From numerical experiments, D. E. Knuth conjectured that 0 < Dn+4 < Dn for a combinatorial sequence (Dn) defined as the difference Dn = Rn − Ln of two definite hypergeometric sums. The conjecture implies an identity of type Ln = bRnc, involving the floor function. We prove Knuth’s conjecture by applying Zeilberger’s algorithm as well as classical hypergeometric machinery.

The main purpose of this article is to solve the connection problems between (p, q)−Fibonacci polynomials and the two polynomials, namely Chebyshev polynomials of third and fourth kinds which are considered as two nonsymmetric polynomials of the Jacobi polynomials. Moreover, the inversion connection formulae for the latter formulae are given. We show that all the connection coefficients are exp...

We define a hypergeometric function over finite fields which is an analogue of the classical generalized hypergeometric series. We prove that this function satisfies many transformation and summation formulas. Some of these results are analogous to those given by Dixon, Kummer and Whipple for the well-poised classical series. We also discuss this function’s relationship to other finite field an...

The generalized hypergeometric function has three singular points 0, 1, and ∞. Connection problem between fundamental solutions at 0 ∞ was completely solved. However, connection 1 been not solved because of holomorphic 1. In this paper, we solve for the function, using analytic continuation integral representation. Namely, all coefficients are products sine cosecant. other words, give a good sy...

Abstract We provide generalizations of two of Euler’s classical transformation formulas for the Gauss hypergeometric function extended to the case of the generalized hypergeometric function r+2Fr+1(x) when there are additional numeratorial and denominatorial parameters differing by unity. The method employed to deduce the latter is also implemented to obtain a Kummer-type transformation formula...