On the two-variable generalized Laguerre polynomials

1. Introduction

The two variable Laguerre polynomials (2VLP) \(L_n(x,y)\) are defined by the series (see [1, 2, 3]) as follows: Equation (1) and specified by the following generating functions: Equation (2) Equation (3) where \(C_0(x)\) denotes the \(0^{th}\) order Tricomi function. The \(n^{th}\) order Tricomi function \(C_n(x)\) is defined as [4]: Equation (4) Also, the (2VLP) \(L_n(x,y)\) satisfy the following properties: Equation (5) where \(L_n(x)\) are the ordinary Laguerre polynomials [5] Equation (6) The two variable associated Laguerre polynomials (2VALP) \(L^{(\alpha)}_n(x,y)\) are defined by the series (see [1, 6]) as follows: Equation (7) and specified by the following generating functions: Equation (8) Equation (9) For \(\alpha=0\), equations (7), (8) and (9) reduces respectively to equations (1), (2) and (3). Also, the (2VALP) \(L^{(\alpha)}_n(x,y)\) satisfy the following properties: Equation (10) where \(L^{(\alpha)}_n(x)\) are the generalized Laguerre polynomials of one variable [5]. Equation (11) Further, the Laguerre polynomials \(L^{(\alpha)}_n(x)\) satisfy the following generating function [5]: Equation (12) The Jacobi polynomials \(P^{(\alpha,\beta)}_n(x)\) [5] are define as: Equation (13) and specified by the following generating function: Equation (14) When \(\alpha=\beta=0\) the polynomials (13) become the Legendre polynomials \(P_n(x)\) [5] Equation (15) Ragab [7] defined the Laguerre polynomials of two variables \(L^{(\alpha,\beta)}(x,y)\) as follows: Equation (16) Chatterjea [8] obtained the following generating function for Ragab polynomials \(L^{(\alpha,\beta)}_n(x,y)\): Equation (17) The aim of the present paper is to introduce the two variable generalized Laguerre polynomials (2VGLP) \({}_GL_n^{(\alpha,\beta)}(x,y)\) as a generalization of the above mentioned Laguerre polynomials \(L_n(x,y)\) and \(L_n^{(\alpha)}(x,y)\). Further, we discuss several properties of these polynomials such as generating functions, relationships with other polynomials, summation formulae and expansions of polynomials.

2. The two variable generalized Laguerre polynomials (2VGLP) \({}_GL^{(\alpha,\beta)}_n(x,y)\)

In this section, we first define the two variable generalized Laguerre polynomials (2VGLP) \({}_GL^{(\alpha,\beta)}_n(x,y)\). We present some generating functions for these polynomials.

Definition 1. The two variables generalized Laguerre polynomials (2VGLP) \({}_GL^{(\alpha,\beta)}_n(x,y)\) are defined as: Equation (18)

Remark 1.

1. For \(\alpha=\beta=0\) the (2VGLP) \({}_GL^{(\alpha,\beta)}_n(x,y)\) in (18) reduces to the (2VLP) \(L(x,y)\).
2. For \(\alpha=0 ~~ or ~~\beta=0\) the (2VGLP) \({}_GL^{(\alpha,\beta)}_n(x,y)\) in (18) reduces to the (2VALP) \(L^{(\alpha)}_n(x,y)\).

Theorem 2. For the two variables generalized Laguerre polynomials (2VGLP) \({}_GL^{(\alpha,\beta)}_n(x,y)\). The following generating functions holds true: Equation (19)

Equation (20) Equation (21)

Proof. (19)
Denoting the left hand side of (19) by \(L\) and using the definition (18}), we get \begin{align*} L &= \sum_{n=0}^{\infty}\sum_{r=0}^{n}\frac{(1+\alpha)_n(1+\beta)_n(-1)^r~y^{n-r}x^rt^n}{(1+\alpha+\beta)_r~n!(n-r)!~r!} \\ &=\sum_{n=0}^{\infty}\sum_{r=0}^{\infty}\frac{(1+\alpha)_{n+r}(1+\beta)_{n+r}(-1)^r~y^nx^rt^{n+r}}{(1+\alpha+\beta)_r~(n+r)!n!~r!} \\ &=\sum_{r=0}^{\infty}\frac{(1+\alpha)_r(1+\beta)_r(-xt)^r}{(1+\alpha+\beta)_r(r!)^2}{}_2F_1[1+\alpha+r,1+\beta+r;1+r;yt]. \end{align*} This completes the proof of (19).

