A Class of Laguerre-Based Generalized Humbert Polynomials

Several mathematicians have extensively investigated polynomials, their extensions, and their applications in various other research areas for a decade. Our paper aims to introduce another such polynomial, namely, Laguerre-based generalized Humbert polynomial, and investigate its properties. In particular, it derives elementary identities, recursive differential relations, additional symmetry identities, and implicit summation formulas.


Introduction
In all the given definitions, let C, R, R + , and N be the sets of complex numbers, real numbers, positive real numbers, and natural numbers, respectively. e two-variable Kampé de Fériet generalized Hermite polynomial (see [1]) is defined as H n (x, y) t n n! . (1) e finite series representation of Hermite polynomial of two variables is given by Substituting y � − 1 and replacing x by 2x, the polynomial in equation (2) reduces to ordinary Hermite polynomial (see [1,2]).
Classical Laguerre polynomial and its orthogonality [3,4] have been studied extensively. Its generalization is given by the two variable Laguerre polynomial. e twovariable Laguerre polynomial L n (x, y) is defined by the following generating function (see [5][6][7][8]): where C 0 (x) is the 0-th order Tricomi function (see [2,[6][7][8]): e explicit expression of two-variable Laguerre polynomial is given as We would now recall the following well-known generating functions, which will be further used in our paper: where P n (x) is the Legendre polynomial of first kind. Also, where U n (x) is the Chebyshev polynomial of the second kind. e following generating function gives the extension of equations (6) and (7): where C ] n (x) is Gegenbauer polynomial.
Substituting ] � 1/2 and ] � 1, equation (8) reduces to Legendre polynomial and Chebyshev polynomial, respectively: where h ] n,m (x) is the Humbert polynomial defined as where m is a positive integer. In 1991, another generalization was given by Milovanović and Djordjević (see [9]), which has the following generating function: where m ∈ N and λ > − 1/2. Also, Generalization of two variables of all the above polynomials and many more was given by Djordjević (see [10]) in the form where For α � 1 and � 1/2, the above polynomial reduces to Chebyshev polynomial of two variables, U m n (x, y), and Legendre polynomial of two variables, P m n (x, y), respectively.
For m � 2 and y � 0, the above polynomial reduces to Gegenbauer polynomial.
Furthermore, by substituting y � 0, the above polynomial reduces to p λ n,m (x), the polynomial defined by Milovanović and Djordjević (see [9]). e three-variable Hermite-Laguerre polynomial H L n (x, y, z) is defined by the following generating function (see [6]): In our paper, we will introduce Laguerre-based generalized Humbert polynomials L G υ,m n (a, b, c; x, y, z).

Definition 1.
e Laguerre-based generalized Humbert polynomials of order ], denoted by L G υ,m n (a, b, c; x, y, z), is defined by the following generating function: where a, b, c, x, y, z, υ ∈ C and m ∈ N. For all the further work, let

Elementary Identities of L G υ,m n (a, b, c; x, y, z)
For our further reference, let us recall the following identities mentioned in the lemma as follows (see [11,12]).

Lemma 1. e following relations hold:
where f(N) and A m,n are complex-and real-valued functions with m, n, N ∈ N 0 and x, y ∈ C. Lemma 1 applies to the convergent double series. For a, b, c, x, y, z, υ ∈ C and m ∈ N, the following relations are satisfied: n (a, b, c; x, y, z) � log k c L G υ,m n− 2k (a, b, c; x, y, z), n, k ∈ N 0 ; 2k ≤ n , 2l (a, b, c; x, y, z), n, k ∈ N 0 ; k + 2l ≤ n , (23) Proof. Differentiating both sides of (16) k times with respect to y and z and then equating the coefficient of t n , we obtain (21) and (22) respectively. Differentiating both sides of (16), with respect to y and z, k times and l times, respectively, and equating the coefficient of t n , we obtain (23).

Differential-Recursive Relations
In this section of the paper, we have derived few differentialrecursive relations involving the Laguerre-based generalized Humbert polynomial in (16), generalized class of Humbert polynomials in (13), and Hermite-Laguerre polynomial in (15). a, b, x, y, z, υ ∈ C, m ∈ N , and k ∈ N 0 . en, the following results hold:

Theorem 2. Let
Proof: . Using (13) and (17), we get Differentiating both sides with respect to t, we obtain Multiplying both sides of (35) by t and then using we get Applying (15), we get Equating the coefficients of t n , we derive (31). Differentiating both sides of (34) k times with respect to z and y and then comparing the coefficient of t n , we obtain the desired results in (32) and (33), respectively.