Rakesh K. Parmar

Bio: Rakesh K. Parmar is an academic researcher from Government College. The author has contributed to research in topics: Hypergeometric function & Generalized hypergeometric function. The author has an hindex of 12, co-authored 45 publications receiving 454 citations.

Extension of extended beta, hypergeometric and confluent hypergeometric functions

Abstract: Recently several authors have extended the Gamma function, Beta function, the hypergeometric function, and the con-fluent hypergeometric function by using their integral representa-tions and provided many interesting properties of their extended functions. Here we aim at giving further extensions of the above-mentioned extended functions and investigating various formulas for the further extended functions in a systematic manner. More-over, our extension of the Beta function is shown to be applied to Statistics and also our extensions find some connections with other special functions and polynomials such as Laguerre polynomials, Macdonald and Whittaker functions.

A Class of Extended Fractional Derivative Operators and Associated Generating Relations Involving Hypergeometric Functions

Abstract: Recently, an extended operator of fractional derivative related to a generalized Beta function was used in order to obtain some generating relations involving the extended hypergeometric functions [1]. The main object of this paper is to present a further generalization of the extended fractional derivative operator and apply the generalized extended fractional derivative operator to derive linear and bilinear generating relations for the generalized extended Gauss, Appell and Lauricella hypergeometric functions in one, two and more variables. Some other properties and relationships involving the Mellin transforms and the generalized extended fractional derivative operator are also given.

Some Families of the Incomplete H-Functions and the Incomplete $$\overline H $$ H ¯ -Functions and Associated Integral Transforms and Operators of Fractional Calculus with Applications

Abstract: Our present investigation is inspired by the recent interesting extensions (by Srivastava et al. [35]) of a pair of the Mellin–Barnes type contour integral representations of their incomplete generalized hypergeometric functions p γ q and p Γ q by means of the incomplete gamma functions γ(s, x) and Γ(s, x). Here, in this sequel, we introduce a family of the relatively more general incomplete H-functions γ (z) and Γ (z) as well as their such special cases as the incomplete Fox-Wright generalized hypergeometric functions p Ψ (γ) [z] and p Ψ (Γ) [z]. The main object of this paper is to study and investigate several interesting properties of these incomplete H-functions, including (for example) decomposition and reduction formulas, derivative formulas, various integral transforms, computational representations, and so on. We apply some substantially general Riemann–Liouville and Weyl type fractional integral operators to each of these incomplete H-functions. We indicate the easilyderivable extensions of the results presented here that hold for the corresponding incomplete $$\overline H $$ -functions as well. Potential applications of many of these incomplete special functions involving (for example) probability theory are also indicated.

A new generalization of gamma, beta hypergeometric and confluent hypergeometric functions

Abstract: The main object of this paper is to present new generalizations of gamma, beta hypergeometric and confluent hypergeometric functions. Some recurrence relations, transformation formulas, operation formulas, differentiation formulas, beta distribution and integral representations are obtained for these new generalizations.

Generalization of extended beta function, hypergeometric and confluent hypergeometric functions

Abstract: The main object of this paper is to present generaliza-tion of extended beta function, extended hypergeometric and con-uent hypergeometric function introduced by Chaudhry et al. andobtained various integral representations, properties of beta func-tion, Mellin transform, beta distribution, dierentiation formulas, transform formulas, recurrence relations, summation formula forthese new generalization.

Confluent Hypergeometric Functions

Abstract: Confluent Hypergeometric Functions By Dr L J Slater Pp ix + 247 (Cambridge: At the University Press, 1960) 65s net

On Fractional Operators and Their Classifications

Abstract: Fractional calculus dates its inception to a correspondence between Leibniz and L’Hopital in 1695, when Leibniz described “paradoxes” and predicted that “one day useful consequences will be drawn” from them. In today’s world, the study of non-integer orders of differentiation has become a thriving field of research, not only in mathematics but also in other parts of science such as physics, biology, and engineering: many of the “useful consequences” predicted by Leibniz have been discovered. However, the field has grown so far that researchers cannot yet agree on what a “fractional derivative” can be. In this manuscript, we suggest and justify the idea of classification of fractional calculus into distinct classes of operators.

Fractional differential equations for the generalized Mittag-Leffler function

Abstract: In this paper, we establish some (presumably new) differential equation formulas for the extended Mittag-Leffler-type function by using the Saigo-Maeda fractional differential operators involving the Appell function $F_{3}(\cdot)$ and results in terms of the Wright generalized hypergeometric-type function ${}_{m+1}\psi^{(\{\kappa_{l}\}_{l\in\mathbb{N}_{0}} )}_{n+1}(z; p)$ recently established by Agarwal. Some interesting special cases are also pointed out.