Proof. (20)
Denoting the left hand side of (20) by \(L\) and using the definition (18), we get \begin{align*} L &= \sum_{n=0}^{\infty}\sum_{r=0}^{n}\frac{(c)_n(-1)^r~y^{n-r}x^rt^n}{(1+\alpha+\beta)_r~(n-r)!~r!} \\ &=\sum_{n=0}^{\infty}\sum_{r=0}^{\infty}\frac{(c)_{n+r}(-1)^r~y^nx^rt^{n+r}}{(1+\alpha+\beta)_r~n!~r!} \\ &=\sum_{r=0}^{\infty}\frac{(c)_r(-xt)^r}{(1+\alpha+\beta)_r~r!}\sum_{n=0}^{\infty}\frac{(c+r)_n(yt)^n}{n!}\\ &=\sum_{r=0}^{\infty}\frac{(c)_r(-xt)^r}{(1+\alpha+\beta)_r~r!}(1-yt)^{-c-r} \\ &=(1-yt)^{-c}{}_1F_1 \left[ \begin{array}{cc} c&;\\ 1+\alpha+\beta&; \end{array} \begin{array}{c} \dfrac{-xt}{1-yt} \end{array} \right]. \end{align*} This completes the proof of (20). Similarly, (21) can be proved.

Special cases of (19),(20) and (21)

1. For \(\alpha=\beta=0\) and \(\beta=0\), (19) reduces respectively to (2) and (8).
2. For \(\alpha=\beta=0\) and \(\beta=0\), (21) reduces respectively to (3) and (9).
3. For \(\alpha=\beta=0\) and \(\beta=0\), (20) reduces respectively to the following well-known generating functions: Equation (22)
   \begin{equation}\label{eq22} \sum_{n=0}^{\infty}\frac{(c)_nL_n(x,y)t^n}{n!} =(1-yt)^{-c}{}_1F_1 \left[c;1;\frac{-xt}{1-yt} \right] \end{equation}

   (22)
   and Equation (23)
   \begin{equation}\label{eq23} \sum_{n=0}^{\infty}\frac{(c)_nL^{(\alpha)}_n(x,y)t^n}{(1+\alpha)_n} =(1-yt)^{-c}{}_1F_1 \left[c;1+\alpha;\frac{-xt}{1-yt} \right]. \end{equation}

   (23)
4. Taking \(c=\beta+1\) and \(c=\alpha+1\) in equation (20) and using Kummer's first theorem [4] Equation (24)
   \begin{equation}\label{eq24} {}_1F_1[a;c;z]=e^z{}_1F_1[c-a;c;-z], \end{equation}

   (24)
   we get respectively the following generating functions: Equation (25)
   \begin{equation}\label{eq25} \sum_{n=0}^{\infty}\frac{n!{}_GL^{(\alpha,\beta)}_n(x,y)t^n}{(1+\alpha)_n} =(1-yt)^{-\beta-1}\exp\left(\frac{-xt}{1-yt}\right) {}_1F_1 \left [ \begin{array}{cc} \alpha&;\\ 1+\alpha+\beta&; \end{array} \begin{array}{c} \dfrac{xt}{1-yt} \end{array} \right] \end{equation}

   (25)
   and Equation (26)
   \begin{equation}\label{eq26} \sum_{n=0}^{\infty}\frac{n!{}_GL^{(\alpha,\beta)}_n(x,y)t^n}{(1+\beta)_n} =(1-yt)^{-\alpha-1}\exp\left(\frac{-xt}{1-yt}\right) {}_1F_1 \left [ \begin{array}{cc} \beta&;\\ 1+\alpha+\beta&; \end{array} \begin{array}{c} \dfrac{xt}{1-yt} \end{array} \right]. \end{equation}

   (26)
5. Taking \(c=\alpha+\beta+1\) in equation (20) and using (8), we have Equation (27)
   \begin{equation}\label{eq27} {}_GL^{(\alpha,\beta)}_n(x,y)=\frac{(1+\alpha)_n(1+\beta)_n}{n!(1+\alpha+\beta)_n}L^{(\alpha+\beta)}_n(x,y). \end{equation}

   (27)
6. Replacing \(x\) by \(xy\) in equation (21) and using (12), we have Equation (28)
   \begin{equation}\label{eq28} {}_GL^{(\alpha,\beta)}_n(xy,y)=y^n\frac{(1+\alpha)_n(1+\beta)_n}{n!(1+\alpha+\beta)_n}L^{(\alpha+\beta)}_n(x). \end{equation}

   (28)

3. Summation formulae

Theorem 3. The following summation formulae for the (2VGLP) \({}_GL^{(\alpha,\beta)}_n(x,y)\) holds true: Equation (29)

Equation (30) Equation (31) Equation (32)

Proof. (29)
From (25), we have Equation (33)

Now, using (8) and (20), we have Equation (34) Equating the coefficient of \(t^n\) on both sides of (34), we get the desired result (29) .
Similarly, we can obtain (30) by using (8) and (23) in (33).

Proof. (31)
From (26), we have Equation (35)

Now, using (8) and (20), we have Equation (36) Equating the coefficient of \(t^n\) on both sides of (36), we get the desired result (31).
Similarly, we can obtain (32) by using (8) and (23) in (35).

Remark 2.

1. For \(\alpha=0\) or \(\beta=0\), the results (29), (30), (31) and (32) reduces to summation formulae for the (2VALP) \(L^{(\alpha)}_n(x,y)\).
2. For \(\alpha=\beta=0\), the results (29), (30), (31) and (32) reduces to summation formulae for the (2VLP) \(L_n(x,y)\).

