Chena Ram

A unified presentation of the Gamma-type functions occurring in diffraction theory and associated probability distributions

In view of their importance and usefulness in diffraction theory and probability distributions, several generalizations of the Gamma-type functions were investigated extensively in recent years. The main object of this paper is to present a systematic study of a unification and generalization of these Gamma-type functions, which is defined here by means of the Fox-Wright hypergeometric function [emailxa0protected]q. A probability density function associated with the generalized Gamma-type function investigated in this paper, together with several other related results in the theory of probability and statistics, are also considered.

A generalization of beta-type distribution involving ω-Lauricella function in several variables

The object of this paper is to define and study a new generalization of beta-type function in the following form: where λ, u, v, a, c, b j (ju2009=u20091, Λ, n)u2009∈u2009C with cu2009≠u20090,−1,−2, Λ; ω, Re(u), Re (au2009+u2009λ), Re (b j u2009+u2009λ)u2009>u20090(ju2009=u20091, Λ, n), | arg v|u2009<u2009π and is the ω-generalization of the Lauricella hypergeometric function of n-variables. The previous defined function provides extensions of the various type of beta-functions studied earlier by many authors, notably by Ben Nakhi and Kalla [Al-Shammery, A.H. and Kalla, S.L., 2000, An extension of some hypergeometric functions of two variables. Revista de la Acade. Canaria de cie., 12, 189–196; Ben Nakhi, Y. and Kalla, S.L., 2002, A generalised beta function and associated probability density. International Journal of Mathematics and Mathematical Science, 8, 467–478.]. A study of this function will provide deeper, general and useful results in the theory of special functions and statistical distributions. A probability density function (p.d.f.) associated with this function is also introduced. The moment generating function, hazard rate function and mean residue life function for this p.d.f. will also be investigated.

On Certain Class of Analytic Functions Involving Linear Operators

Invoking the Hadamard product (or convolution) , a class of univalent functions has been introduced. In the present paper we obtain necessary and sufficient conditions and some important properties for the analytic functions for its belongingness to certain class of functions. The distortion inequalities, closer theorems, radii of close-to-convexity, radii of starlikeness and radii of convexity are obtained for the same class of functions. Some properties involving Hadamard product are also obtained.

Generalized gamma-type functions involving Kummer’s confluent hypergeometric function and associated probability distributions

The object of this paper is to define and study a new generalization of the generalized gamma-type function in the form where Φ (α, β; z) is the well known Kummer’s confluent hypergeometric function and 2 R 1 (·) is a special case of Wright’s generalized hypergeometric function. This generalization provides unification and extension of the various generalizations given earlier by Kobayashi, Al-Musallam and Kalla and Virchenko et al. Incomplete generalized gamma function corresponding to the above defined generalized gamma function are also defined and their basic properties are investigated, which extend the existing results in the field of generalized gamma-type functions. We study a class of probability density functions involving Kummer’s confluent hypergeometric function and 2 R 1 (·).