Watson’s Basic Analogue of Ramanujan’s Entry $40$ and Its Generalization

Abstract

Abstract. We generalize Watson's q-analogue of Ramanujan's Entry 40 continued fraction by deriving solutions to a 10 φ 9 series contiguous relation and applying Pincherle's theorem. Watson's result is recovered as a special terminating case, while a limit case yields a new continued fraction associated with an 8 φ 7 series contiguous relation.

Essays on the theory of elliptic hypergeometric functions

“…For fixed values of parameters, in the limit p → 0 one obtains the terminating continued fraction of Gupta and Masson [77] (see Corollary 3.3) described by a very-well poised balanced 10 ϕ 9 -series. Further specification of parameters leads to the continued fraction of Watson which, in its turn, is a q-analogue of the famous Ramanujan continued fraction (see the details in [78]). …”

Flowers Which We Cannot Yet See Growing in Ramanujan’s Garden of Hypergeometric Series, Elliptic Functions, and q’S

“…G. N. Watson [63] and D. P. Gupta and Masson [36] have established q-analogues of Ramanujan's most general theorem on continued fractions for quotients of gamma functions [6, p. 163, Entry 40]. In some cases, we know q-continued fractions which share the same features, but which apparently are not q-analogues.…”

The last of the hypergeometric continued fractions

“…The proof of this contiguous relation is based on extending the contiguous relations derived in [5] for a terminating φ to nonterminating complementary pairs Φ . This is made possible by using the transformation in exercise 2.30 of Gasper and Rahman [2] to replace the transformation of exercise 2.19 used in [5].…”

“…This is made possible by using the transformation in exercise 2.30 of Gasper and Rahman [2] to replace the transformation of exercise 2.19 used in [5].…”

“…There is a remarkable hidden symmetry that yields additional solutions [5]. If we make the replacements…”