4. Expansions of polynomials

Theorem 4. The following expansions of Jacobi polynomials \(P^{(\alpha,\beta)}_n(x)\) in terms of the (2VGLP) \({}_GL^{(\alpha,\beta)}_n(x,y)\) holds true: Equation (37)

Equation (38) Equation (39)

Proof. (37)
Taking the generating function (14), replacing \(\alpha\) and \(\beta\) by \(\alpha+\lambda\) and \(\beta+\mu\) respectively and using (21), we have \begin{eqnarray*} \sum_{n=0}^{\infty}\frac{P^{(\alpha+\lambda,\beta+\mu)}_n(x)t^n}{(\alpha+\lambda+1)_n(\beta+\mu+1)_n}&=&\left(\sum_{n=0}^{\infty}\frac{n!{}_GL^{(\alpha,\lambda)}_n(\frac{1}{2}(1-x),y)t^n}{(1+\alpha)_n(1+\lambda)_n}\right)\left(\sum_{r=0}^{\infty}\frac{r!{}_GL^{(\beta,\mu)}_r(-\frac{1}{2}(1+x),-y)t^r}{(1+\beta)_r(1+\mu)_r}\right) \\ &=&\sum_{n=0}^{\infty}\sum_{r=0}^{n}\frac{(n-r)!r!{}_GL^{(\alpha,\lambda)}_{n-r}(\frac{1}{2}(1-x),y){}_GL^{(\beta,\mu)}_r(-\frac{1}{2}(1+x),-y)t^n}{(1+\alpha)_{n-r}(1+\lambda)_{n-r}(1+\beta)_r(1+\mu)_r}. \end{eqnarray*} Now, equating the coefficient of \(t^n\) from both sides, we get the desired result (37).
Similarly, we can prove (38) and (39).

Remark 3. For \(\lambda=\mu=0\), the results (37), (38) and (39) reduces to the expansions of Jacobi polynomials \(P^{(\alpha,\beta)}_n(x)\) in terms of the (2VALP) \(L^{(\alpha)}_n(x,y)\). \end{remark}

Remark 4. For \(\alpha=\beta=\lambda=\mu=0\), the results (37), (38) and (39) reduces to the following summation formulae for the classical Legendre polynomials \(P_n(x)\) in terms of the (2VLP) \(L_n(x,y)\): Equation (40)

Equation (41) Equation (42)

Remark 5. The results (40) and (41) are known results of Khan and Al-Gonah [9].

Theorem 5. The following expansions of Ragab polynomials \(L^{(\alpha,\beta)}_n(x,y)\) in terms of the (2VGLP) \({}_GL^{(\alpha\beta)}_n(x,y)\) holds true: Equation (43)

Equation (44) Equation (45) Equation (46)

Proof. (43)
Taking the generating function (17), replacing \(\alpha\) and \(\beta\) by \(\alpha+\lambda\) and \(\beta+\mu\) respectively and using (21), we have \begin{align*} \sum_{n=0}^{\infty}\frac{n!L^{(\alpha+\lambda,\beta+\mu)}_n(x,y)t^n}{(\alpha+\lambda+1)_n(\beta+\mu+1)_n}=&\left(\sum_{n=0}^{\infty}\frac{n!{}_GL^{(\alpha,\lambda)}_n(x,y)t^n}{(1+\alpha)_n(1+\lambda)_n}\right)\left(\sum_{r=0}^{\infty}\frac{r!{}_GL^{(\beta,\mu)}_r(y,-y)t^r}{(1+\beta)_r(1+\mu)_r}\right)\left(\sum_{s=0}^{\infty}\frac{t^s}{s!}\right) \\ =&\sum_{n=0}^{\infty}\sum_{r=0}^{n}\sum_{s=0}^{n-r}\frac{(n-r-s)!r!{}_GL^{(\alpha,\lambda)}_{n-r-s}(x,y){}_GL^{(\beta,\mu)}_r(y,-y)t^n}{(1+\alpha)_{n-r-s}(1+\lambda)_{n-r-s}(1+\beta)_r(1+\mu)_rs!} \end{align*} Now, equating the coefficient of \(t^n\) from both sides, we get the desired result (43).
Similarly, we can prove (44), (45) and (46).

Remark 6. For \(\lambda=\mu=0\), the results (43), (44), (45) and (46) reduces to the expansions of Ragab polynomials \(L^{(\alpha,\beta)}_n(x,y)\) in terms of the (2VALP) \(L^{(\alpha)}_n(x,y)\).

5. Conclusion

In this paper, the two variable generalized Laguerre polynomials (2VGLP) \({}_GL_n^{(\alpha,\beta)}(x,y)\) are introduced and certain properties of these polynomials are deduced. The results of this paper are important tool for discuss certain properties of other polynomials.

Author Contributions

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

Competing Interests

The author(s) do not have any competing interests in the manuscript.