x+1\right) } \,,\end{equation*} where

  \(\Psi _{1}=x^{n+1}(x-198)(x^{2}-6x+1)H_{n}^{3}+x^{n+1}(x^{2}-6x+1)H_{n-1}^{3}-3x^{n+1}(x^{2}-198x+1)H_{n+1}\)

  \(+18x^{n+1}(x^{2}-198x+33)H_{n}-3x^{n+1}(x^{2}-6x+1)H_{n-1}-8(27x^{3}-595x^{2}+177x-1) \,,\)

  and

  if \( \left( x^{2}-6x+1\right) \left( x^{2}-198x+1\right) =0,\) i.e., \(x=99+70\sqrt{2}\) or \(x=99-70\sqrt{2}\) or \(x=3+2\sqrt{2}\) or \(x=3-2 \sqrt{2}\) then

  \begin{equation*} \sum\limits_{k=0}^{n}x^{k}H_{k}^{3}=\frac{\Psi _{2}}{4\left( x^{3}-153x^{2}+595x-51\right) } \,,\end{equation*} where

  \(\Psi _{2}=x^{n}(n(x-198)(x^{2}-6x+1)+4x^{3}-612x^{2}+2378x-198)H_{n}^{3}+x^{n}(n(x^{2}-6x+1)+3x^{2}-12x+1)H_{n-1}^{3}-3x^{n}(n(x^{2}-198x+1)\)

  \(+3x^{2}-396x+1)H_{n+1}+18x^{n}(n(x^{2}-198x+33)+3x^{2}-396x+33)H_{n} -3x^{n}(n(x^{2}-6x+1)\)

  \(+3x^{2}-12x+1)H_{n-1}-8(81x^{2}-1190x+177).\)

• (b) \((m=2,\) \(j=0)\)

  If \( \left( x^{2}-34x+1\right) \left( x^{2}-39202x+1\right) \neq 0,\) i.e., \(x\neq 19601+13860\sqrt{2},\) \(x\neq 19601-13860\sqrt{2},\) \(x\neq 17+12 \sqrt{2},\) \(x\neq 17-12\sqrt{2},\) then

  \begin{equation*} \sum\limits_{k=0}^{n}x^{k}H_{2k}^{3}=\frac{\Psi _{1}}{\left( x^{2}-34x+1\right) \left( x^{2}-39202x+1\right) } \,,\end{equation*} where

  \(\Psi _{1}=x^{n+1}(x-39202)(x^{2}-34x+1)H_{2n}^{3}+x^{n+1}(x^{2}-34x+1)H_{2n-2}^{3}-3x^{n+1}(x^{2}-39202x+1)H_{2n+2}+102x^{n+1}(x^{2}-39202x+1153)H_{2n}\)

  \(-3x^{n+1}(x^{2}-34x+1)H_{2n-2}-8(4913x^{3}-666435x^{2}+34323x-1) \,,\)

  and

  if \( \left( x^{2}-34x+1\right) \left( x^{2}-39202x+1\right) =0,\) i.e., \(x=19601+13860\sqrt{2}\) or \(x=19601-13860\sqrt{2}\) or \( x=17+12 \sqrt{2}\) or \(x=17-12\sqrt{2}\) then

  \begin{equation*} \sum\limits_{k=0}^{n}x^{k}H_{2k}^{3}=\frac{\Psi _{2}}{4\left( x^{3}-29427x^{2}+666435x-9809\right) } \,,\end{equation*} where

  \(\Psi _{2}=x^{n}(n(x-39202)(x^{2}-34x+1)+2(2x^{3}-58854x^{2}+1332869x-19601))H_{2n}^{3}+x^{n}(n(x^{2}-34x+1)\)

  \(+3x^{2}-68x+1)H_{2n-2}^{3}-3x^{n}(n(x^{2}-39202x+1)+3x^{2}-78404x+1)H_{2n+2} +102x^{n}(n(x^{2}-39202x+1153)\)

  \(+3x^{2}-78404x+1153)H_{2n}-3x^{n}(n(x^{2}-34x+1)+3x^{2}-68x+1)H_{2n-2}-24(4913x^{2}-444290x+11441). \)

• (c) \((m=2,\) \(j=1)\)

