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...

The mathematical properties of the generalized hypergeometric function are now well established, due largely to the efforts of Bailey, Watson, Slater, and many others. There are available a number of books on the subject ['], [6] which contain extensive bibliographies together with an outline of the history of the investigations. A large portion of the published work concerning these functions ...

We survey the applications of an elementary identity used by Euler in one of his proofs of the Pentagonal Number Theorem. Using a suitably reformulated version of this identity that we call Euler’s Telescoping Lemma, we give alternate proofs of all the key summation theorems for terminating Hypergeometric Series and Basic Hypergeometric Series, including the terminating Binomial Theorem, the Ch...

We present the fermionic representation for the q-deformed hypergeometric functions related to Schur polynomials. For q = 1 it is known that these hypergeometric functions are related to zonal spherical polynomials for GL(N,C)/U(N) symmetric space. We show that multivariate hypergeometric functions are tau-functions of the KP and of the two-dimensional Toda lattice hierarchies. The variables of...

We introduce gamma structures on regular hypergeometric D–modules in dimension 1 as special one–parametric systems of solutions on the compact subtorus. We note that a balanced gamma product is in the Paley–Wiener class and show that the monodromy with respect to the gamma structure is expressed algebraically in terms of the hypergeometric exponents. We compute the hypergeometric monodromy expl...