Explicit Values for Ramanujan's Theta Function ϕ(q)

This paper provides a survey of particular values of Ramanujan's theta function $\varphi(q)=\sum_{n=-\infty}^{\infty}q^{n^2}$, when $q=e^{-\pi\sqrt{n}}$, where $n$ is a positive rational number. First, descriptions of the tools used to evaluate theta functions are given. Second, classical values are briefly discussed. Third, certain values due to Ramanujan and later authors are given. Fourth, the methods that are used to determine these values are described. Lastly, an incomplete evaluation found in Ramanujan's lost notebook, but now completed and proved, is discussed with a sketch of its proof.

Ramanujan’s theta function identities and the relations between sums of squares and sums of triangular numbers

For positive integers [Formula: see text], [Formula: see text] and [Formula: see text], let [Formula: see text] denote the number of representations of a nonnegative integer [Formula: see text] as [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text] are nonnegative integers, and let [Formula: see text] denote the number of representations of [Formula: see text] as [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text] are integers. Recently, Sun proved a number of relations between [Formula: see text] and [Formula: see text] along with numerous conjectures on such relations. In this work, we confirm several conjectures of Sun by using Ramanujan’s theta function identities.

Proofs of some conjectures of Sun on the relations between sums of squares and sums of triangular numbers

In a recent paper, Sun posed six conjectures on the relations between [Formula: see text] and [Formula: see text], where [Formula: see text] denotes the number of representations of [Formula: see text] as [Formula: see text], where [Formula: see text] are positive integers, [Formula: see text] are arbitrary nonnegative integers, and [Formula: see text] denotes the number of representations of [Formula: see text] as [Formula: see text], where this time [Formula: see text] are integers. In this paper, we prove Sun’s six conjectures by using Ramanujan’s theta function identities.

Exact generating functions for the number of partitions into distinct parts

Let [Formula: see text] denote the number of partitions of a non-negative integer into distinct (or, odd) parts. We find exact generating functions for [Formula: see text], [Formula: see text] and [Formula: see text]. We deduce some congruences modulo 5 and 25. We employ Ramanujan’s theta function identities and some identities for the Rogers–Ramanujan continued fraction.

Some Congruence Properties of a Restricted Bipartition Function cN(n)

Let cN(n) denote the number of bipartitions (λ,μ) of a positive integer n subject to the restriction that each part of μ is divisible by N. In this paper, we prove some congruence properties of the function cN(n) for N=7, 11, and 5l, for any integer l≥1, by employing Ramanujan’s theta-function identities.

Some New Explicit Values of Quotients of Theta-Function ϕ(q) and Applications to Ramanujan's Continued Fractions

We find some new explicit values of the parameter hk,n for positive real numbers k and n involving Ramanujan's theta-function ϕ(q) and give some applications of these new values for the explicit evaluations of Ramanujan's continued fractions. In the process, we also establish two new identities for ϕ(q) by using modular equations.

ON THE QUATERNARY FORMS x2+y2+2z2+3t2, x2+2y2+2z2+6t2, x2+3y2+3z2+6t2 AND 2x2+3y2+6z2+6t2

Formulas are proved for the number of representations of a positive integer by each of the four quaternary quadratic forms x2+y2+2z2+3t2, x2+2y2+2z2+6t2, x2+3y2+3z2+6t2 and 2x2+3y2+6z2+6t2. As a consequence of these formulas, each of the four series [Formula: see text] is determined in terms of Ramanujan's theta function.