  If \( \left( x^{2}-34x+1\right) \left( x^{2}-39202x+1\right) \neq 0,\) i.e., \(x\neq 19601+13860\sqrt{2},\) \(x\neq 19601-13860\sqrt{2},\) \(x\neq 17+12 \sqrt{2},\) \(x\neq 17-12\sqrt{2},\) then

  \begin{equation*} \sum\limits_{k=0}^{n}x^{k}H_{2k+1}^{3}=\frac{\Psi _{1}}{ \left( x^{2}-34x+1\right) \left( x^{2}-39202x+1\right) } \,,\end{equation*} where

  \(\Psi _{1}=x^{n+1}(x-39202)(x^{2}-34x+1)H_{2n+1}^{3}+x^{n+1}(x^{2}-34x+1)H_{2n-1}^{3}-3x^{n+1}(x^{2}-39202x+1)H_{2n+3}\)

  \(+102x^{n+1}\left( (x^{2}-39202x+1153)\right) H_{2n+1}-3x^{n+1}(x^{2}-34x+1) H_{2n-1}-216(x-1)(x^{2}-3298x+1) \,,\)

  and

  if \( \left( x^{2}-34x+1\right) \left( x^{2}-39202x+1\right) =0,\) i.e., \(x=19601+13860\sqrt{2}\) or \(x=19601-13860\sqrt{2}\) or \( x=17+12 \sqrt{2}\) or \(x=17-12\sqrt{2}\) then

  \begin{equation*} \sum\limits_{k=0}^{n}x^{k}H_{2k+1}^{3}=\frac{\Psi _{2}}{4\left( x^{3}-29427x^{2}+666435x-9809\right) } \,,\end{equation*} where

  \(\Psi _{2}=x^{n}(n(x-39202)(x^{2}-34x+1)+2(2x^{3}-58854x^{2}+1332869x-19601))H_{2n+1}^{3}+x^{n}(n(x^{2}-34x+1)+3x^{2}-68x+1)H_{2n-1}^{3}\)

  \(-3x^{n}(n(x^{2}-39\,202x+1)+3x^{2}-78404x+1)H_{2n+3}+102x^{n}(n(x^{2}-39202x+1153)+3x^{2}-78404x+1153)H_{2n+1}\)

  \(-3x^{n}(n(x^{2}-34x+1)+3x^{2}-68x+1)H_{2n-1}-216(3x^{2}-6598x+3299). \)

• (d) \((m=-1,\) \(j=0)\)

  If \( \left( x^{2}-6x+1\right) \left( x^{2}-198x+1\right) \neq 0,\) i.e., \(x\neq 99+70\sqrt{2},\) \(x\neq 99-70\sqrt{2},\) \(x\neq 3+2\sqrt{2},\) \( x\neq 3-2\sqrt{2},\) then

  \begin{equation*} \sum\limits_{k=0}^{n}x^{k}H_{-k}^{3}=\frac{\Psi _{1}}{ \left( x^{2}-6x+1\right) \left( x^{2}-198x+1\right) } \,,\end{equation*} where

  \(\Psi _{1}=x^{n+1}(x^{2}-6x+1)H_{-n+1}^{3}+x^{n+1}(x-198)(x^{2}-6x+1)H_{-n}^{3}-3x^{n+1}(x^{2}-6x+1)H_{-n+1}\)

  \(+18x^{n+1}(x^{2}-198x+33)H_{-n}-3x^{n+1}(x^{2}-198x+1) H_{-n-1}-8(27x^{3}-595x^{2}+177x-1) \,,\)

  and

  if \( \left( x^{2}-6x+1\right) \left( x^{2}-198x+1\right) =0,\) i.e., \(x=99+70\sqrt{2}\) or \(x=99-70\sqrt{2}\) or \(x=3+2\sqrt{2}\) or \(x=3-2 \sqrt{2}\) then

  \begin{equation*} \sum\limits_{k=0}^{n}x^{k}H_{-k}^{3}=\frac{\Psi _{2}}{4\left( x^{3}-153x^{2}+595x-51\right) } \,,\end{equation*} where

  \(\Psi _{2}=x^{n}(n(x^{2}-6x+1)+3x^{2}-12x+1)H_{-n+1}^{3}+x^{n}(n(x-198)(x^{2}-6x+1)+2(2x^{3}-306x^{2}+1189x-99))H_{-n}^{3}\)

  \(-3x^{n}(n(x^{2}-6x+1)+3x^{2}-12x+1)H_{-n+1}+18x^{n}(n(x^{2}-198x+33)+3x^{2}-396x+33)H_{-n} \)

