Ajay Kumar Shukla

SOME REMARKS ON GENERALIZED MITTAG-LEFFLER FUNCTION

The principal aim of the paper is to establish the function and its properties by using Fractional Calculus. We also obtained some integral representations of the function which is recently introduced by Shukla and Prajapati[6].

ON SOME PROPERTIES OF A CLASS OF POLYNOMIALS SUGGESTED BY MITTAL

The object of this paper is to establish some generating relations by using operational formulae for a class of polynomials T (α+s−1) kn (x) defined by Mittal. We have also derived finite summation formulae for (1.6) by employing operational techniques. In the end several special cases are discussed.

Some properties of Wright-type generalized hypergeometric function via fractional calculus

This paper is devoted to the study of a Wright-type hypergeometric function (Virchenko, Kalla and Al-Zamel in Integral Transforms Spec. Funct. 12(1):89-100, 2001) by using a Riemann-Liouville type fractional integral, a differential operator and Lebesgue measurable real or complex-valued functions. The results obtained are useful in the theory of special functions where the Wright function occurs naturally.MSC: 33C20, 33E20, 26A33, 26A99.

A general class of polynomials associated with generalized Mittag–Leffler function

Several authors have defined polynomial sets by means of general Rodrigues formulae in different forms. In this article, an attempt is made to provide an elegant unification of several classes of polynomials. We investigate a general class of polynomials by means of a generalized Rodrigues formula defined as (2) associated with a generalized Mittag–Leffler function , which is recently introduced by Shukla and Prajapati [Shukla, A.K. and Prajapati, J.C., On a generalisation of Mittag–Leffler function and its properties. Journal of Mathematical Analysis and Appplications, article in press]. We have also derived several families of generating relations, finite summation formulae for equation (2) by employing operational techniques and bilateral-generating relation. In the end, several special cases have been discussed.

SOME PROPERTIES OF GENERALIZED HYPERGEOMETRIC FUNCTION

In present paper, we obtain functions R t(c,�,a,b) and R t(c, µ,a,b) by using generalized hypergeometric function. A recurrence re- lation, integral representation of the generalized hypergeometric function 2R1(a,b;c;� ;z) and some special cases have also been discussed.

An extension of Sheffer polynomials

Sheffer [Some properties of polynomial sets of type zero, Duke Math. J. 5 (1939), pp.590-622] studied polynomial sets zero type and many authors investigated various properties and its applications. In the sequel to the study of Sheffer Polynomials, an attempt is made to generalize the Sheffer polynomials by using partial differential operator.

On the Laguerre transform in two variables

The main objective of the present article is to introduce the Laguerre transform of two variables. We obtained some basic operational properties and also examined the MAPLE implementation to obtain the aforementioned transform.

GENERALIZATION OF A CLASS OF POLYNOMIALS

In [1], R. Andre-Jeannin considered a class of polynomials Un(p, q; x) defined by Un(P> ft ) = ( + P)U„-i(P, ft ) ~ qUn-iiP, ft x), n>\9 with initial values U0(p, q;x) = 0 and U{(p, q;x) = l. Particular cases of Un(p,q;x) are: the well-known Fibonacci polynomials Fn(x); the Pell polynomials Pn(x) (see [4]); the Fermat polynomials of the first kind fl(x) (see [5], [3]); and the Morgan-Voyce polynomials of the second kind Bn(x) (see [2]). In this paper we shall consider the polynomials

On some generalized Sister Celine's polynomials

„(p, q; x) defined by

Note on generalized hypergeometric function

Certain generalizations of Sister Celines polynomials are given which include most of the known polynomials as their special cases. Besides, generating functions and integral representations of these generalized polynomials are derived and a relation between generalized Laguerre polynomials and generalized Batemans polynomials is established.