Abstract
In the present work, we prove few new Dedekind eta-function identities of level 6 discovered by Somos in two different methods. Also during this process, we give an alternate method to Somos’s Dedekind eta-function identities of level 6 proved by B. R. Srivatsa Kumar et. al. As an application of this, we establish colored partition identities.
Highlights
Throughout the paper, we use the standard q-series notation and f (−qk) is defined asY ∞ f (−qk) :=∞ = (1 − qnk) n=1 k ∈ Z, |q| < 1.Ramanujan’s modular equation involve quotients of the function f (−qk) at certain arguments
Somos [14] used computer to discover around 6277 new elegant Dedekind eta-function identities of various levels without offering the proof
He has a large list of eta-product identities and he runs PARI/GP scripts and it works as a sophisticated programmable calculator
Summary
Throughout the paper, we use the standard q-series notation and f (−qk) is defined as. Ramanujan’s modular equation involve quotients of the function f (−qk) at certain arguments. Somos [14] used computer to discover around 6277 new elegant Dedekind eta-function identities of various levels without offering the proof. He has a large list of eta-product identities and he runs PARI/GP scripts and it works as a sophisticated programmable calculator. The purpose of this paper is to prove Somos’s Dedekind eta-function identities of level 6 by employing Ramanujan’s modular equations of degree 3 in two different methods. We provide an alternate method to Somos’s Dedekind eta-function identities of level 6 proved by Srivatsa Kumar et al [17].
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