A Class of Laguerre-Based Generalized Humbert Polynomials

Abstract

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.

1. Introduction

In all the given definitions, let be the sets of complex numbers, real numbers, positive real numbers, and natural numbers, respectively.

The two-variable Kampé de Fériet generalized Hermite polynomial (see [1]) is defined as

The finite series representation of Hermite polynomial of two variables is given by

Substituting and replacing by , 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. The two-variable Laguerre polynomial is defined by the following generating function (see [5–8]):where is the 0-th order Tricomi function (see [2, 6–8]):

The 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 is the Legendre polynomial of first kind. Also,where is the Chebyshev polynomial of the second kind.

The following generating function gives the extension of equations (6) and (7):where is Gegenbauer polynomial.

Substituting and , equation (8) reduces to Legendre polynomial and Chebyshev polynomial, respectively:where is the Humbert polynomial defined aswhere is a positive integer.

In 1991, another generalization was given by Milovanović and Djordjević (see [9]), which has the following generating function:where and . Also,

Generalization of two variables of all the above polynomials and many more was given by Djordjević (see [10]) in the formwhere

For and , the above polynomial reduces to Chebyshev polynomial of two variables, , and Legendre polynomial of two variables, , respectively.

For and , the above polynomial reduces to Gegenbauer polynomial.

Furthermore, by substituting , the above polynomial reduces to , the polynomial defined by Milovanović and Djordjević (see [9]).

The three-variable Hermite–Laguerre polynomial is defined by the following generating function (see [6]):

In our paper, we will introduce Laguerre-based generalized Humbert polynomials

Definition 1. The Laguerre-based generalized Humbert polynomials of order , denoted by , is defined by the following generating function:where and .
For all the further work, let

2. Elementary Identities of

For our further reference, let us recall the following identities mentioned in the lemma as follows (see [11, 12]).

Lemma 1. The following relations hold:where and are complex- and real-valued functions with and . Lemma 1 applies to the convergent double series.

Theorem 1. For and , the following relations are satisfied:

Proof. Differentiating both sides of (16) times with respect to and and then equating the coefficient of , we obtain (21) and (22) respectively.
Differentiating both sides of (16), with respect to and , times and times, respectively, and equating the coefficient of , we obtain (23).
Using the right-hand side of (13) and (15) in (17), with the help of (18) for , we obtain (24).
Using equations (1), (4), and (13) in (17) with the help of (18) for and then equating the coefficient of , we obtain (25).
Using (5) and (13) in (17) with the help of (18) for , we obtain (26). From (17), we haveUsing (13) and (15) in (29), we derive (27).
Again, from (17), we getUsing (1), (4), and (13) in (30) and rearranging the equations, we get the desired result in (28).

3. Differential-Recursive Relations

In this section of the paper, we have derived few differential-recursive relations involving the Laguerre-based generalized Humbert polynomial in (16), generalized class of Humbert polynomials in (13), and Hermite–Laguerre polynomial in (15).

Theorem 2. Let , , and . Then, the following results hold:

Proof: . Using (13) and (17), we getDifferentiating both sides with respect to , we obtainMultiplying both sides of (35) by and then usingwe getApplying (15), we getEquating the coefficients of , we derive (31).
Differentiating both sides of (34) times with respect to and and then comparing the coefficient of , we obtain the desired results in (32) and (33), respectively.

4. Symmetry Identities

In this section, we derive few additional symmetric identities for Laguerre-based generalized Humbert polynomials which are summarized in Theorem 3.

Theorem 3. For , , and , the following identities hold:

Proof. Using polynomials involved in (1), (13), and (17) along with equation (18), we can prove the identities mentioned in Theorem 3.

5. Implicit Summation Formula

The following theorem establishes the implicit summation formula of Laguerre-based generalized Humbert polynomial.

Theorem 4. Let , , and . Then,

Proof. Replacing by in (16), we haveUsing the binomial expansion of with the help of (19), we getMultiplying both the sides of (48) with , we getOn expanding the exponential function with the aid of (20), we get the following equation:Substituting equation (50) in (49), we get a quadruple series as follows:Comparing the coefficient of on both sides in (51), we get the desired result in (46).

6. Special Cases

Let us now investigate few special cases of Laguerre-based generalized Humbert polynomial summarized as follows: Case 1: where is the Hermite–Laguerre polynomial (see [6]). Case 2: where is the Hermite polynomial (see [1]). Case 3: where is the generalized Humbert polynomial (see [10, 13]). Case 4: where is the Gegenbauer polynomial (see [13, 14]). Case 5: letting and in equation (55), respectively, we get where is the Chebyshev polynomial of second kind (see [13, 14]). Also, where is the Legendre polynomial of first kind (see [13, 14]). Case 6: where is the Hermite–Gegenbauer polynomial (see [14]). Case 7: letting and in equation (58), respectively, we get where is the Hermite–Legendre polynomial of one variable (see [14]). Also, where is the Hermite–Chebyshev polynomial of one variable (see [14]). Case 8: where is the Hermite-based generalized Humbert polynomial. Case 9: where is the Laguerre-based Legendre polynomials of two variables. Also, where is the Laguerre-based Chebyshev polynomials of two variables. Using these special cases to the identities derived in previous sections, we can derive the corresponding identities. To state an example, we have derived an implicit summation formula for Hermite-based generalized Humbert polynomial in the following corollary.

Corollary 1. Let , , and . Then,

7. Conclusion

This study has defined a new polynomial class, namely, Laguerre-based generalized Humbert polynomials. We have derived recursive relations, additional symmetry identities, and implicit summation formulas for these special functions. We have also defined few essential special cases of this class; a few of them are Laguerre-based Legendre, Laguerre-based Chebyshev, Hermite-based generalized Humbert polynomial etc. Using these special cases to the identities derived, we get the corresponding identities.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.