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In this paper, the author proves some basic hypergeometric series which utilizes the same ideas that Margaret Jackson used to give a proof of Ramanujan’s [Formula: see text] summation formula.

Recently, Andrews defined the combinatorial objects which he called singular overpartitions and proved that these singular overpartitions, which depend on two parameters k and i, can be enumerated by the function $\overline{C}_{k, i}(n)$ which gives the number of overpartitions of n in which no part is divisible by k and only parts ≡ ±i (mod k) may be overlined. He also proved that $\overline{C}_{3, 1}(9n + 3) \equiv \overline{C}_{3, 1}(9n + 6) \equiv 0 ({\rm mod} 3)$. Chen, Hirschhorn and Sellers then found infinite families of congruences modulo 3 and modulo powers of 2 for $\overline{C}_{3, 1}(n)$, $\overline{C}_{6, 1}(n)$ and $\overline{C}_{6, 2}(n)$. In this paper, we find new congruences for $\overline{C}_{3, 1}(n)$ modulo 4, 18 and 36, infinite families of congruences modulo 2 and 4 for $\overline{C}_{8, 2}(n)$, congruences modulo 2 and 3 for $\overline{C}_{12, 2}(n)$, $\overline{C}_{12, 4}(n)$, and congruences modulo 2 for $\overline{C}_{24, 8}(n)$ and $\overline{C}_{48, 16}(n)$. We use simple p-d...

In his famous paper, "On certain arithmetical functions", Ramanujan offers for the first time the Euler product of the Dirichlet series in which the coefficients are given by Ramanujan's tau-function. In his lost notebook, Ramanujan records further Euler products for L-series attached to modular forms, and, typically, does not record proofs for these claims. In this semi-expository article, for the Euler products appearing in his lost notebook, we provide or sketch proofs using elementary methods, binary quadratic forms, and modular forms.

Ramanujan's famous congruences for the partition function modulo powers of 5, 7, and 11 have inspired much further research. For example, in 2002 Lovejoy and Ono found subprogressions of 5jn + β5(j) for which Ramanujan's congruence mod 5j could be strengthened to a statement modulo 5j+1. Here we provide the analogous results modulo powers of 7 and 11. We require the arithmetic properties of two special elliptic curves.

Two congruences are proved for an infinite family of Appell–Lerch sums. As corollaries, special cases imply congruences for some of the mock theta functions of order two, six and eight.

This paper summarizes the development of Ramanujan expansions of arithmetic functions since Ramanujan's paper in 1918, following Carmichael's mean-value-based concept from 1932 up to 1994. A new technique, based on the concept of related arithmetic functions, is introduced that leads to considerable extensions of preceding results on Ramanujan expansions. In particular, very short proofs of theorems for additive and multiplicative functions going far beyond previous borders are presented, and Ramanujan expansions that formerly have been considered mysterious are explained.

In this paper, we study the divisibility of the function a(n) defined by $\sum_{n\geq 0} a(n) q^n := (q;q)^{-1}_\infty (q^2; q^2)^{-1}_\infty $. In particular, we prove certain "Ramanujan type cong...

In this paper, we prove infinite families of congruences for coefficients of harmonic Maass forms whose coefficients encode Dyson's rank. This generalizes the earlier joint work of the author with Ken Ono.

Using Ramanujan's differential equations for Eisenstein series and an idea from Ramanujan's unpublished manuscript on the partition function p(n) and the tau function τ(n), we provide simple proofs of Ramanujan's congruences for p(n) modulo 5, 7, and 11.