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The Scientific World Journal
Volume 2013 (2013), Article ID 187452, 12 pages
http://dx.doi.org/10.1155/2013/187452
Research Article

On a Family of Multivariate Modified Humbert Polynomials

1Department of Mathematics, Faculty of Science, Ankara University, Tandoğan, 06100 Ankara, Turkey
2Department of Mathematics, Faculty of Science, Gazi University, Teknikokullar, 06500 Ankara, Turkey

Received 29 April 2013; Accepted 6 June 2013

Academic Editors: F. J. Garcia-Pacheco and V. Privman

Copyright © 2013 Rabia Aktaş and Esra Erkuş-Duman. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper attempts to present a multivariable extension of generalized Humbert polynomials. The results obtained here include various families of multilinear and multilateral generating functions, miscellaneous properties, and also some special cases for these multivariable polynomials.

1. Introduction

The generalized Humbert polynomials are generated by [1] where is a positive integer and other parameters are unrestricted in general (see also [2, pages 77, 86] and [3, 4]). This definition includes many well-known special polynomials such as Humbert, Louville, Gegenbauer, Legendre, Tchebycheff, Pincherle, and Kinney polynomials.

In this paper, we consider the following multivariable extension of the generalized Humbert polynomials which are completely different from the polynomials introduced in [5]. This class of polynomials is generated by where ,  , and is a positive integer. It follows from (2) that where , is the Pochhammer symbol.

The aim of this paper is to derive various families of multilinear and multilateral generating functions and to give several recurrence relations and expansions in the series of orthogonal polynomials for the family of multivariable polynomials given explicitly by (3). We present some special cases of our results and also obtain some other properties for these special cases.

2. Bilinear and Bilateral Generating Functions

In this section, with the help of the similar method as considered in [59], we derive several families of bilinear and bilateral generating functions for the family of multivariable polynomials generated by (2) and given explicitly by (3).

We begin by stating the following theorem.

Theorem 1. Corresponding to an identically nonvanishing function of complex variables and of complex order , let where ; . Then, for ; ; ; , , one has provided that each member of (5) exists.

Proof. For convenience, let denote the first member of the assertion (5) of Theorem 1. Replacing by , we may write that which completes the proof.

By using a similar idea, we also get the next result immediately.

Theorem 2. For a nonvanishing function of complex variables    and for , , , ,   let where ; ; . Then, one has provided that each member of (8) exists.

3. Special Cases

As an application of the above theorems, when the multivariable function ,  ,  , , is expressed in terms of simpler functions of one and more variables, then we can give further applications of the above theorems. We first set in Theorem 1, where the Chan-Chyan-Srivastava polynomials [10] are generated by We are thus led to the following result which provides a class of bilateral generating functions for the Chan-Chyan-Srivastava polynomials and the family of multivariable polynomials given explicitly by (3).

Corollary 3. If , , , then, one has provided that each member of (11) exists.

Remark 4. Using the generating relation (10) for the Chan-Chyan-Srivastava polynomials and getting ,  , , we find that

On the other hand, choosing and ,  , in Theorem 2, we obtain the following class of bilinear generating functions for the family of multivariable polynomials given explicitly by (3).

Corollary 5. If where ; ; , , then, we get provided that each member of (15) exists.

Furthermore, for every suitable choice of the coefficients , if the multivariable functions and ,  , are expressed as an appropriate product of several simpler functions, the assertions of Theorems 1 and 2 can be applied in order to derive various families of multilinear and multilateral generating functions for the family of multivariable polynomials given explicitly by (3).

4. Some Miscellaneous Properties

In this section, we now discuss some further properties of the family of multivariable polynomials given by (3). We start with the following theorems.

Theorem 6. Let be a family of functions generated by The following relations hold: for ,  ; , and for , ; , . Also, one finds that for , ; , and for ,; , .

Proof. Fix . Then, by differentiating (16) with respect to and , after making necessary calculations we obtain that Comparing the coefficients of , we obtain (17) and (18) for the fixed . Similarly, if we differentiate (16) with respect to and , we can find the relation (19).

