Asymptotic expansions of the hypergeometric function with two large parameters - Application to the partition function of a lattice gas in a field of traps

Mislav Cvitković, Ana Sunčana Smith, Jayant Pande

Research output: Contribution to journalArticlepeer-review

16 Scopus citations

Abstract

The canonical partition function of a two-dimensional lattice gas in a field of randomly placed traps, like many other problems in physics, evaluates to the Gauss hypergeometric function 2F1(a, b; c; z) in the limit when one or more of its parameters become large. This limit is difficult to compute from first principles, and finding the asymptotic expansions of the hypergeometric function is therefore an important task. While some possible cases of the asymptotic expansions of 2F1 (a, b; c; z) have been provided in the literature, they are all limited by a narrow domain of validity, either in the complex plane of the variable or in the parameter space. Overcoming this restriction, we provide new asymptotic expansions for the hypergeometric function with two large parameters, which are valid for the entire complex plane of z except for a few specific points. We show that these expansions work well even when we approach the possible singularity of 2F1 (a, b; c; z), |z| = 1, where the current expansions typically fail. Using our results we determine asymptotically the partition function of a lattice gas in a field of traps in the different possible physical limits of few/many particles and few/many traps, illustrating the applicability of the derived asymptotic expansions of the HGF in physics.

Original languageEnglish
Article number265206
JournalJournal of Physics A: Mathematical and Theoretical
Volume50
Issue number26
DOIs
StatePublished - 7 Jun 2017
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2017 IOP Publishing Ltd Printed in the UK.

Keywords

  • asymptotic expansion
  • hypergeometric function
  • large parameters
  • method of steepest descent
  • partition function
  • special functions

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