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Heine's method and to transformation formulas.

Gaurav Bhatnagar1

  • 1Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria.

The Ramanujan Journal
|July 2, 2019
PubMed
Summary

This study extends Heine's transformation formula to multiple basic series, yielding bibasic extensions and recovering Ramanujan's q-transformations. It also generalizes Andrews' q-Lauricella function, enabling n-fold to m-fold sum transformations.

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Area of Science:

  • Mathematics
  • Combinatorics
  • Special Functions

Background:

  • Heine's transformation formula is fundamental in the theory of basic hypergeometric series.
  • Previous work includes bibasic extensions of Heine's formula and Andrews' q-Lauricella function.

Purpose of the Study:

  • To apply Heine's method to multiple basic series over the root system of type A.
  • To derive bibasic extensions of Heine's transformation formula and related identities.
  • To generalize Andrews' q-Lauricella function and obtain new transformation formulas.

Main Methods:

  • Application of Heine's method to multiple basic series.
  • Derivation of bibasic extensions of Heine's transformation formula.
  • Generalization of the q-Lauricella function.
Keywords:
Bibasic Heine transformation formulaRamanujan’sq-Lauricella functions

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Main Results:

  • A bibasic extension of Heine's formula is derived, implicitly present in Andrews' 1966 work.
  • Extensions of Ramanujan's q-transformations are recovered as special cases.
  • General transformation formulas are obtained, including transformations of n-fold sums into m-fold sums.

Conclusions:

  • The study provides a unified framework for deriving bibasic transformation formulas.
  • The results offer new insights into the theory of basic hypergeometric series and their generalizations.
  • The work establishes connections between classical identities and more general q-series identities.