Theorem 7. If is a family of functions generated by (16), then it satisfies the relations for , ; , and for , ; , .

Proof. By comparing the derivatives of (16) with respect to and , we have which implies that for , ; , and for ,; , . Thus, the proof is completed.

As a result of these theorems, if we choose then we can give some recurrence relations for the family of multivariable polynomials given explicitly by (3). In the view of Theorem 6, we get

Corollary 8. For the family of multivariable polynomials generated by (2), the following relations hold for ,; , . Also, for , ; , , we have

Corollary 9. By summing the relations given by (28) and (29), respectively, for , we get for , ; , .

Similarly, as a consequence of Theorem 7, we can give the next result at once.

Corollary 10. Other recurrence relations for the family of multivariable polynomials are for , ; , and for , ; , .

Theorem 11. The generating relation (2) yields the following addition formula for the family of multivariable polynomials given by (3)

Proof. From (2), we have Replacing by , ,   the right-hand side of the last equality is Comparing the coefficients of completes the proof.

We now give expansions of the family of multivariable polynomials given explicitly by (3) in series of Legendre, Gegenbauer, Hermite, and Laguerre polynomials.

Theorem 12. Expansions of in series of Legendre, Gegenbauer, Hermite, and Laguerre polynomials are as follows

Proof. By (2), we get If we use the result in [11, page 181] we can write that Getting instead of in the last equality, we have Comparing the coefficients of gives the desired relation.
In a similar manner, in (39), using the following results, respectively, [11, page 283 (36), page 194 (4), page 207 (2)] one can easily obtain the other expansions of in series of Gegenbauer, Hermite, and Laguerre polynomials.

5. The Special Cases of (x, y, m) and Some Properties

In this section, we discuss some special cases of the family of multivariable polynomials and give their several properties.

5.1. The Case of , , in (2)

This case gives a multivariable analogue of Gegenbauer polynomials which reduces to two variables Gegenbauer polynomials given by [12] for . Equation (44) yields the following formula:

It follows that is a polynomial of degree with respect to the fixed variable .  Thus, is a polynomial of total degree with respect to the variables . Equation (45) also yields where is a polynomial of degree with respect to the variables . In (44), by getting and , we have

Similarly, for ,   we get

Taking and, , in (44), we obtain

Theorem 13. For the polynomials , one has and if at least one of , , is odd; then

Proof. If we set all , in (44), we have On the other hand, we get By comparing the coefficients of , we obtain the desired.

From the theorems and corollaries given in Section 4, we can give some other properties of .

Remark 14. By Corollaries 8 and 9, for the family of multivariable polynomials generated by (44), the following relations: hold for ,.

Remark 15. From Theorem 11, the multivariable polynomials satisfy the following addition formula:

Remark 16. As a result of Theorem 12, expansions of in series of Legendre, Gegenbauer, Hermite, and Laguerre polynomials are as follows:

We now give a hypergeometric representation for the multivariable polynomials given by (44).

Theorem 17. The multivariable polynomials have the following hypergeometric representation: where is a Lauricella function of -variable defined by [13] (see also [2])

Proof. We have the following results from [11]: In view of these relations, the equality (45) can be written as follows: where . The proof is completed.

Remark 18. For Theorem 17 reduces to the known result for two variable Gegenbauer polynomials given by [12].

5.2. The Case of , , in (2)

This case yields which is a different unification from that in [8].

Remark 19. From Corollaries 8 and 9, the multivariable polynomials satisfy for , ; , .

Remark 20. As result of Theorem 11, the multivariable polynomials have the following addition formula:

5.2.1. The Case of , in Section 5.2

In this case, we get a -variable analogue of Lagrange polynomials (see [14]) which is different from that in [10]

They are given explicitly by

Remark 21. From Corollaries 8 and 9, the multivariable polynomials satisfy for and

Remark 22. By Theorem 11, the multivariable polynomials have the following addition formula:

Remark 23. From Theorem 12, expansions of in series of Legendre, Gegenbauer, Hermite, and Laguerre polynomials are given by

We can discuss some generating functions for the multivariable polynomials