# Top PDF Finite double integrals involving multivariable Aleph-function ### Finite double integrals involving multivariable Aleph-function

In this paper we establish two finite double integrals involving the multivariable Aleph-function with general arguments. Our integrals are quite general in character and a number of new integrals can be deduced as particular cases. Several interesting special cases of our main findings have also been mentioned briefly. We will study the particular cases of multivariable I-function, Aleph-function of two variables and I-function of two variables. ### Generalized Finite Integral Involving the Multiple Logarithm-Function, a General Class of Polynomials,the Multivariable Aleph Function, the Multivariable I-Function II

In the present paper we evaluate a generalized finite integral involving the product of the multiple logarithm function, the multivariable Aleph- function, the multivariable I-function defined by Prasad and general class of polynomials of several variables. The importance of the result established in this paper lies in the fact they involve the Aleph-function of several variables which is sufficiently general in nature and capable to yielding a large of results merely by specializating the parameters their in. ### Generalized finite integral involving a sequence of functions, a general class of polynomials,the multivariable Aleph-function and the multivariable I-function I

The function Aleph of several variables generalize the multivariable I-function recently study by C.K. Sharma and Ahmad , itself is an a generalisation of G and H-functions of multiple variables. The multiple Mellin-Barnes integral occuring in this paper will be referred to as the multivariables Aleph-function throughout our present study and will be defined and represented as follows. ### Finite integral involving the spheroidal function, a class of polynomials multivariable Aleph-functions and Fresnel integral

In this paper we have evaluated a generalized finite integral involving the multivariable Aleph-functions,the Fresnel integral function, a class of polynomials of several variables and the spheroidal function.The integral established in this paper is of very general nature as it contains Multivariable Aleph-function, which is a general function of several variables studied so far. Thus, the integral established in this research work would serve as a key formula from which, upon specializing the parameters, as many as desired results involving the special functions of one and several variables can be obtained. ### Some Integrals Involving Multivariable Gimel-Function

The importance of our all the results lies in their manifold generality. Firstly, in view of general arguments utilized in these double integrals, we can obtain a large simpler double or single finite integrals, Secondly by specialising the various parameters as well as variables in the generalized multivariable Gimel-function, we get a several formulae involving remarkably wide variety of useful functions ( or product of such functions) which are expressible in terms of E, F, G, H, I, Aleph-function of one and several variables and simpler special functions of one and several variables. Hence the formulae derived in this paper are most general in character and may prove to be useful in several intersting cases appearing in literature of Pure and Applied Mathematics and Mathematical Physics. ### Certain finite finite integral involving sequence of functions, a general class of polynomials and multivariable Aleph-functions IV

In this paper we have evaluated a finite integral involving the multivariable Aleph-functions, a class of polynomials of several variables and the general of sequence of functions.The integral established in this paper is of very general nature as it contains Multivariable Aleph-function, which is a general function of several variables studied so far. Thus, the integral established in this research work would serve as a key formula from which, upon specializing the parameters, as many as desired results involving the special functions of one and several variables can be obtained. ### Certain finite finite integral involving sequence of functions, a general class of polynomials and multivariable Aleph-functions V

In the present paper we evaluate a finite integral with involving the product of sequence of functions, a hyperbolic sinus function of general argument, product of two multivariable Aleph-functions and general class of polynomials of several variables. The importance of the result established in this paper lies in the fact they involve the Aleph-function of several variables which is sufficiently general in nature and capable to yielding a large of results merely by specializating the parameters their in. ### Generalized Elliptic-Type Integrals and Generating Functions with Multivariable Aleph-Function

which includes most of the known generalized and unified families of elliptic type integrals (including those discussed in (1.1) through (1.9)). For more details also see [17 ,27, 26, 1, 2, 24]. Upon a closer examination of the above equation. (1.10), it can be seen that the family of elliptic-type integral can be put in to the following form involving Euler-type integral: ### Multiple Integrals Involving A Extension of The Hurwitz-Lerch Zeta Function, Class of Pol,Ynomials, Multivariable I-Function, Multivariable Aleph-Function and Product of Two Jacobi Polynomials

The function Aleph of several variables is an extension the multivariable I-function recently studied by C.K. Sharma and Ahmad  , itself is a generalisation of G and H-functions of multiple variables. The multiple Mellin-Barnes integral occurring in this paper will be referred to as the multivariables Aleph-function throughout our present study and will be defined and represented as follows. ### Some Finite Double Integral Formulae Involving Generalized Multivariable Gimel-Function

To prove (3.1), on the lef hand side of (2.1), using the series representation of with the help of (1.15) and expressing the generalized multivariable Gimel-function as Mellin-Barnes multiple integrals contour with the the help of (1.1), interchanging the order of summation and integration which is justifed under the conditions mentioned above, we get (say I ) ### A Study of Unified Finite Integrals Involving Generalized Modified Bessel Function of Third Kind, (λŋµ,ʋ) General Class of Polynomials and the Multivariable Gimel-Function