  \(-3x^{n}(n(x^{2}-198x+1)+3x^{2}-396x+1)H_{-n-1}-8(81x^{2}-1190x+177).\)

• (e) \((m=-2,\) \(j=0)\)

  If \( \left( x^{2}-34x+1\right) \left( x^{2}-39202x+1\right) \neq 0,\) i.e., \(x\neq 19601+13860\sqrt{2},\) \(x\neq 19601-13860\sqrt{2},\) \(x\neq 17+12 \sqrt{2},\) \(x\neq 17-12\sqrt{2},\) then

  \begin{equation*} \sum\limits_{k=0}^{n}x^{k}H_{-2k}^{3}=\frac{\Psi _{1}}{ \left( x^{2}-34x+1\right) \left( x^{2}-39202x+1\right) } \,,\end{equation*} where

  \(\Psi _{1}=x^{n+1}(x^{2}-34x+1)H_{-2n+2}^{3}+x^{n+1}(x-39202)(x^{2}-34x+1)H_{-2n}^{3}-3x^{n+1}(x^{2}-34x+1)H_{-2n+2}+102x^{n+1}(x^{2}-39202x+1153)H_{-2n}\)

  \(-3x^{n+1}(x^{2}-39202x+1)H_{-2n-2}-8(4913x^{3}-666435x^{2}+34323x-1) \,,\)

  and

  if \( \left( x^{2}-34x+1\right) \left( x^{2}-39202x+1\right) =0,\) i.e., \(x=19601+13860\sqrt{2}\) or \(x=19601-13860\sqrt{2}\) or \( x=17+12 \sqrt{2}\) or \(x=17-12\sqrt{2}\) then

  \begin{equation*} \sum\limits_{k=0}^{n}x^{k}H_{-2k}^{3}=\frac{\Psi _{2}}{4\left( x^{3}-29427x^{2}+666435x-9809\right) } \,,\end{equation*} where

  \(\Psi _{2}=x^{n}(n(x^{2}-34x+1)+3x^{2}-68x+1)H_{-2n+2}^{3}+x^{n}(n(x-39202)(x^{2}-34x+1)+2(2x^{3}-58854x^{2}+1332869x-19601))H_{-2n}^{3}\)

  \(-3x^{n}(n(x^{2}-34x+1)+3x^{2}-68x+1)H_{-2n+2}+102x^{n}(n(x^{2}-39202x+1153)+3x^{2}-78404x+1153)H_{-2n}\)

  \(-3x^{n}(n(x^{2}-39202x+1)+3x^{2}-78404x+1)H_{-2n-2}-24(4913x^{2}-444290x+11441). \)

• (f) \((m=-2,\) \(j=1)\)

  If \( \left( x^{2}-34x+1\right) \left( x^{2}-39202x+1\right) \neq 0,\) i.e., \(x\neq 19601+13860\sqrt{2},\) \(x\neq 19601-13860\sqrt{2},\) \(x\neq 17+12 \sqrt{2},\) \(x\neq 17-12\sqrt{2},\) then

  \begin{equation*} \sum\limits_{k=0}^{n}x^{k}H_{-2k+1}^{3}=\frac{\Psi _{1}}{\left( x^{2}-34x+1\right) \left( x^{2}-39202x+1\right) } \,,\end{equation*} where

  \(\Psi _{1}=x^{n+1}(x^{2}-34x+1)H_{-2n+3}^{3}+x^{n+1}(x-39202)(x^{2}-34x+1)H_{-2n+1}^{3}\)

  \(-3x^{n+1}(x^{2}-34x+1)H_{-2n+3}+102x^{n+1}(x^{2}-39202x+1153)H_{-2n+1}\)

  \(-3x^{n+1}(x^{2}-39202x+1)H_{-2n-1}-216(35937x^{3}-1329571x^{2}+39235x-1) \,,\)

  and

  if \( \left( x^{2}-34x+1\right) \left( x^{2}-39202x+1\right) =0,\) i.e., \(x=19601+13860\sqrt{2}\) or \(x=19601-13860\sqrt{2}\) or \( x=17+12 \sqrt{2}\) or \(x=17-12\sqrt{2}\) then

  \begin{equation*} \sum\limits_{k=0}^{n}x^{k}H_{-2k+1}^{3}=\frac{\Psi _{2}}{4\left( x^{3}-29427x^{2}+666435x-9809\right) } \,,\end{equation*} where

