We consider the applicability (or terminating condition) of the well-known Zeilberger’s algorithm and give the complete solution to this problem for the case where the original hypergeometric term F(n; k) is a rational function. We specify a class of identities ∑n k=0 F(n; k) = 0; F(n; k)∈C(n; k), that cannot be proven by Zeilberger’s algorithm. Additionally, we give examples showing that the s...

In 1992, Koepf [J. Symb. Comp. 13 (1992) 581] introduced an algorithm for computing a Formal Power Series of a given function using generalized hypergeometric series and a recurrence equation of hypergeometric type. The main aim of this paper is to develop a new algorithm for computing exact power series solutions of second order linear differential equations with polynomial coefficients, near ...

The main object of this paper is to give a di erent approach to proof of generating matrix functions for Laguerre matrix polynomials. We also obtain the hypergeometric matrix representations, addition theorem, nite summation formula and an integral representation for Laguerre matrix polynomials. We get the relations between Laguerre, Legendre and Hermite matrix polynomials. We get the generatin...

which has since been known as Kummer's theorem. This appears to be the simplest relation involving a hypergeometric function with argument ( — 1). All the relations in the theory of hypergeometric series rF8 which have analogues in the theory of basic series are those in which the argument is ( + 1 ) . Apparently, there has been no successful at tempt to establish the basic analogue of any form...

Nonpolynomial basic hypergeometric eigenfunctions of the Askey–Wilson second order difference operator are known to be expressible as very-well-poised 8φ7 series. In this paper we use this fact to derive various basic hypergeometric and theta function identities. We relate most of them to identities from the existing literature on basic hypergeometric series. This leads for example to a new der...

We continue our study of the construction of analytical coefficients of the epsilon-expansion of hypergeometric functions and their connection with Feynman diagrams. In this paper, we apply the approach of obtaining iterated solutions to the differential equations associated with hypergeometric functions to prove the following result: Theorem 1: The epsilon-expansion of a generalized hypergeome...

We describe a multiplicative normal form for rational functions which exhibits the shift structure of the factors, and investigate its properties. On the basis of this form we propose an algorithm which, given a rational function R, extracts a rational part F from the product of consecutive values of R: ∏n−1 k=n0 R(k) = F (n) ∏n−1 k=n0 V (k) where the numerator and denominator of the rational f...

The eastern society is rich in terms of science and technology. Mathematics considered as the base science. history shows that Hindu mathematics. evolution mathematics can be studied from time ‘patiganita’ to latest form Geometric series an important tool arithmetic which now developed hypergeometric function. Hypergeometric function advance used solve differential equations second order. purpo...

The transformation which assigns to a linear operator L the recurrence satis ed by coe cient se quences of the polynomial series in its kernel is shown to be an isomorphism of the corresponding operator algebras We use this fact to help factoring q di erence and recurrence operators and to nd nice power series solutions of linear di erential equations In particular we characterize generalized h...

We define a function in terms of quotients of the p-adic gamma function which generalizes earlier work of the author on extending hypergeometric functions over finite fields to the p-adic setting. We prove, for primes p > 3, that the trace of Frobenius of any elliptic curve over Fp, whose jinvariant does not equal 0 or 1728, is just a special value of this function. This generalizes results of ...