The main object of this paper is to consider the asymptotic distribution of the zeros of certain classes of the Clausenian hypergeometric 3F2 functions and polynomials. Some classical analytic methods and techniques are used here to analyze the behavior of the zeros of the Clausenian hypergeometric polynomials: 3F2(−n, τn+ a, b; τn+ c,−n+ d; z), where n is a nonnegative integer. Some families o...

The aim of this paper is to obtain explicit expressions of the generalized hypergeometric function r+2Fr+1 [ a, b, 1 2 (a+ b+ j + 1), (fr +mr) (fr) ; 1 2 ] for j = 0,±1, . . . ,±5, where r pairs of numeratorial and denominatorial parameters differ by positive integers mr. The results are derived with the help of an expansion in terms of a finite sum of 2F1( 1 2 ) functions and a generalization ...

We derive extensions of the classical summation theorems of Kummer and Watson for the generalized hypergeometric series where r pairs of numeratorial and denominatorial parameters differ by positive integers. The results are obtained with the help of a generalization of Kummer’s second summation theorem for the 2F1 series given recently by Rakha and Rathie [Integral Transforms and Special Funct...

In this short communication, we present a new limit relation that reduces pseudo-Jacobi polynomials directly to Hermite polynomials. The proof of is based upon 2F1-type hypergeometric transformation formulas, which are applicable even and odd separately. This opens the way studying exactly solvable harmonic oscillator models in quantum mechanics terms

We prove a general identity for a 3F2 hypergeometric function over a finite field Fq, where q is a power of an odd prime. A special case of this identity was proved by Greene and Stanton in 1986. As an application, we prove a finite field analogue of Clausen’s Theorem expressing a 3F2 as the square of a 2F1. As another application, we evaluate an infinite family of 3F2(z) over Fq at z = −1/8. T...

In the paper, the authors establish two identities to express the generating function of the Chebyshev polynomials of the second kind and its higher order derivatives in terms of the generating function and its derivatives each other, deduce an explicit formula and an identities for the Chebyshev polynomials of the second kind, derive the inverse of an integer, unit, and lower triangular matrix...

We give elementary derivations of some classical summation formulae for bilateral (basic) hypergeometric series. In particular, we apply Gauß’ 2F1 summation and elementary series manipulations to give a simple proof of Dougall’s 2H2 summation. Similarly, we apply Rogers’ nonterminating 6φ5 summation and elementary series manipulations to give a simple proof of Bailey’s very-well-poised 6ψ6 summ...

In previous work we introduced and studied a function R(a+, a−, c; v, v̂) that is a generalization of the hypergeometric function 2F1 and the Askey–Wilson polynomials. When the coupling vector c ∈ C is specialized to (b, 0, 0, 0), b ∈ C, we obtain a function R(a+, a−, b; v, 2v̂) that generalizes the conical function specialization of 2F1 and the q-Gegenbauer polynomials. The function R is the joi...

The well-known Bailey’s transform is extended. Using the extended transform, we derive hitherto undiscovered ordinary and q-hypergeometric identities and discuss their particular cases of importance, namely, two new q-sums for Saalschützian 4Φ3, new double series Rogers-Ramanujan-type identities of modulo 81, discrete extension of the q-analogs of two quadratic transformations of 2F1, and two n...