The Ramanujan Journal

In his last letter to Hardy, Ramanujan defined 17 functions F(q), where |q| < 1. He called them mock theta functions, because as q radially approaches any point e 2πir (r rational), there is a theta function F r(q) with F(q) − F r(q) = O(1). In this paper we obtain the transformations of Ramanujan's fifth and seventh order mock theta functions under the modular group generators τ → τ + 1 and τ → −1/τ, where q = e πiτ. The transformation formulas are more complex than those of ordinary theta functions. A definition of the order of a mock theta function is also given.

The main result of this paper is a generalization of a conjecture of Guoniu Han, originally inspired by an identity of Nekrasov and Okounkov. Our result states that if F is any symmetric function (say over ℚ) and if $$\Phi_n(F)=\frac{1}{n!}\sum_{\lambda\vdash n}f_\lambda^2F(h_u^2:u\in\lambda),$$ where h u denotes the hook length of the square u of the partition λ of n and f λ is the number of standard Young tableaux of shape λ, then Φ n (F) is a polynomial function of n. A similar result is obtained when F(h 2 :u∈λ) is replaced with a function that is symmetric separately in the contents c u of λ and the shifted parts λ i +n−i of λ.