Gosper's Lambert series identities of level $14$

TL;DR

This work advances Gosper's Lambert series identities by deriving two level-$14$ identities involving the $q$-constant $Π_q$ through a special case of Bailey's $_6ψ_6$ summation and a modular-function framework on $Γ_0(14)$. It constructs auxiliary quantities $z$, $g$ and related generalized $η$-quotients to obtain algebraic relations via genus-one cusp-order analysis, leading to explicit cubic polynomial relations between the Lambert-series variables. The main results are two cubic relations that encode the level-$14$ identities, demonstrated through elimination of modular-function variables and careful q-series expansions. The methods illustrate how genus considerations on $Γ_0(N)$ guide polynomial elimination and extend the modular-function approach to Gosper-type Lambert series identities beyond previously treated levels.

Abstract

We derive two Gosper's Lambert series identities of level $14$ which involve the $q$-constant $Π_q$ using a special case of Bailey's $_6ψ_6$ summation formula and certain propeties of $η$-quotients and generalized $η$-quotients on the congruence subgroup $Γ_0(14)$.

Theorems & Definitions (24)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Remark 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5