Finite Integral Formulas Involving Aleph Function

abstract: In the present work we derive various integral formulas involving אfunction multiplied with algebraic functions and special functions.

(1.2) Here Γ denotes the familiar Gamma function; The integration path L = L i γ ∞ (γ ∈ R) extends from γ − i∞ to γ + i∞; The poles of the Gamma functions Γ (1 − a j − A j s) (j, n ∈ N; 1 ≤ j ≤ n) do not coincide with those of Γ (b j + B j s) (j, m ∈ N; 1 ≤ j ≤ m); The parameters p k , q k ∈ N 0 satisfy the conditions 0 ≤ n ≤ p k , 1 ≤ m ≤ q k , τ k > 0 (1 ≤ k ≤ r); The parameters A j , B j , A jk , B jk > 0 and a j , b j , a jk , b jk ∈ C; The empty product in (1.2) is (as usual) understood to be unity.The existence conditions for the defining integral (1.1) are given below: and where 2).Namely, in the ℵ-functions, the kernel Ω m,n p k ,q k ,τ k ; r (s), parameter couples (a j , A j ) 1,n , (b j , B j ) 1,m build the Gamma function terms exclusively in the numerator, and [τ j (a jk , A jk )] n+1,p k , [τ j (b jk , B jk )] n+1,q k build the linear combination exclusively in the denominator, while, for the H m,n p,q [z], both upper (a j , A j ) 1,p and lower couples of parameters (b j , B j ) 1,q play roles in forming both numerator and denominator terms according to m and n.Remark 1.2.Setting τ j = 1 (j ∈ 1, r := {1, 2, . . ., r}) in (1.1) yields the Ifunction (see [19]) whose further special case when r = 1 reduces to the familiar H-function (see [8,9]).

Definition 1. Gamma Function:
The simplest interpretation of the gamma function is simply the generalization of Definition 2. Beta Function: Also known as the Euler Integral of the First Kind, the Beta function B(p, q) is in important relationship in factorial calculus.Its solution not is only defined through the use of multiple Gamma Functions, but furthermore shares a form that is characteristically similar to the Fractional Differintegral of many functions, particularly polynomials of the form t α and the Mittag-Leffler Function.The Beta Integral and its solution in terms of the Gamma function as given following: p−1 t q−1 dt = Γ (p) Γ (q) Γ (p + q) = B (q, p) , (p, q ∈ ℜ) . (1.8)

Integrals involving ℵ-function with Algebraic function
In this section we calculate the ℵ-function with some algebraic functions.
We also have In this section we derive integral formulas involving Aleph function multiplied with Jacobi polynomials.
Next we use the following formula: where α > −1 and β > −1.Also, 3 F 2 is the special case of generalized hypergeometric series.
and with δ = 1 2 − 1 2 z as the independent variable the above differential equation becomes as following:

.3)
Finite Integral Formulas Involving Aleph Function
(4.4) where P µ ν (z) is known as the Legendre function of the first kind [1].Next, we derive the integrals with Legendre function.

Integrals involving ℵ-function and Hypergeometric function
The hypergeometric function defined for c > 0 as [10].
where (a) n , (b) n and (c) n are the Pochhammer symbols which are defined as follows: by putting x = t + 1 ⇒ dx = dt, then we get which provided |arg z| < 1 2 πΩ.

Integrals involving Aleph function and Bessel Maitland function
The Bessel Maitland function (also known as Wright generalized Bessel function) defined as following [2]: x ρ−σs J µ ν (x) dx ds.

Concluding Remarks
The results obtained here are basic in nature and are likely to find useful applications in the study of simple and multiple variable hypergeometric series which in turn are useful in statistical mechanics, electrical networks and probability theory.If we follow Remark 1.2 then all given results can be written in the form of I-function and H-function.

Finite
Integral Formulas Involving Aleph Function 179 the factorial for all real numbers.The definition of gamma function is given by

8 .
Some Special Cases (i).If we replace δ by η − 1 and put µ = ν = ρ = σ = 0, then the integral formula (3.6) transform to the following integral involving product of Legendre Finite Integral Formulas Involving Aleph Function 191 polynomial and Aleph function:

Remark 8 . 1 .
If we set τ j = 1 j ∈ 1, r and put r = 1, m = 1, n = p k = p, q k = q+ 1, b 1 = 0, B 1 = 1, a j = 1−a j , b j k = 1−b j , B j k = B j ,then Aleph function reduces toWright's generalized hypergeometric function p ψ q and we can easily obtain derived integrals in the form of Wright's generalized hypergeometric function.