In his last letter to Hardy, Ramanujan defined 17 functions M(q), | q | < 1, which he called mock θ-functions. He observed that as q radially approaches any root of unity ζ at which M(q) has an… Expand

The q-shifted factorial defined by (a : qk)n = (1 − a) (1 − aqk)(1 − aq2k)... (1 − aq(n − 1)k) appears in the terms of basic hypergeometric series. Complete asymptotic expansions as q → 1 of some… Expand

Abstract In his last letter to Hardy, Ramanujan defined 17 functions $F\left( q \right)$ , where $\left| q \right|<1$ . He called them mock theta functions, because as $q$ radially approaches any… Expand

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 e2πir (r rational), there is a theta… Expand

The problem of distinguishing prime numbers from composite numbers (. . .) is known to be one of the most important and useful in arithmetic. (. . .) The dignity of the science itself seems to… Expand

Abstract We obtain complete asymptotic expansions for certain binomial sums, including the Apery numbers. In general, binomial sums cannot be expressed by closed formulae, but they do satisfy… Expand