  \(\Psi _{2}=x^{n}(n(x^{2}-34x+1)+3x^{2}-68x+1)H_{-2n+3}^{3}+x^{n}(n(x-39202)(x^{2}-34x+1)+4x^{3}-117708x^{2}+2665738x-39202)H_{-2n+1}^{3}\)

  \(-3x^{n}(n(x^{2}-34x+1)+3x^{2}-68x+1)H_{-2n+3} +102x^{n}(n(x^{2}-39202x+1153)+3x^{2}-78404x+1153)H_{-2n+1}\)

  \(-3x^{n}(n(x^{2}-39202x+1)+3x^{2}-78404x+1)H_{-2n-1}-216(107811x^{2}-2659142x+39235). \)

From the above proposition, we have the following corollary which gives sum formulas of Lucas-balancing numbers (take \(W_{n}=C_{n}\) with \( C_{0}=1,C_{1}=3 \));

Corollary 4. For \(n\geq 0,\) Lucas-balancing numbers have the following properties:

• (a) \((m=1,\) \(j=0)\)

  If \( \left( x^{2}-6x+1\right) \left( x^{2}-198x+1\right) \neq 0,\) i.e., \(x\neq 99+70\sqrt{2},\) \(x\neq 99-70\sqrt{2},\) \(x\neq 3+2\sqrt{2},\) \( x\neq 3-2\sqrt{2},\) then

  \begin{equation*} \sum\limits_{k=0}^{n}x^{k}C_{k}^{3}=\frac{\Psi _{1}}{4\left( x^{2}-6x+1\right) \left( x^{2}-198x+1\right) } \,,\end{equation*} where

  \(\Psi _{1}=4x^{n+1}(x-198)(x^{2}-6x+1)C_{n}^{3}+4x^{n+1}(x^{2}-6x+1)C_{n-1}^{3}-3x^{n+1}(x^{2}-198x+1)C_{n+1}\)

  \(+18x^{n+1}(x^{2}-198x+33)C_{n}-3x^{n+1}(x^{2}-6x+1)C_{n-1}-4(27x^{3}-595x^{2}+177x-1) \,,\)

  and

  if \( \left( x^{2}-6x+1\right) \left( x^{2}-198x+1\right) =0,\) i.e., \(x=99+70\sqrt{2}\) or \(x=99-70\sqrt{2}\) or \(x=3+2\sqrt{2}\) or \(x=3-2 \sqrt{2}\) then

  \begin{equation*} \sum\limits_{k=0}^{n}x^{k}C_{k}^{3}=\frac{\Psi _{2}}{16\left( x^{3}-153x^{2}+595x-51\right) } \,,\end{equation*} where

  \(\Psi _{2}=4x^{n}(n(x-198)(x^{2}-6x+1)+4x^{3}-612x^{2}+2378x-198)C_{n}^{3}+4x^{n}(n(x^{2}-6x+1)+3x^{2}-12x+1)C_{n-1}^{3}\)

  \(-3x^{n}(n(x^{2}-198x+1)+3x^{2}-396x+1)C_{n+1}+18x^{n}(n(x^{2}-198x+33)+3x^{2}-396x+33)C_{n}-3x^{n}(n(x^{2}-6x+1)\)

  \(+3x^{2}-12x+1)C_{n-1}-4(81x^{2}-1190x+177). \)

• (b) \((m=2,\) \(j=0)\)

  If \( \left( x^{2}-34x+1\right) \left( x^{2}-39202x+1\right) \neq 0,\) i.e., \(x\neq 19601+13860\sqrt{2},\) \(x\neq 19601-13860\sqrt{2},\) \(x\neq 17+12 \sqrt{2},\) \(x\neq 17-12\sqrt{2},\) then

  \begin{equation*} \sum\limits_{k=0}^{n}x^{k}C_{2k}^{3}=\frac{\Psi _{1}}{4\left( x^{2}-34x+1\right) \left( x^{2}-39202x+1\right) } \,,\end{equation*} where

  \(\Psi _{1}=4x^{n+1}(x-39202)(x^{2}-34x+1)C_{2n}^{3}+4x^{n+1}(x^{2}-34x+1)C_{2n-2}^{3}-3x^{n+1}(x^{2}-39202x+1)C_{2n+2}\)

