Praveen Agarwal

Some results on the extended beta and extended hypergeometric functions

The purpose of this paper is to present a systematic study of some extended special functions like B b ; Â? , λ α , β x , y , 2 F 1 α , β ; Â? , λ z ; b and p F q α , β ; Â? , λ z ; b . We obtain various properties of these extended functions and establish their some connections with the Laguerre polynomial and Foxs H-function. Furthermore, we also establish the extended Riemann-Liouville type fractional integral operator and extended Kober type fractional integral operators.

Generating functions for the generalized Gauss hypergeometric functions

Formulas and identities involving many well-known special functions (such as the Gamma and Beta functions, the Gauss hypergeometric function, and so on) play important roles in themselves and in their diverse applications. Various families of generating functions have been established by a number of authors in many different ways. In this paper, we aim at establishing some (presumably new) generating functions for the generalized Gauss type hypergeometric type function F p ( α , β ; ? , µ ) ( a , b ; c ; z ) which is introduced here. We also present some special cases of the main results of this paper.

CERTAIN NEW INTEGRAL FORMULAS INVOLVING THE GENERALIZED BESSEL FUNCTIONS

Abstract. A remarkably large number of integral formulas involving avariety of special functions have been developed by many authors. Alsomany integral formulas involving various Bessel functions have been pre-sented. Very recently, Choi and Agarwal derived two generalized integralformulas associated with the Bessel function J ν (z) of the first kind, whichare expressed in terms of the generalized (Wright) hypergeometric func-tions. In the present sequel to Choi and Agarwal’s work, here, in thispaper, we establish two new integral formulas involving the generalizedBessel functions, which are also expressed in terms of the generalized(Wright) hypergeometric functions. Some interesting special cases of ourtwo main results are presented. We also point out that the results pre-sented here, being of general character, are easily reducible to yield manydiverse new and known integral formulas involving simpler functions. 1. Introduction and preliminariesA remarkably large number of integral formulas involving a variety of spe-cial functions have been developed by many authors (see, e.g., [5], [7] and [9];for a very recent work, see also [6]). Many integral formulas involving prod-ucts of Bessel functions have been developed and play an important role inseveral physical problems. In fact, Bessel functions are associated with a widerange of problems in diverse areas of mathematical physics, for example, thosein acoustics, radio physics, hydrodynamics, and atomic and nuclear physics.These connections of Bessel functions with various other research areas haveled many researchers to the field of special functions. Among many propertiesof Bessel functions, they also have investigated some possible extensions of theBessel functions. A useful generalization w

Certain unified integrals associated with Bessel functions

A remarkably large number of integral formulas involving a variety of special functions have been developed by many authors. Very recently, Ali gave three interesting unified integrals involving the hypergeometric function F12. Using Ali’s method, in this paper, we present two generalized integral formulas involving the Bessel function of the first kind Jν(z), which are expressed in terms of the generalized (Wright) hypergeometric functions. Some interesting special cases of our main results are also considered.MSC:33B20, 33C20, 33B15, 33C05.

The general solution for impulsive differential equations with Riemann-Liouville fractional-order q ∈ (1,2)

Abstract In this paper we consider the generalized impulsive system with Riemann-Liouville fractional-order q ∈ (1,2) and obtained the error of the approximate solution for this impulsive system by analyzing of the limit case (as impulses approach zero), as well as find the formula for a general solution. Furthermore, an example is given to illustrate the importance of our results.

On Fractional Integral Inequalities Involving Hypergeometric Operators

Here we aim at establishing certain new fractional integral inequalities involving the Gauss hypergeometric function for synchronous functions which are related to the Chebyshev functional. Several special cases as fractional integral inequalities involving Saigo, Erdelyi-Kober, and Riemann-Liouville type fractional integral operators are presented in the concluding section. Further, we also consider their relevance with other related known results.

Certain Grüss type inequalities involving the generalized fractional integral operator

A remarkably large number of Grüss type fractional integral inequalities involving the special function have been investigated by many authors. Very recently, Kalla and Rao (Matematiche LXVI(1):57-64, 2011) gave two Grüss type inequalities involving the Saigo fractional integral operator. Using the same technique, in this paper, we establish certain new Grüss type fractional integral inequalities involving the Gauss hypergeometric function. Moreover, we also consider their relevances for other related known results.MSC: 26D10, 26A33.

Fractional Integration of the Product of Two Multivariables H-Function and a General Class of Polynomials

A significantly large number of earlier works on the subject of fractional calculus give interesting account of the theory and applications of fractional calculus operators in many different areas of mathematical analysis (such as ordinary and partial differential equations, integral equations, special functions, summation of series, etc.). The main object of the present paper is to study and develop the Saigo operators. First, we establish two results that give the images of the product of two multivariables H-function and a general class of polynomials in Saigo operators. On account of the general nature of the Saigo operators, multivariable H-functions and a general class of polynomials a large number of new and Known Images involving Riemann-Liouville and Erde’lyi-Kober fractional integral operators and several special functions notably generalized Wright hypergeometric function, Mittag-Leffler function, Whittaker function follow as special cases of our main findings. Results given by Kilbas, Kilbas and Sebastian, Saxena et al. and Gupta et al., follow as special cases of our findings.

Certain Hermite-Hadamard type inequalities via generalized k -fractional integrals

Some Hermite-Hadamard type inequalities for generalized k-fractional integrals (which are also named (k,s)

Extension of the fractional derivative operator of the Riemann-Liouville

(k,s)