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The Scientific World Journal
Volume 2013 (2013), Article ID 187452, 12 pages
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.
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.
The generalized Humbert polynomials are generated by  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 . 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 [5–9], 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.
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  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
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.
Corollary 8. For the family of multivariable polynomials generated by (2), the following relations hold for ,; , . Also, for , ; , , we have
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 , ; , .
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)
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 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).
5.2. The Case of , , in (2)
This case yields which is a different unification from that in .
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
They are given explicitly by
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 .
Theorem 24. For the polynomials , the following generating function holds true for