Abstract
We derive a summation formula for the terminating hypergeometric series \[{}_4F_3\left[\!\!\begin{array}{c}-m,a,b,1+c\\1+a+m,1+a-b,c\end{array}\!\!;1\right],\] where $m$ denotes a nonnegative integer. Using this summation formula, we establish a reduction formula for the Srivastava-Daoust double hypergeometric function with arguments $z$ and $-z$. Special cases of this reduction formula lead to several reduction formulas for the hypergeometric functions ${}_{p+1}F_p$ with quadratic arguments when $p=2,3$ and 4 by employing series rearrangement techniques. A general double series identity involving a bounded sequence of arbitrary complex numbers is also given.
Highlights
In our investigations, we shall Z− ∪ {0} = {0, −1, −2, −3, · ··use the }
In the following corollaries we present some cases where the Srivastava–Daoust function in (4.1) reduces to a generalised hypergeometric function with a quadratic argument which can be expressed in terms of lower-order hypergeometric functions with linear argument
We conclude our present investigation by observing that several further interesting hypergeometric summation formulas for terminating series 4F3(1), reduction formulas for the Gaussian hypergeometric functions 3F2, 4F3 and 5F4 with the argument −Z and general double-series identity can be obtained in an analogous manner
Summary
A natural generalization of the Gaussian hypergeometric series 2F1[α, β; γ; z] is accomplished by introducing an arbitrary number of numerator and denominator parameters. Assuming that none of the numerator and denominator parameters is zero or a negative integer, the pFq(z) function defined by Equation (1.1) converges for |z| < ∞ (p ≤ q ), |z| < 1 ( p = q + 1) and |z| = 1 ( p = q + 1 and R(s) > 0 ), where s is the parametric excess defined by The consideration of special cases of this last result enables a few reduction formulas for the generalised hypergeometric function p+1Fp ( p = 2, 3, 4 ) with quadratic arguments to be deduced using a series rearrangement technique.
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