M. A. Pathan

Lie-theoretic generating relations of Hermite 2D polynomials

In this paper, we derive some generating relations involving Hermite 2D polynomials (H2DP) Hm,n(U;x,y), of two variables with an arbitrary 2D matrix U as parameter using Lie-theoretic approach. Certain (known or new) generating relations for the polynomials related to H2DP are also obtained as special cases.

On multiindices and multivariables presentation of the Voigt functions

This paper aims at presenting multiindices and multivariables study of the unified (or generalized) Voigt functions which play an important role in the several diverse field of physics such as astrophysical spectroscopy and the theory of neutron reactions. Some expressions (representations) of these functionl are given in terms of familiar special functions of multivariables. Further representations and series expansions involving multidimensional classical polynomials (Laguerre and Hermite) of mathematical physics are established.

Some Properties of Two Variable Laguerre Polynomials Via Lie Algebra

In this paper we derive a result concerning eigenvector for the product of two operators defined on a Lie algebra of endomorphisms of a vector space. Further using this result, we deduce some properties of two variable Laguerre polynomials.

Origin of certain generating relations of Hermite matrix functions from the view point of Lie algebra

Weisner’s [McBride, E.B., 1971, Obtaining Generating Functions (New york: Springer-Verlag).] group theoretic method is utilized to obtain new generating relations for Hermite matrix functions T n (x, A) studied by Jodar and Defez by giving suitable interpretations to the index (n). A few special cases of interest are also discussed, which would inevitably yield many new and known results of theory of special functions.

Representation of a Lie Algebra {\cal G}(0,1) and Three Variable Generalized Hermite Polynomials H_{n} (x,y,z)

In this paper, the authors address the problem of framing three variable generalized Hermite polynomials (G.H.P.), H_{n} (x, y, z) , into the context of the representation \uparrow_{\omega, \mu} of a Lie algebra {\cal G}(0,1), thus stressing the mathematical relevance of G.H.P. and representations of Lie algebras. Generating relations involving G.H.P. and associated Laguerre polynomials are obtained following the ideas and methods suggested by Miller.

Some connections between generalized Voigt functions with the different parameters

The Voigt functions play an important role in several diverse fields of physics such as astrophysical spectroscopy and the theory of neutron reactions. Motivated by the contributions toward the unification (and generalization) of these functions, in this paper, we establish some connections between generalized Voigt functions with the different parameters involving one and two-variable hypergeometric functions. Further, we derive various other interesting results as applications of these connections.

A note on hypergeometric polynomials

In a paper which appeared in this journal, Manocha and Sharma [6] obtained some results of Carlitz [4], Halim and Salain [5] and generalized a few of them by using fractional derivatives. The present paper is concerned with some erroneous results of this paper [6]. Many more sums of the product of hypergeometric polynomials are also obtained.

Evaluations of certain Euler type integrals

Abstract In this paper, we obtain the evaluations of certain Euler type integrals. Further, we establish a theorem on extended beta function and apply this to obtain evaluations of some integrals in terms of extended beta function B ( α , β ; A ) . Furthermore, we derive a number of new results as applications of these evaluations. We extend also some results of this paper to the multi-variable case.

Some bilateral generating relations involving Gegenbauer polynomials

In this paper, we obtain three interesting bilateral generating functions for Gegenbauer polynomials C n b (x) associated with hypergeometric polynomials 2 F 1, 1 F 2 and 3 F 2. Our results are obtained with the help of series rearrangement technique.

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 Kampe de Feriet, Srivastava, Srivastava–Daoust, Chaundy, Fasenmyer, Cohen, Pasternack, Khandekar, Rainville and other multiple Gaussian hypergeometric polynomials scattered in the literature of special functions.