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.

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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.

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The **function** **Aleph** of several variables generalize the **multivariable** I-**function** recently study by C.K. Sharma and Ahmad [7], 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.

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.

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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.

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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.

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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.

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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:

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The **function** **Aleph** of several variables is an extension the **multivariable** I-**function** recently studied by C.K. Sharma and Ahmad [3] , 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.

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 )

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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.

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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.

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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

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 )

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can also be derived from the general sequence of **function** defined by Agarwal and Choubey [1], it unifies and extends a number of classical polynomials studied by various research workers such as Gould and Hooper [3], Gradshtiyn and Ryshk [4], Krall and Frink [5], Singh and Srivastava [7] etc. Moreover, it can be expressed in the following series form as:

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.

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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 [2]. Keywords:Multivariable **Aleph**-**function**, general class of polynomials, multiple **integrals**, generalized incomplete hypergeometric **function**, **multivariable** I-**function**, **multivariable** H-**function**

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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.

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To prove the theorems 1 and 2, expressing the **multivariable** Beth-**function** with the help of (1.1), interchanging the order of integrations which is justified under the conditions stated with the **integrals**, evaluating the inner integral with the help of lemmae 5 and 6 respectively and Interpreting the resulting expression with the help of (1.1), we obtain the desired theorems 1 and 2.

The I-**function** of several variables defined by Prasad [4] presented in this paper, is quite basic in nature. Therefore , on specializing the parameters of this **function**, we may obtain various other special functions o several variables such as **multivariable** I-**function** ,**multivariable** Fox's H-**function**, Fox's H-**function** , Meijer's G-**function**, Wright's generalized Bessel **function**, Wright's generalized hypergeometric **function**, MacRobert's E-**function**, generalized hypergeometric **function**, Bessel **function** of first kind, modied Bessel **function**, Whittaker **function**, exponential **function** , binomial **function** etc. as its special cases, and therefore, various unified integral presentations can be obtained as special cases of our results

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