  \(+102x^{n+1}(x^{2}-39202x+1153)C_{2n}-3x^{n+1}(x^{2}-34x+1)C_{2n-2}-4(4913x^{3}-666\,435x^{2}+34\,323x-1) \,,\)

  and

  if \( \left( x^{2}-34x+1\right) \left( x^{2}-39202x+1\right) =0,\) i.e., \(x=19601+13860\sqrt{2}\) or \(x=19601-13860\sqrt{2}\) or \( x=17+12 \sqrt{2}\) or \(x=17-12\sqrt{2}\) then

  \begin{equation*} \sum\limits_{k=0}^{n}x^{k}C_{2k}^{3}=\frac{\Psi _{2}}{16\left( x^{3}-29427x^{2}+666435x-9809\right) } \,,\end{equation*} where

  \(\Psi _{2}=4x^{n}(n(x-39202)(x^{2}-34x+1)+2(2x^{3}-58854x^{2}+1332869x-19601))C_{2n}^{3}+4x^{n}(n(x^{2}-34x+1)\)

  \(+3x^{2}-68x+1)C_{2n-2}^{3}-3x^{n}(n(x^{2}-39202x+1)+3x^{2}-78404x+1)C_{2n+2} \)

  \(+102x^{n}(n(x^{2}-39202x+1153)+3x^{2}-78404x+1153)C_{2n}-3x^{n}(n(x^{2}-34x+1)+3x^{2}-68x+1)C_{2n-2}-12(4913x^{2}-444290x+11441). \)

• (c) \((m=2,\) \(j=1)\)

  If \( \left( x^{2}-34x+1\right) \left( x^{2}-39202x+1\right) \neq 0,\) i.e., \(x\neq 19601+13860\sqrt{2},\) \(x\neq 19601-13860\sqrt{2},\) \(x\neq 17+12 \sqrt{2},\) \(x\neq 17-12\sqrt{2},\) then

  \begin{equation*} \sum\limits_{k=0}^{n}x^{k}C_{2k+1}^{3}=\frac{\Psi _{1}}{4 \left( x^{2}-34x+1\right) \left( x^{2}-39202x+1\right) } \,,\end{equation*} where

  \(\Psi _{1}=4x^{n+1}(x-39202)(x^{2}-34x+1)C_{2n+1}^{3}+4x^{n+1}(x^{2}-34x+1)C_{2n-1}^{3}-3x^{n+1}(x^{2}-39202x+1)C_{2n+3}\)

  \(+102x^{n+1}\left( (x^{2}-39202x+1153)\right) C_{2n+1}-3x^{n+1}(x^{2}-34x+1)C_{2n-1}-108(x-1)(x^{2}-3298x+1) \,,\)

  and

  if \( \left( x^{2}-34x+1\right) \left( x^{2}-39202x+1\right) =0,\) i.e., \(x=19601+13860\sqrt{2}\) or \(x=19601-13860\sqrt{2}\) or \( x=17+12 \sqrt{2}\) or \(x=17-12\sqrt{2}\) then

  \begin{equation*} \sum\limits_{k=0}^{n}x^{k}C_{2k+1}^{3}=\frac{\Psi _{2}}{16\left( x^{3}-29427x^{2}+666435x-9809\right) } \,,\end{equation*} where

  \(\Psi _{2}=4x^{n}(n(x-39202)(x^{2}-34x+1)+2(2x^{3}-58854x^{2}+1332869x-19601))C_{2n+1}^{3}+4x^{n}(n(x^{2}-34x+1)\)

  \(+3x^{2}-68x+1)C_{2n-1}^{3}-3x^{n}(n(x^{2}-39\,202x+1)+3x^{2}-78404x+1)C_{2n+3}+102x^{n}(n(x^{2}-39202x+1153)\)

  \(+3x^{2}-78404x+1153)C_{2n+1}-3x^{n}(n(x^{2}-34x+1)+3x^{2}-68x+1)C_{2n-1}-108(3x^{2}-6598x+3299). \)

• (d) \((m=-1,\) \(j=0)\)

