A universal identity for theta functions of degree eight and applications

International audience Previously, we proved an identity for theta functions of degree eight, and several applications of it were also discussed. This identity is a natural extension of the addition formula for the Weierstrass sigma-function. In this paper we will use this identity to reexamine our work in theta function identities in the past two decades. Hundreds of results about elliptic modular functions, both classical and new, are derived from this identity with ease. Essentially, this general theta function identity is a theta identities generating machine. Our investigation shows that many well-known results about elliptic modular functions with different appearances due to Jacobi, Kiepert, Ramanujan and Weierstrass among others, actually share a common source. This paper can also be seen as a summary of my past work on theta function identities. A conjecture is also proposed.

A mock theta function identity related to the partition rank modulo 3 and 9

We prove a new mock theta function identity related to the partition rank modulo 3 and 9. As a consequence, we obtain the [Formula: see text]-dissection of the rank generating function modulo [Formula: see text]. We also evaluate all of the components of the rank–crank differences modulo [Formula: see text]. These are analogous to conjectures of Lewis [The generating functions of the rank and crank modulo 8, Ramanujan J. 18 (2009) 121–146] on rank–crank differences modulo [Formula: see text], first proved by Mortenson [On ranks and cranks of partitions modulo 4 and 8, J. Combin. Theory Ser. A 161 (2019) 51–80].

Combinatorial proof of a partial theta function identity of Warnaar

By constructing a sign-reversing involution, we prove Warnaar’s identity involving a partial theta function, which plays many important roles in the study of asymptotic behaviors and quantum modularities in number theory. We also obtain an Euler-like theorem for a certain kind of unimodal sequences from Warnaar’s identity.