Some families of generating functions for the extended Srivastava polynomials

Almost four decades ago, H.M. Srivastava considered a general family of univariate polynomials, the Srivastava polynomials, and initiated a systematic investigation for this family [10]. In 2001, B. González, J. Matera and H.M. Srivastava extended the Srivastava polynomials by inserting one more parameter [4]. In this study we obtain a family of linear generating functions for these extended polynomials. Some illustrative results including Jacobi, Laguerre and Bessel polynomials are also presented. Furthermore, mixed multilateral and multilinear generating functions are derived for these polynomials.     

Some families of generating functions for a class of bivariate polynomials

The main purpose of this paper is to present various families of generating functions for a class of polynomials in two variables. Furthermore, several general classes of bilinear, bilateral or mixed multilateral generating functions are obtained for these polynomials.     

New families of generating functions for certain class of three-variable polynomials

Recently, Srivastava, Özarslan and Kaanoglu have introduced certain families of three and two variable polynomials, which include Lagrange and Lagrange-Hermite polynomials, and obtained families of two-sided linear generating functions between these families [H.M. Srivastava, M.A. Özarslan, C. Kaanoglu, Some families of generating functions for a certain class of three-variable polynomials, Integr. Transform. Spec. Funct. iFirst (2010) 1-12]. The main object of this investigation is to obtain new two-sided linear generating functions between these families by applying certain hypergeometric transformations. Furthermore, more general families of bilinear, bilateral, multilateral finite series relationships and generating functions are presented for them.     

Some generating functions of Jacobi polynomials

In this paper, Weisner’s group-theoretic method of obtaining generating functions is utilized in the study of Jacobi polynomialsP> _n ^(a,ß)(x) by giving suitable interpretations to the index (n) and the parameter (β) to find out the elements for constructing a six-dimensional Lie algebra.     

Two variable Lagrange expansion and new families of mixed generating functions

We show that a new method based on the combination of the two variable Lagrange expansion and of operational techniques can be exploited to derive new families of mixed generating functions for many variable many index special functions.

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Sunto In questo lavoro si fa uso di un nuovo metodo, basato sulla combinazione della espansione bidimensionale di Lagrange e di tecniche operazionali, per ottenere nuove famiglie di funzioni generatrici miste per funzioni speciali a più indici e a più variabili.

     

Bilateral generating functions for Jacobi polynomials

In this paper, we have obtained three theorems on generating functions. We derive from these theorems a large number of bilateral generating functions for Jacobi polynomials. Certain interesting expansions of triple hypergeometric series are also obtained from one of the theorems.     

Some generating functions of modified Gegenbauer polynomials

L Weisner's group theoretic method has been introduced in the study of special function. In this paper we obtain two differential operators, one of which simultaneously raises the index and lowers the parameter of modified Gegenbauer polynomialsC _n ^v+n (x) by unity and the other acts onC _n ^v+n(x) in the reversed way by suitable interpretation to the indexn and the parameterv ofC _n ^v+n(x) . We have also found out the extended form of the groups generated by the operatorsA _ij(i,j=1,2). We have also derived some novel generating functions ofC _n ^v+n (x) from which several special generating functions can be easily derived.     

New generating functions for Jacobi and related polynomials

In this paper the authors prove a generalization of certain generating functions for Jacobi and related polynomials, given recently by H. M. Srivastava. The method used is due to Pólya and Szegö, and it is based on Rodrigues' formula for the Jacobi polynomials and Lagrange's expansion theorem. A number of special and limiting cases of the main result will give rise to a class of generating functions for ultraspherical, Laguerre and Bessel polynomials.     

Bivariate generating functions for Rogers-Szegö polynomials

In the present paper, we utilize the general q-exponential operators to derive several Carlitz type bivariate generating functions for Rogers-Szegö polynomials. Moreover, we give an equivalent expansion formula of a certain bivariate generating functions for Rogers-Szegö polynomials and propose an open problem.     

Series identities and associated families of generating functions

The authors investigate several families of double-series identities as well as their (known or new) consequences involving various hypergeometric functions in one and two variables. A number of associated generating-function relationships, involving certain classes of hypergeometric polynomials, are also considered.     

Homogeneous $$$$-difference euations and generating functions for $$$$-hypergeometric polynomials

In this paper we show how to deduce several types of generating functions for \(q\)-hypergeometric polynomials by the method of homogeneous \(q\)-difference equations. In addition, we build relations between transformation formulas and homogeneous \(q\)-difference equations. Moreover, we generalize the Andrews–Askey integral from the perspective of \(q\)-integrals by the method of homogeneous \(q\)-difference equations.     

Some families of generating functions for the multiple orthogonal polynomials associated with modified Bessel K-functions

The main object of this paper is to derive several substantially more general families of bilinear, bilateral, and mixed multilateral finite-series relationships and generating functions for the multiple orthogonal polynomials associated with the modified Bessel K-functions also known as Macdonald functions. Some special cases of the above statements are also given.     

Newton functions generating symmetric fields and irreducibility of Schur polynomials

We prove (Theorem 1.1) that if e₀>>e_r>0 are coprime integers, then the Newton functions , i=0,…,r, generate over the field of symmetric rational functions in X₁,…,X_r. This generalizes a previous result of us for r=2. This extension requires new methods, including: (i) a study of irreducibility and Galois-theoretic properties of Schur polynomials (Theorem 3.1), and (ii) the study of the dimension of the varieties obtained by intersecting Fermat hypersurfaces (Theorem 4.1). We shall also observe how these results have implications to the study of zeros of linear recurrences over function fields; in particular, we give (Theorem 4.2) a complete classification of the zeros of recurrences of order four with constant coefficients over a function field of dimension 1.     

Generalized Christoffel functions for power orthogonal polynomials

In this paper we extend the Christoffel functions to the case of power orthogonal polynomials. The existence and uniqueness as well as some properties are given.     

An extension of bilateral generating functions of modified Laguerre polynomials

In this note a theorem concerning the extension of bilateral generating functions of the modified Laguerre polynomials is derived. Some applications of the theorem are also pointed out.     

On normal families and differential polynomials for meromorphic functions

We consider the normality criterion for a families F meromorphic in the unit disc Δ, and show that if there exist functions a(z) holomorphic in Δ, a(z)≠1, for each z∈Δ, such that there not only exists a positive number ε₀ such that |an(a(z)−1)−1|?ε₀ for arbitrary sequence of integers an(n∈N) and for any z∈Δ, but also exists a positive number B>0 such that for every f(z)∈F, B|f^′(z)|?|f(z)| whenever f(z)f^″(z)−a(z)(f^′2(z))=0 in Δ. Then is normal in Δ.     

New class of generating functions associated with generalized hypergeometric polynomials

In this paper authors prove a general theorem on generating relations for a certain sequence of functions. Many formulas involving the families of generating functions for generalized hypergeometric polynomials are shown here to be special cases of a general class of generating functions involving generalized hypergeometric polynomials and multiple hypergeometric series of several variables. It is then shown how the main result can be applied to derive a large number of generating functions involving hypergeometric functions of Kampé de Fériet, Srivastava, Srivastava-Daoust, Chaundy, Fasenmyer, Cohen, Pasternack, Khandekar, Rainville and other multiple Gaussian hypergeometric polynomials scattered in the literature of special functions.     

Finding new families of rank-one convex polynomials

We introduce a method to find, in a systematic way, rank-one convex polynomials. We show how it works in several examples. It can also be applied to convexity along general cones.