[Paper Review] Further (p,q)-identities related to divisor functions

Victor J. W. Guo, Jiang Zeng|arXiv (Cornell University)|Dec 23, 2013

Advanced Mathematical Identities24 references3 citations

TL;DR

This paper introduces a novel generalization of (p,q)-identities connected to divisor functions using basic hypergeometric functions and partial fraction decomposition, extending prior work by Uchimura, Dilcher, Van Hammer, Prodinger, and Chen-Fu. It establishes a new identity linking Lambert series to Eulerian polynomials, offering deeper algebraic structure in q-series and divisor function theory.

ABSTRACT

Using basic hypergeometric functions and partial fraction decomposition we give a new kind of generalization of identities due to Uchimura, Dilcher, Van Hammer, Prodinger, and Chen-Fu related to divisor functions. An identity relating Lambert series to Eulerian polynomials is also proved.

Motivation & Objective

• To extend known (p,q)-identities related to divisor functions using advanced special functions.
• To generalize identities previously established by Uchimura, Dilcher, Van Hammer, Prodinger, and Chen-Fu.
• To explore connections between Lambert series and Eulerian polynomials in the context of divisor functions.
• To provide a unified framework using basic hypergeometric functions and partial fraction decomposition for such identities.

Proposed method

• Employing basic hypergeometric functions to construct generalized (p,q)-identities involving divisor functions.
• Applying partial fraction decomposition to analyze and derive new identities from rational functions in q-series.
• Using generating function techniques to relate divisor sums to polynomial structures.
• Deriving identities that link Lambert series expressions to Eulerian polynomials through algebraic manipulation.
• Establishing identities valid for formal power series in p and q, with convergence considerations in the context of q-series.
• Leveraging known results on divisor functions to build new identities with enhanced generality.

Experimental results

Research questions

• RQ1 How can basic hypergeometric functions be used to generalize existing (p,q)-identities involving divisor functions?
• RQ2 What new connections exist between Lambert series and Eulerian polynomials in the context of divisor sums?
• RQ3 In what ways do partial fraction decompositions facilitate the derivation of new identities in q-series?
• RQ4 Can the identities of Uchimura, Dilcher, Van Hammer, Prodinger, and Chen-Fu be extended to a broader (p,q)-framework?
• RQ5 What structural insights emerge from relating divisor functions to Eulerian polynomials via Lambert series?

Key findings

• A new class of generalized (p,q)-identities for divisor functions is derived using basic hypergeometric functions and partial fraction decomposition.
• The paper establishes a direct identity linking Lambert series to Eulerian polynomials, extending known results in q-series theory.
• The generalized identities subsume and extend previous results by Uchimura, Dilcher, Van Hammer, Prodinger, and Chen-Fu.
• The method provides a systematic approach to constructing identities involving divisor functions in (p,q)-settings.
• The connection between Lambert series and Eulerian polynomials reveals deeper algebraic symmetry in divisor sum identities.
• The framework enables the derivation of identities valid in formal power series rings, with implications for modular forms and arithmetic functions.