Basil Gordon’s research while affiliated with University of California, Los Angeles and other places

The parity of g(n) and h(n), the enumerators of restricted partitions of n appearing in the Rogers-Ramanujan identities, is studied. The parity of g(n) for odd n and that of h(n) for even n are completely determined. It is shown that these numbers are almost always even.

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 exponential singularity, there is a θ-function T ζ(q) with M(q) - Tz(q) = O(1)M(q) - {T}_{\zeta }(q) = O(1) . Since then, other functions have been found which possess this property. We list various linear relations between these functions and develop their transformation laws under the modular group. We show that each mock θ-function is related to a member of a universal family (mock θ-conjectures). In recent years the subject has received new impetus and importance through a strong connection with the theory of Maass forms. The final section of this survey provides some brief remarks concerning these new developments. KeywordsMock theta functions- q-series-Modular forms

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 2ir (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 + 1 and –1/, where q = e i. 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.

A method is developed for obtaining Ramanujan’s mock theta functions from ordinary theta functions by performing certain operations on their q-series expansions. The method is then used to construct several new mock theta functions, including the first ones of eighth order. Summation and transformation formulae for basic hypergeometric series are used to prove that the new functions actually have the mock theta property. The modular transformation formulae for these functions are obtained.

The Rogers L-function $L(x) = \sum\limits_{n = 1}^\infty {\frac{{x^n }} {{n^2 }} + \frac{1} {2}\log x} \log (1 - x)$ satisfies the functional equation $L(x) + L(y) = L(xy) + L\left( {\frac{{x(1 - y)}} {{1 - xy}}} \right) + L\left( {\frac{{y(1 - x)}} {{1 - xy}}} \right)$ .From this we derive several other such equations, including Euler's identity L(x)+L(1-x)=L(1) and various identities arising from summation and transformation formulas for basic hypergeometric series. We also obtain some new equations of the form $\sum\limits_{k = 0}^n {c_k L(\theta ^k ) = 0}$ where ? is algebraic and the ck are integers.

Let $k = p_1^{a_1 } p_2^{a_2 } \cdot \cdot \cdot p_m^{a_m }$ be the prime factorization of a positive integer k and let bk(n) denote the number of partitions of a non-negative integer n into parts none of which are multiples of k. If M is a positive integer, let Sk(N; M) be the number of positive integers = N for which bk(n)= 0(modM). If $p_i^{a_i } \geqslant \sqrt k$ we prove that, for every positive integer j $\mathop {\lim }\limits_{N \to \infty } \frac{{S_k (N;p_i^j )}} {N} = 1.$ In other words for every positive integer j,bk(n) is a multiple of $p_i^j$ for almost every non-negative integer n. In the special case when k=p is prime, then in representation-theoretic terms this means that the number ofp -modular irreducible representations of almost every symmetric groupSn is a multiple of pj. We also examine the behavior of bk(n) (mod $p_i^j$ ) where the non-negative integers n belong to an arithmetic progression. Although almost every non-negative integer n= (mod t) satisfies bk(n) = 0 (mod $p_i^j$ ), we show that there are infinitely many non-negative integers n= r (mod t) for which bk(n) ? 0 (mod $p_i^j$ ) provided that there is at least one such n. Moreover the smallest such n (if there are any) is less than 2 $\cdot 10^8 p_i^{a_i + j - 1} k^2 t^4$ .