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On asymptotics for lacunary partition functions

Alexander E Patkowski

Abstract

We give asymptotic analysis of power series associated with lacunary partition functions. New partition theoretic interpretations of some basic hypergeometric series are offered as examples.

On asymptotics for lacunary partition functions

Abstract

We give asymptotic analysis of power series associated with lacunary partition functions. New partition theoretic interpretations of some basic hypergeometric series are offered as examples.

Paper Structure

This paper contains 3 sections, 6 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Proofs of Theorems
  3. Conclusion

Key Result

Lemma \oldthetheorem

([3]) Recall $\Gamma(\delta)$ denotes the classical Gamma function. Let $(a_n)$ be a real sequence, and $\sum_{n\le x}a_n\sim x^{\delta}h(x),$$\delta$ positive, as $x\rightarrow\infty.$ Suppose $x^{\delta}h(x)$ tends to a positive value or $+\infty$ when $x\rightarrow\infty.$ Then, as $z\rightarrow1.$

Theorems & Definitions (10)

  • Lemma \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Lemma \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • proof : Proof of Theorem \ref{['thm:thm1']}
  • proof : Proof of Theorem \ref{['thm:thm2']}
  • proof : Proof of Theorem \ref{['thm:thm3']}