Extended Landen's formulas and double product theta functions

TL;DR

This paper extends Landen's formulas to relate theta functions with scaled moduli to products of simpler theta functions and demonstrates how double products of genus 1 theta functions can be expressed as sums of genus 2 theta functions, with applications to known identities.

Contribution

It introduces a generalized Landen's formula for arbitrary positive integers and expresses double products of genus 1 theta functions as sums of genus 2 theta functions, expanding the theoretical framework.

Findings

Extended Landen's formulas for arbitrary p

Double product of genus 1 theta functions expressed as genus 2 sums

Applications include the Borwein identities

Abstract

For an arbitrary positive integer $p$, Landen's formula is extended to express theta function with modulus $p\tau$ by $p$ product of theta functions with $\tau$, which is applied to several examples. Next it is shown that double product of theta functions of genus $g=1$ is written by a sum of $g=2$ theta functions, which is a subset having a special period matrix of $\tau_{11}=\tau_{22}$. Several applied examples are shown, which include the cubic identity of Borwein and Borwein.