The sixth, eighth, ninth, and tenth powers of Ramanujan’s theta function

Abstract:

In his Lost Notebook, Ramanujan claimed that the “circular” summation of the $n$-th powers of the symmetric theta function $f(a,b)$ satisfies a factorization of the form $f(a,b)F_{n}(ab)$. Moreover, Ramanujan recorded identities expressing $F_{2}(q)$, $F_{3}(q)$, $F_{4}(q)$, $F_{5}(q)$, and $F_{7}(q)$ in terms of his theta functions $\varphi (q)$, $\psi (q)$, and $f(-q)$. Ramanujan’s claims were proved by Rangachari, and later (via elementary methods) by Son. In this paper we obtain similar identities for $F_{6}(q)$, $F_{8}(q)$, $F_{9}(q)$, and $F_{10}(q)$.