  If \( \left( x^{2}-6x+1\right) \left( x^{2}-198x+1\right) \neq 0,\) i.e., \(x\neq 99+70\sqrt{2},\) \(x\neq 99-70\sqrt{2},\) \(x\neq 3+2\sqrt{2},\) \( x\neq 3-2\sqrt{2},\) then

  \begin{equation*} \sum\limits_{k=0}^{n}x^{k}C_{-k}^{3}=\frac{\Psi _{1}}{4 \left( x^{2}-6x+1\right) \left( x^{2}-198x+1\right) } \,,\end{equation*} where

  \(\Psi _{1}=4x^{n+1}(x^{2}-6x+1)C_{-n+1}^{3}+4x^{n+1}(x-198)(x^{2}-6x+1)C_{-n}^{3}-3x^{n+1}(x^{2}-6x+1)C_{-n+1}\)

  \(+18x^{n+1}(x^{2}-198x+33)C_{-n}-3x^{n+1}(x^{2}-198x+1)C_{-n-1}-4(27x^{3}-595x^{2}+177x-1) \,,\)

  and

  if \( \left( x^{2}-6x+1\right) \left( x^{2}-198x+1\right) =0,\) i.e., \(x=99+70\sqrt{2}\) or \(x=99-70\sqrt{2}\) or \(x=3+2\sqrt{2}\) or \(x=3-2 \sqrt{2}\) then

  \begin{equation*} \sum\limits_{k=0}^{n}x^{k}C_{-k}^{3}=\frac{\Psi _{2}}{16\left( x^{3}-153x^{2}+595x-51\right) } \,,\end{equation*} where

  \(\Psi _{2}=4x^{n}(n(x^{2}-6x+1)+3x^{2}-12x+1)C_{-n+1}^{3}+4x^{n}(n(x-198)(x^{2}-6x+1)+2(2x^{3}-306x^{2}+1189x-99))C_{-n}^{3}\)

  \(-3x^{n}(n(x^{2}-6x+1)+3x^{2}-12x+1)C_{-n+1}+18x^{n}(n(x^{2}-198x+33)+3x^{2}-396x+33)C_{-n}\)

  \(-3x^{n}(n(x^{2}-198x+1)+3x^{2}-396x+1)C_{-n-1}-4(81x^{2}-1190x+177). \)

• (e) \((m=-2,\) \(j=0)\)

  If \( \left( x^{2}-34x+1\right) \left( x^{2}-39202x+1\right) \neq 0,\) i.e., \(x\neq 19601+13860\sqrt{2},\) \(x\neq 19601-13860\sqrt{2},\) \(x\neq 17+12 \sqrt{2},\) \(x\neq 17-12\sqrt{2},\) then

  \begin{equation*} \sum\limits_{k=0}^{n}x^{k}C_{-2k}^{3}=\frac{\Psi _{1}}{4 \left( x^{2}-34x+1\right) \left( x^{2}-39202x+1\right) } \,,\end{equation*} where

  \(\Psi _{1}=4x^{n+1}(x^{2}-34x+1)C_{-2n+2}^{3}+4x^{n+1}(x-39202)(x^{2}-34x+1)C_{-2n}^{3}-3x^{n+1}(x^{2}-34x+1)C_{-2n+2}\)

  \(+102x^{n+1}(x^{2}-39202x+1153) C_{-2n}-3x^{n+1}(x^{2}-39202x+1)C_{-2n-2}-4(4913x^{3}-666435x^{2}+34323x-1) \,,\)

  and

  if \( \left( x^{2}-34x+1\right) \left( x^{2}-39202x+1\right) =0,\) i.e., \(x=19601+13860\sqrt{2}\) or \(x=19601-13860\sqrt{2}\) or \( x=17+12 \sqrt{2}\) or \(x=17-12\sqrt{2}\) then

  \begin{equation*} \sum\limits_{k=0}^{n}x^{k}C_{-2k}^{3}=\frac{\Psi _{2}}{16\left( x^{3}-29427x^{2}+666435x-9809\right) } \,,\end{equation*} where

  \(\Psi _{2}=4x^{n}(n(x^{2}-34x+1)+3x^{2}-68x+1)C_{-2n+2}^{3}+4x^{n}(n(x-39202)(x^{2}-34x+1)\)

  \(+2(2x^{3}-58854x^{2}+1332869x-19601))C_{-2n}^{3}-3x^{n}(n(x^{2}-34x+1)+3x^{2}-68x+1)C_{-2n+2}+102x^{n}(n(x^{2}-39202x+1153)\)

