Some new proofs of modular relations for the Göllnitz-Gordon functions

Abstract

In this paper, we find new proofs of modular relations for the Göllnitz-Gordon functions established earlier by S.-S. Huang and S.-L. Chen. We use Schröter's formulas and some simple theta-function identities of Ramanujan to establish the relations. We also find some new modular relations of the same nature.

New theta-function identities and general theorems for the explicit evaluations of Ramanujan’s continued fractions

“…Yuttanan [19] also proved that For further references on K (q) see Chan and Huang [7], Vasuki and Kumar [18], and Baruah and Saikia [3]. In this paper, we prove some theta-function identities analogous to (1.7)-(1.10) for the continued fractions T (q) and W (q) which are defined, respectively, as T (q) := q 1 − q 2 + q 4 1 − q 6 + q 8 1 − q 10 + · · · , |q| < 1.…”

“…(ii) Setting n = 1 in (5.10), employing in (2.19) and simplifying, we obtain dividing second factor of (5.13) by y 2 and rearranging the terms, we obtain 4,25 , and choosing the appropriate root, we complete the proof of (ii).…”

Modular relations for the nonic analogues of the Rogers–Ramanujan functions with applications to partitions

“…Applying Theorem 6.5 with m = 1 and p = 9, we find that φ α,β,1,9 = 2q (α+β)/24 f −q α f −q β g (9,1) β g (9,1) α + g (9,2) β g (9,2) α + g (9,3) β g (9,3) α + g (9,4) β g (9,4) α .…”

“…They also extracted partition theoretic results from some of their relations. N.D. Baruah et al [2] also found new proofs for the relations which involve only the Göllnitz-Gordon functions by using Schröter's formulas and some theta function identities found in Ramanujan's notebooks [17]. In the process, they also found some new relations.…”

Some Modular Relations Analogues to the Ramanujan’s Forty Identities with Its Applications to Partitions

“…S.-L. Chen and Huang [17] have derived some new modular relations for the Göllnitz-Gordon functions. N. D. Baruah, J. Bora and N. Saikia [18] offered new proofs of many of these identities by using Schröter's formulas and some theta-function identities found in Ramanujan's notebooks, as well as establishing some new relations. Gugg [14] found new proofs of modular relations, which involve only S(q) and T (q).…”