A generalization of a modular identity of Rogers

Abstract

In a handwritten manuscript published with his lost notebook, Ramanujan stated without proofs forty identities for the Rogers-Ramanujan functions. Most of the elementary proofs given for these identities are based on Schröter-type theta function identities in particular, the identities of L.J. Rogers. We give a generalization of Rogers's identity that also generalizes similar formulas of H. Schröter, and of R. Blecksmith, J. Brillhart, and I. Gerst. Applications to modular equations, Ramanujan's identities for the Rogers-Ramanujan functions as well as new identities for these functions are given. © 2009 Elsevier Inc. All rights reserved

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

“…To prove some of our results, we use Corollary 3.2 found in [34]. Following Yesilyurt [34], we define…”

Some new modular relations for the Rogers–Ramanujan type functions of order eleven with applications to partitions

“…236-237]. In the years since then, the 40 identities have been established in a series of papers by Darling [12] in 1921, Rogers [18] in 1921, Watson [19] in 1933, Bressoud [11] in 1977, Biagioli [9] in 1989 and Yesilyurt [20,21] in 2009 and 2012. Although all the forty identities can be proved by using the theory of modular forms, the method employed by Biagioli [9] is more instructive to find proofs that Ramanujan might have found.…”

“…Although all the forty identities can be proved by using the theory of modular forms, the method employed by Biagioli [9] is more instructive to find proofs that Ramanujan might have found. Outside the theory of modular forms, Roger's method in [18], which was generalized by Bressoud [11] and also extended by Yesilyurt [20,21], is the only general method that has been devised for proving identities from Ramanujan's list. For other details, proofs and further references, see the excellent monograph by Berndt et al [8].…”

On Rogers–Ramanujan functions, binary quadratic forms and eta-quotients

“…Then, by[15, Th. 1] with x = y = 1 and q replaced by q 1/2 , we have(2.12) R(ǫ, δ, l, t, α, β, m, p, λ) = ∞ u,v=−∞ (−1) δv+ǫu q (λU 2 +2αmUV +pαV 2 )/8 ,where U := 2u + t and V := 2v + l. From this representation it follows that [15, Cor.…”

“…15) and (2.11), we have(2,1,9) = (9, 2, 1) = R(0, 0, 0, 0, 1, 71, 1, 4, 18)= ϕ(q 2 )ϕ(Q 2 ) + 2q 9 ψ(q)ψ(Q) + 4q 36 ψ(q 4 )ψ(Q 4 ). (4.28) From (2.15), (2.13), and (2.11), we deduce that (3, 1, 6) = R(0, 0, 0, 0, 1, 71, 1, 12, 6) = R(0, 0, 0, 0, 1, 71, −5, 16, 6) = ϕ(q 8 )ϕ(Q 8 ) + 2q 3 f (q 3 , q 13 )f (Q 7 , Q 9 ) + 2q 10 f (q 2 , q 14 )f (Q 6 , Q 10 ) + 2q 20 f (q 7 , q 9 )f (Q 5 , Q 11 ) + 2q 36 ψ(q 4 )ψ(Q 4 ) + 2q 57 f (q, q 15 )f (Q 3 , Q 13 ) + 2q 80 f (q 6 , q 10 )f (Q 2 , Q 14 ) + 2q 109 f (q 5 , q 11 )f (Q, Q 15 ) + 4q 144 ψ(q 16 )ψ(Q 16 ).…”