Research
Fractional Integration of the Product of two Multivariable Gimel-Functions and a General Class of Polynomials
A significantly large number of earlier works on the subject of fractional calculus give the 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, the summation of series, etc.). The object of the present paper is to study and develop the Saigo-Maeda operators. First, we establish four results that give the images of the product of two multivariable Gimel-functions and a general class of multivariable polynomials in Saigo- Maeda operators. On account of the general nature of the Saigo-Maeda operators, multivariable Gimel-functions and a class multivariable polynomials a large number of new and known theorems involving Riemann-Liouville and Erdelyi- Kober fractional integral operators and several special functions.
On a General Class of Multiple Eulerian Integrals with Multivariable Gimel-Function
Recently, Raina and Srivastava [11] and Srivastava and Hussain [16] have provided closed-form expressions for a number of a Eulerian integral about the multivariable H-functions. Motivated by these recent works, we aim at evaluating a general multiple Eulerian integrals involving the product of multivariable I-function defined by Prathima et al.
An Unified Study of Some Multiple Integrals with Multivariable Gimel-Function
In this paper, we first evaluate a unified and general finite multiple integral whose integrand involves the product of the functions and the multivariable Gimel-function occurring in the integrand involve the product of factors of the form , while that of occurring herein involves a finite series of such factors. On account of the most general nature of the functions occurring in the integrand of our main integral, a large number of new and known integrals can easily be obtained from ot merely by specializing the functions and parameters involved therein. At the end of this study, we illustrate a new integral whose integrand involves a product of the Jacobi polynomial, the Appell’s function and the Bessel function
Certain Fractional Derivative Formulae Involving the Product of a General Class of Polynomials and the Multivariable Gimel-Function
In the present paper, we obtain three unified fractional derivative formulae. The first involves the product of a general class of polynomials and the multivariable Gimel-function. The second involves the product of a general class of polynomials and two multivariable Gimel-functions and has been obtained with the help of the generalized Leibniz rule for fractional derivatives. The last fractional derivative formulae also involves the product of a general class of polynomials and the multivariable Gimel-function but it is obtained by the application of the first fractional derivative formulae twice and, it involve two independents variables instead of one. The polynomials and the functions involved in all our fractional derivative formulae as well as their arguments which are of the type. The formulae are the very general character and thus making them useful in applications. In the end, we shall give a particular case.
Selberg Integral Involving the Product of Multivariable Special Functions
The Selberg integral was an integral first evaluated by Selberg in 1944. The aim of the present paper is to estimate generalized Selberg integral. It involves the product of the general class of multivariable polynomials, multivariable I-function and modified multivariable H-function. The result is believed to be new and is capable of giving a large number of integrals involving a variety of functions and polynomials as its cases. We shall see several corollaries and particular cases at the end.
Unified Fractional Derivative Formulae for the Multivariable Aleph-Function
The object of this paper is to derive three unified fractional derivatives formulae for the Saigo-Maeda operators of fractional integration. The first formula deals with the product of a general class of multivariable polynomials and the multivariable Aleph-function. The second concerns the multivariable polynomials and two multivariable Aleph-functions with the help of the Leibniz rule for fractional derivatives. The last relation also implies the product of a class of multivariable polynomials and the multivariable Aleph-function but it is obtained by the application of the first formula twice and it implicates two independents variables instead of one. The polynomials and the functions have their arguments of the type are quite general nature. These formulae, besides being on very general character have been put in a compact form avoiding the occurrence of infinite series and thus making them put in applications. Our findings provide unifications and extensions of some (known and new) results. We shall give several corollaries and particular cases.
An Unified Study of Some Multiple Integrals
In this paper, we first evaluate unified finite multiple integrals whose integrand involves the product of the generalized hypergeometric function, general class of multivariable polynomials , the series expansion of multivariable A-function, a sequence of functions and the multivariable I-function. The arguments occurring in the integrand involve the product of factors of the form while that of occurring herein involves a finite series of such coefficients. On account of the most general nature of the functions happening in the integrand of our integral, a large number of new and known integrals can be obtained from it merely by specializing the functions and parameters involved here. At the end, we shall see two corollaries.
On a General Class of Multiple Eulerian Integrals with Multivariable I-Functions
Recently, Raina and Srivastava and Srivastava and Hussain have provided closed-form expressions for a number of a Eulerian integral involvingmultivariable H-functions. Motivated by these recent works, we aim at evaluating a general class of multiple Eulerian integrals concerning theproduct of two multivariable I-functions defined by Prathima et al. [6], a class of multivariable polynomials and the spheroidal function. Theseintegrals will serve as a capital formula from which one can deduce numerous integrals.
On a General Class of Multiple Eulerian Integrals with Multivariable Aleph-Functions
Recently, Raina and Srivastava [5] and Srivastava and Hussain [12] have provided closed-form expressions for a number of a Eulerian integral involving multivariable H-functions. Motivated by these recent works, we aim at evaluating a general class of multiple Eulerians integral involving the product of two multivariable Aleph-functions, a class of multivariable polynomials and the general sequence of functions. These integrals will serve as a capital formula from which one can deduce numerous integrals.
On a General Class of Multiple Eulerian Integrals with Multivariable A-Functions
Recently, Raina and Srivastava and Srivastava and Hussain have provided closed-form expressions for a number of an Eulerian integral involving multivariable H-functions. Motivated by these recent works, we aim at evaluating a general class of multiple Eulerian integrals concerning the product of two multivariable A-functions defined by Gautam et Asgar[4], a class of multivariable polynomials and the extension of the Hurwitz-Lerch Zeta function. These integrals will serve as a fundamental formula from which one can deduce numerous useful integrals.
