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 n -th powers of the symmetric theta function f ( a , b ) f(a,b) satisfies a factorization of the form f ( a , b ) F n ( a b ) f(a,b)F_{n}(ab) . Moreover, Ramanujan recorded identities expressing F 2 ( q ) F_{2}(q) , F 3 ( q ) F_{3}(q) , F 4 ( q ) F_{4}(q) , F 5 ( q ) F_{5}(q) , and F 7 ( q ) F_{7}(q) in terms of his theta functions φ ( q ) \varphi (q) , ψ ( q ) \psi (q) , and f ( − q ) 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_{6}(q) , F 8 ( q ) F_{8}(q) , F 9 ( q ) F_{9}(q) , and F 10 ( q ) F_{10}(q) .