[Paper Review] Some Observations on Lambert series, vanishing coefficients and dissections of infinite products and series

James Mc Laughlin|arXiv (Cornell University)|Jun 27, 2019

Advanced Mathematical Identities15 references4 citations

TL;DR

This paper demonstrates that vanishing coefficients and m-dissections in infinite q-products—previously proven via complex series manipulations—follow directly from parameter specialization in a single identity derived from Ramanujan's ${}_1\psi_1$ summation. It establishes new q-series identities, including parameterized expansions involving an integer m, and applies the method to the Fine function, unifying and simplifying prior results.

ABSTRACT

Andrews and Bressoud, Alladi and Gordon, and others, have proven, in a number of papers, that the coefficients in various arithmetic progressions in the series expansions of certain infinite $q$-products vanish. In the present paper it is shown that these results follow automatically (simply by specializing parameters) in an identity derived from a special case of Ramanujan's $_1ψ_1$ identity. Likewise, a number of authors have proven results about the $m$-dissections of certain infinite $q$-products using various methods. It is shown that many of these $m$-dissections also follow automatically (again simply by specializing parameters) from this same identity alluded to above. Two identities that mat be considered as extensions of two Identities of Ramanujan are also derived. It is also shown how applying similar ideas to certain other Lambert series gives rise to some rather curious $q$-series identities, such as, for any positive integer $m$, \begin{multline*} {\displaystyle \frac{\left(q,q,a,\frac{q}{a},\frac{b q}{d}, \frac{dq}{b}, \frac{aq}{b d}, \frac{b d q}{a};q ight)_{\infty }} {\left(b,\frac{q}{b},d,\frac{q}{d},\frac{a}{b},\frac{bq}{a},\frac{a}{d},\frac{dq}{a};q ight)_{\infty }}} = \sum _{r=0}^{m-1} q^r \frac{ \left(q^m,q^m,a q^{2 r},\frac{q^{m-2 r}}{a},\frac{b q^m}{d},\frac{d q^m}{b}, \frac{a q^m}{b d},\frac{b dq^m}{a};q^m ight){}_{\infty }} {\left(b q^r,\frac{q^{m-r}}{b},d q^r,\frac{q^{m-r}}{d},\frac{a q^r}{b},\frac{b q^{m-r}}{a},\frac{a q^r}{d}, \frac{dq^{m-r}}{a};q^m ight){}_{\infty }} \end{multline*} and \begin{equation*} (aq;q)_{\infty}\sum_{n=1}^{\infty} \frac{n a^n q^{n}}{(q;q)_n} = \sum_{r=1}^{m}(aq^{r};q^m)_{\infty}\sum_{n=1}^{\infty} \frac{na^n q^{n r}}{(q^m;q^m)_n}. \end{equation*} Applications to the Fine function $F(a,b;t)$ are also considered.

Motivation & Objective

• To unify and simplify existing proofs of vanishing coefficients in arithmetic progressions of q-product expansions.
• To show that m-dissections of infinite q-products follow automatically from parameter specialization in a single general identity.
• To derive new, unusual q-series identities involving an integer parameter m.
• To extend the method to the Fine function F(a,b;t) and related hypergeometric series.
• To demonstrate that complex results in q-series can be derived systematically from a single foundational identity.

Proposed method

• Utilizes a special case of Ramanujan’s ${}_1\psi_1$ summation formula as the core identity.
• Applies parameter specialization to derive identities for vanishing coefficients and m-dissections.
• Employs arithmetic progression decomposition via substitution n → nm + r to generate m-dissection identities.
• Applies the same method to Bailey pairs and the ${}_6\psi_6$ summation to derive new q-series expansions.
• Uses functional equations and series rearrangements to connect different q-series representations.
• Applies the framework to the Fine function F(a,b;t), deriving parameterized identities from known Rogers-Fine-type identities.

Experimental results

Research questions

• RQ1 Can vanishing coefficient results in q-products be derived uniformly from a single general identity?
• RQ2 Do m-dissections of infinite q-products follow from parameter specialization in a known summation formula?
• RQ3 What new q-series identities emerge when applying the method to other Lambert series or hypergeometric identities?
• RQ4 Can the method be extended to functions like the Fine function F(a,b;t)?
• RQ5 How do parameterized identities involving an integer m arise from such general frameworks?

Key findings

• The identity \( \frac{(q,q,a,q/a,bq/d,dq/b,aq/bd,bdq/a;q)_\infty}{(b,q/b,d,q/d,a/b,bq/a,a/d,dq/a;q)_\infty} = \sum_{r=0}^{m-1} q^r \frac{(q^m,q^m,aq^{2r},q^{m-2r}/a,bq^m/d,dq^m/b,aq^m/bd,bdq^m/a;q^m)_\infty}{(bq^r,q^{m-r}/b,dq^r,q^{m-r}/d,aq^r/b,bq^{m-r}/a,aq^r/d,dq^{m-r}/a;q^m)_\infty} \) is derived as a new extension of Ramanujan’s identity.
• For any positive integer m, the identity \( (aq;q)_\infty \sum_{n=1}^\infty \frac{na^n q^n}{(q;q)_n} = \sum_{r=1}^m (aq^r;q^m)_\infty \sum_{n=1}^\infty \frac{na^n q^{nr}}{(q^m;q^m)_n} \) holds, providing a parameterized q-series expansion.
• Vanishing coefficient results, such as those in Theorem 1.1, are shown to follow automatically by specializing parameters in the general identity, bypassing prior complex manipulations.
• The 3-dissection of the product $ \frac{(q^7,q^5;q^{12})_\infty}{(q,q^{11};q^{12})_\infty} $ is recovered as a special case, confirming Corollary 1.2.
• Infinite product identities, such as $ \frac{(q^5,q^5;q^5)_\infty}{(q,q^4;q^5)_\infty} = \frac{1}{1-q^2} \prod_{k=1}^\infty \left(1 + q^{2^{k-1}} \cdots \right) $, are derived via iteration of the main identity.
• The equivalence of six different q-series representations of the same function (e.g., $ h_1(a,q) = \cdots = h_6(a,q) $) is established and extended to m-parameter versions.