Extended Landen's formulas and double product theta functions

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.