Generalizing Ramanujan's J Functions (1109.5957v2)

Abstract: We generalize Ramanujan's expansions of the fractional-power Euler functions (q^{1/5})_{\infty} = J_1 - q^{1/5} + q^{2/5} J_2 {\infty} and (q^{1/7}){\infty} = J_1 + q^{1/7} J_2 - q^{2/7} + q^{5/7} J_3 {\infty} to (q^{1/N}){\infty}, where N is a prime number greater than 3. We show that there are exactly (N+1)/2 non-zero J functions in the expansion of (q^{1/N})_{\infty}, that one of these functions has the form +-q^{X_0}, that all others have the form +-q^{X_k} times the ratio of two Ramanujan theta functions, and that the product of all the non-zero J's is +-q^Z, where Z and the X's denote non-negative integers.