Two theorems of Gauss and allied identities proved arithmetically

George E. Andrews

Vol. 41 (1972), No. 3, 563–578

DOI: 10.2140/pjm.1972.41.563

Abstract

The product formulae of Gauss for the theta functions 𝜃₄(0,q) and (1∕2)(−q)^−1∕8𝜃₂(0,(−q)^1∕2) have been derived in many ways by analytic means. In this paper these formulae are derived by enumerating certain types of partitions. The enumeration technique is shown to be applicable to more general results, and several important theorems in basic hypergeometric series are proved from suitable enumerations of partitions.

Mathematical Subject Classification 2000

Primary: 10A45

Secondary: 05A17

Milestones

Received: 22 March 1971

Revised: 31 May 1971

Published: 1 June 1972