The integral formulae involving in this paper are double fold generality in term of variables and parameters. By specializing the various parameters and variables involved, these formulae can suitably be applied to derive the corresponding results involving wide variety of useful functions (or product of several such functions) which can be expressed in terms of E, F, G, H, I, Aleph-function of one and several variables and simpler special functions of one and several variables. Hence the formulae derived in this paper are most general in character and may prove to be useful in several interesting cases appearing in literature of Pure and Applied Mathematics and Mathematical Physics. ### Integral involving a generalized multiple-index Mittag-Leffler function,hyperbolic functions, a class of polynomials multivariable Aleph-function and multivariable I-function I

In this paper we have evaluated a generalized finite integral involving the generalized multiple-index Mittag-Leffler function, the hyperbolic functions, the multivariable Aleph-function, a class of polynomials of several variables a sequence of functions and the multivariable I-function defined by Prasad. The integral established in this paper is of very general nature as it contains Multivariable Aleph-function, which is a general function of several variables studied so far. Thus, the integral established in this research work would serve as a key formula from which, upon specializing the parameters, as many as desired results involving the special functions of one and several variables can be obtained. ### On integration of certain products involving general polynomials, Aleph-function and the multivariable Aleph-function

We use the series form concerning the polynomial of several variables and Aleph-function of one variable with help of (1.1) and (1.5) respectively and expressing the multivariable Aleph-function in multiple Mellin-Barnes integrals. Interchange the series and Mellin-Barnes integrals due to absolute convergence of the series and integrals involved in due to absolute convergence of the series and integrals involved in the process. Now evaluate the inner ### On Finite Double Integrals Involving a General Multivariable Polynomial and Generalized Multivariable Gimel-Function

To establish the integral (2.1), we first use the series representation of the multivariable polynomial with the help of (1.13) and express the generalized multivariable Gimel-function as Mellin-Barnes multiple integrals contour with the the help of (1.1), interchanging the order of summation and integration which is justified under the conditions mentioned above, we have (say I ) ### Some Finite Integrals Of Generalized Polynomial Sets And The Multivariable Aleph -Function With Applications

can also be derived from the general sequence of function defined by Agarwal and Choubey , it unifies and extends a number of classical polynomials studied by various research workers such as Gould and Hooper , Gradshtiyn and Ryshk , Krall and Frink , Singh and Srivastava  etc. Moreover, it can be expressed in the following series form as: ### A Finite Double Integral Involving a General Multivariable Polynomial and Multivariable Gimel-Function

The importance of our all the results lies in their manifold generality. Firstly, in view of general arguments utilized in these double integrals, we can obtain a large simpler double or single finite integrals. Secondly by specialising the various parameters as well as variables in the generalized multivariable polynomials, we obtain a large number of formulae involving simpler special functions ( ultraspherical -Gegenbauer, Legendre, Tchebyshev, Bateman’s, Hermite, Laguerre polynomials and others). Thirdy by specialising the various parameters as well as variables in the multivariable Gimel-function, we get a several formulae involving remarkably wide variety of useful functions ( or product of such functions) which are expressible in terms of E, F, G, H, I, Aleph-function of one and several variables and simpler special functions of one and several variables. Hence the formulae derived in this paper are most general in character and may prove to be useful in several intersting cases appearing in literature of Pure and Applied Mathematics and Mathematical Physics. ### Multiple Integrals Transformation about The Generalized Incomplete Hypergeometric Function,a General Class of Polynomials and The Multivariable Aleph-Functions

In the present paper we evaluate a generalized multiple integrals transformation involving the product of rhe generalized incomplete hypergeometric function, the multivariable Aleph-functions, and general class of polynomials of several variables. The importance of the result established in this paper lies in the fact they involve the Aleph-function of several variables which is sufficiently general in nature and capable to yielding a large of results merely by specializating the parameters their in.. We shall the case concerning the multivariable I-function defined by Sharma and Sharma . Keywords:Multivariable Aleph-function, general class of polynomials, multiple integrals, generalized incomplete hypergeometric function, multivariable I-function, multivariable H-function ### Double Integrals Involving Multivariable Gimel-Function, General Class of Polynomials and Biorthogonal Polynomial

The importance of our all the results lies in their manifold generality. Firstly, in view of general arguments utilized in these double integrals, we can obtain a large simpler double or single finite integrals. Secondly by specialising the various parameters as well as variables in the multivariable Gimel-function, we get a several formulae involving remarkably wide variety of useful functions ( or product of such functions) which are expressible in terms of E, F, G, H, I, Aleph-function of one and several variables and simpler special functions of one and several variables. Hence the formulae derived in this paper are most general in character and may prove to be useful in several intersting cases appearing in literature of Pure and Applied Mathematics and Mathematical Physics.  