  \(+3x^{2}-78404x+1153)C_{-2n}-3x^{n}(n(x^{2}-39202x+1)+3x^{2}-78404x+1)C_{-2n-2}-12(4913x^{2}-444290x+11441). \)

• (f) \((m=-2,\) \(j=1)\)

  If \( \left( x^{2}-34x+1\right) \left( x^{2}-39202x+1\right) \neq 0,\) i.e., \(x\neq 19601+13860\sqrt{2},\) \(x\neq 19601-13860\sqrt{2},\) \(x\neq 17+12 \sqrt{2},\) \(x\neq 17-12\sqrt{2},\) then

  \begin{equation*} \sum\limits_{k=0}^{n}x^{k}C_{-2k+1}^{3}=\frac{\Psi _{1}}{4\left( x^{2}-34x+1\right) \left( x^{2}-39202x+1\right) } \,,\end{equation*} where

  \(\Psi _{1}=4x^{n+1}(x^{2}-34x+1)C_{-2n+3}^{3}+4x^{n+1}(x-39202)(x^{2}-34x+1)C_{-2n+1}^{3}-3x^{n+1}(x^{2}-34x+1)C_{-2n+3}+102x^{n+1}(x^{2}-39202x+1153)C_{-2n+1}\)

  \(-3x^{n+1}(x^{2}-39202x+1)C_{-2n-1}-108(35937x^{3}-1329571x^{2}+39235x-1) \,,\)

  and

  if \( \left( x^{2}-34x+1\right) \left( x^{2}-39202x+1\right) =0,\) i.e., \(x=19601+13860\sqrt{2}\) or \(x=19601-13860\sqrt{2}\) or \( x=17+12 \sqrt{2}\) or \(x=17-12\sqrt{2}\) then

  \begin{equation*} \sum\limits_{k=0}^{n}x^{k}C_{-2k+1}^{3}=\frac{\Psi _{2}}{16\left( x^{3}-29427x^{2}+666435x-9809\right) } \,,\end{equation*} where

  \(\Psi _{2}=4x^{n}(n(x^{2}-34x+1)+3x^{2}-68x+1)C_{-2n+3}^{3}+4x^{n}(n(x-39202)(x^{2}-34x+1)+4x^{3}-117708x^{2}+2665738x-39202)C_{-2n+1}^{3}\)

  \(-3x^{n}(n(x^{2}-34x+1)+3x^{2}-68x+1)C_{-2n+3}+102x^{n}(n(x^{2}-39202x+1153)+3x^{2}-78404x+1153)C_{-2n+1}-3x^{n}(n(x^{2}-39202x+1)\)

  \(+3x^{2}-78404x+1)C_{-2n-1}-108(107811x^{2}-2659142x+39235). \)

Taking \(x=1\) in the last two corollaries we get the following corollary;

Corollary 5. For \(n\geq 0,\) balancing\ numbers, modified Lucas-balancing numbers and Lucas-balancing\ numbers have the following properties:

