Four Dimensional Joint Moments due to Dirichlet Density and Their Applications in Summability of Quadruple Hypergeometric Functions

abstract: Using the Exton’s multiple joint moments in four dimensional spaces due to Dirichlet density and a generalization of Bosanquet and Kestelman theorem , we prove some theorems in summability of the series containing quadruple hypergeometric functions. These theorems generalize some well known generating functions and multiplication theorems involving product of hypergeometric functions of one and more variables. We discuss some other applications and establish several interesting particular cases. Finally, we obtain an approximation formula of the series involving Exton’s quadruple hypergeometric function K11.

In our investigation, we consider the Exton's quadruple hypergeometric function K 11 (., ., ., .)defined by following Euler type integral formula (see Exton [6]) c, c, d; x, y, z, t) (1.3) For the sake of our present investigation, we define the Dirichlet density (1.2) for four dimensional spaces in the form (1.4) We also present following generalization of Bosanquet and Kestelman Theorem (see [1]): Again, in this region g n (x 1 , ..., x k ) be a sequence of multivariable measurable function and if for any constant η there exists Further, with help of the Eqns.(1.5) and (1.7) we may write  [11].
Here in our work, we introduce and investigate Exton's multiple joint moments in four dimensional spaces due to Dirichlet density and then employ these extended spaces to summability of the series involving quadruple hypergeomtric functions.Relevant connections of the results presented here with those that were obtained in earlier works are also indicated precisely.Finally, in Section 5, we obtain an approximation formula of series involving Exton's quadruple hypergeometric function K 11 .

Four Dimensional Joint Moments for Dirichlet Density and Their
Quadruple Summable Series In this section, on using Eqn.(1.1), we consider four dimensional Exton type joint moments due to Dirichlet density given in Eqn.(1.4) in the following form: Next, we present a theorem to get the sum of the series involving joint moments given in Eqn.(2.1): , also all conditions given in Eqns.(1.4) and (2.1) are satisfied, then the quadruple series
Hence, this is the Theorem.✷

Applications in Summability of the Series Involving Quadruple Hypergeometric Functions
In this section, we obtain the sum of series involving Exton's quadruple hypergeometric function K 11 (., ., ., .) on presenting following theorems:  c, c, c, d is summable and equal to the formula provided that max{|h 1 α|, |h 2 β|, |h 3 γ|} < 1.

4)
Proof: In this Theorem, to get thte result (3.4), we use the technique of Theorem 3.1 and make an appeal to the result due to Srivastava [13] (see also Srivastava and Daoust [15] and Srivastava and Manocha [17, p. 165]).
Here in Eqn.(3.4), F P : R; U Q : S; V x, y is two variable Kampe' de Fe'riet function [9] (see also Srivastava Proof: It is easy to observe that by making an appeal to the result due to Srivastava [14, p. 26 is summable and equal to Theorem 3.5.If all conditions given in Theorem 2.1 are satisfied, then for Dirichlet density defined in (1.4), |T | < 1 and any λ ∈ C, the series is summable and equal to

Other Applications with Special Cases
With a view to describing and illustrating some special cases involving the results of known and unknown bilinear and bilateral functions, we begin this section by presenting the following Theorem: Theorem 4.1.(Converse of Theorem 1.1 in four dimensional spaces).In the region R given by x ≥ 0, y ≥ 0, z ≥ 0, t ≥ 0 and x α + y β + z γ + t δ ≤ 1, let g n (x, y, z, t)be a sequence of four dimensional measurable function due to following Dirichlet measure Take h 2 → 0 in both sides of Eqn.(4.5) to get another bilateral relation Again on replacing b 3 by χ and h 3 by h3 χ and then take χ → ∞, in both sides of Eqn.(4.6), we get Again with the help of the Theorem 4.1 in the Theorems 3.2, 3.3 and 3.4, we may find the generating functions of Srivastava [13] (see also Srivastava and Daoust [15] ), Srivastava [14, p. 26 Eqn.(1.2)], Khan [10, p. 181] respectively.

Approximation Formula
In this section, we obtain an approximation formula for the summation of the series consisting Exton's quadruple hypergeometric function K 11 (., ., ., .).
To obtain this formula we make an appeal to the following theorems due to T. M. Flett [Proc.Edinburgh, Math.Soc. ( 2