Continued fraction proofs of m-versions of some identities of Rogers–Ramanujan–Slater type

Abstract

We derive two general transformations for certain basic hypergeometric series from the recurrence formulae for the partial numerators and denominators of two q-continued fractions previously investigated by the authors.By then specializing certain free parameters in these transformations, and employing various identities of Rogers-Ramanujan type, we derive m-versions of these identities. Some of the identities thus found are new, and some have been derived previously by other authors, using different methods.By applying certain transformations due to Watson, Heine and Ramanujan, we derive still more examples of such m-versions of Rogers-Ramanujan-type identities.

On some continued fraction expansions of the Rogers–Ramanujan type

“…s 2k = q 8k 2 +4k n≥0 z n q n 2 +2n(2k+1) (−q 8 ; q 8 ) k−1 (q 2 ; q 2 ) n (−q 2 ; q 2 ) n+4k−1 (−q 4 ; q 8 ) k , k ≥ 1, s 2k+1 = q 8k 2 +12k+4 n≥0 z n q n 2 +4n(k+1) (−q 4 ; q 8 ) k (q 2 ; q 2 ) n (−q 2 ; q 2 ) n+4k+1 (−q 8 ; q 8 ) k , z 2 G(z) F (z) = z 2 1 + zq + z 2 q 4 (1 + q 4 ) (1 + q 2 )(1 + q 4 ) − zq 7 + z 2 q 12 (1 + q 8 ) (1 + q 4 )(1 + q 6 )(1 + q 8 ) + zq 5 (1 − q 12 ) + z 2 q 20 (1 + q 4 )(1 + q 12 ) (1 + q 8 )(1 + q 10 )(1 + q 12 ) + zq 7 (1 − q 20 ) + z 2 q 28 (1 + q 8 )(1 + q 16 ) (1 + q 12 )(1 + q 14 )(1 + q 16 ) + zq 9 (1 − q 28 ) + · · · + z 2 q 8n+4 (1 + q 4n−4 )(1 + q 4n+4 ) (1 + q 4n )(1 + q 4n+2 )(1 + q 4n+4 ) + zq 2n+3 (1 − q 8n+4 ) + · · · . Theorem 4.5 Let…”

“…Some related kinds of continued fraction identity of Ramanujan were widely studied in the literature, see [2-8, 13-16, 22, 23]. Recently, in [9], Bowman et al used two q-continued fractions given in [8] to derive two general transformations, and then obtained m-versions of several identities of Rogers-Ramanujan-Slater type. Inspired by the idea in [27], we find that the method which resembles the method due to Viskovatoff [20, pp.…”