1. • (a) \(\sum\limits_{k=0}^{n}B_{k}^{3}=\frac{1}{6272} (6304B_{n}^{3}-32B_{n-1}^{3}-147B_{n+1}+738B_{n}-3B_{n-1}+112).\)
   • (b) \(\sum\limits_{k=0}^{n}B_{2k}^{3}=\frac{1}{1254400} (1254432B_{2n}^{3}-32B_{2n-2}^{3}-3675B_{2n+2}+121278B_{2n}-3B_{2n-2}+15120). \)
   • (c) \(\sum\limits_{k=0}^{n}B_{2k+1}^{3}= \frac{1}{1254400} (1254432B_{2n+1}^{3}-32B_{2n-1}^{3}-3675B_{2n+3}+ 121278B_{2n+1}-3B_{2n-1}+7280).\)
   • (d) \(\sum\limits_{k=0}^{n}B_{-k}^{3}=\frac{1}{6272} (6304B_{-n}^{3}-32B_{-n+1}^{3}-3B_{-n+1}+738B_{-n}-147B_{-n-1}-112).\)
   • (e) \(\sum\limits_{k=0}^{n}B_{-2k}^{3}=\frac{1}{1254400} (-32B_{-2n+2}^{3}+1254432B_{-2n}^{3}-3B_{-2n+2}+121278B_{-2n}-3675B_{-2n-2}-15120). \)
   • (f) \(\sum\limits_{k=0}^{n}B_{-2k+1}^{3}=\frac{1}{1254400} (-32B_{-2n+3}^{3}+1254432B_{-2n+1}^{3}- 3B_{-2n+3}+121278B_{-2n+1}-3675B_{-2n-1}+1247120).\)
2. • (a) \(\sum\limits_{k=0}^{n}H_{k}^{3}=\frac{1}{196} (197H_{n}^{3}-H_{n-1}^{3}+147H_{n+1}-738H_{n}+3H_{n-1}+784).\)
   • (b) \(\sum\limits_{k=0}^{n}H_{2k}^{3}=\frac{1}{39200} (39201H_{2n}^{3}-H_{2n-2}^{3}+3675H_{2n+2}-121278H_{2n}+3H_{2n-2}+156800).\)
   • (c) \(\sum\limits_{k=0}^{n}H_{2k+1}^{3}=\frac{1}{39200} (39201H_{2n+1}^{3}-H_{2n-1}^{3}+3675H_{2n+3}-121278 H_{2n+1}+3H_{2n-1}).\)
   • (d) \(\sum\limits_{k=0}^{n}H_{-k}^{3}=\frac{1}{196} (197H_{-n}^{3}-H_{-n+1}^{3}+3H_{-n+1}-738H_{-n}+147H_{-n-1}+784).\)
   • (e) \(\sum\limits_{k=0}^{n}H_{-2k}^{3}=\frac{1}{39200} (-H_{-2n+2}^{3}+39201H_{-2n}^{3}+3H_{-2n+2}-121278H_{-2n}+3675H_{-2n-2}+156800). \)
   • (f) \(\sum\limits_{k=0}^{n}H_{-2k+1}^{3}=\frac{1}{39200} (-H_{-2n+3}^{3}+39201H_{-2n+1}^{3}+3H_{-2n+3}-121278H_{-2n+1}+3675H_{-2n-1}+8467200). \)
3. • (a) \(\sum\limits_{k=0}^{n}C_{k}^{3}=\frac{1}{784} (788C_{n}^{3}-4C_{n-1}^{3}+147C_{n+1}-738C_{n}+3C_{n-1}+392).\)
   • (b) \(\sum\limits_{k=0}^{n}C_{2k}^{3}=\frac{1}{156800} (156804C_{2n}^{3}-4C_{2n-2}^{3}+3675C_{2n+2}-121278C_{2n}+3C_{2n-2}+78400).\)
   • (c) \(\sum\limits_{k=0}^{n}C_{2k+1}^{3}=\frac{1}{156800} (156804C_{2n+1}^{3}-4C_{2n-1}^{3}+3675C_{2n+3}-121278 C_{2n+1}+3C_{2n-1}).\)
   • (d \(\sum\limits_{k=0}^{n}C_{-k}^{3}=\frac{1}{784} (-4C_{-n+1}^{3}+788C_{-n}^{3}+3C_{-n+1}-738C_{-n}+147C_{-n-1}+392).\)
   • (e) \(\sum\limits_{k=0}^{n}C_{-2k}^{3}=\frac{1}{156800} (-4C_{-2n+2}^{3}+156804C_{-2n}^{3}+3C_{-2n+2}-121278C_{-2n}+3675C_{-2n-2}+78400). \)
   • (f) \(\sum\limits_{k=0}^{n}C_{-2k+1}^{3}=\frac{1}{156800} (-4C_{-2n+3}^{3}+156804C_{-2n+1}^{3}+3C_{-2n+3}-121278C_{-2n+1}+3675C_{-2n-1}+4233600). \)

3. Conclusions

Recently, there have been so many studies of the sequences of numbers in the literature and the sequences of numbers were widely used in many research areas, such as architecture, nature, art, physics and engineering. In this work, sum identities were proved. The method used in this paper can be used for the other linear recurrence sequences, too. We have written sum identities in terms of the generalized balancing sequence, and then we have presented the formulas as special cases the corresponding identity for the balancing, modified Lucas-balancing and Lucas-balancing numbers. All the listed identities in the corollaries may be proved by induction, but that method of proof gives no clue about their discovery. We give the proofs to indicate how these identities, in general, were discovered.

Author Contributions:

All authors contributed equally to this work. They all read and approved the last version of the manuscript.

Conflicts of Interest:

The authors declare no conflict of interest.

Data Availability:

No data is required for this research.

Funding Information:

No funding is available for this research.