Ajit Singh

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math.NT2024

Derivatives of theta functions as Traces of Partition Eisenstein series

Tewodros Amdeberhan, Ken Ono, Ajit Singh

In his "lost notebook'', Ramanujan used iterated derivatives of two theta functions to define sequences of $q$ -series ${U_{2 t} (q)}$ and ${V_{2 t} (q)}$ that he claimed to be quas…

math.NT2024

Generalized cubic partitions

Tewodros Amdeberhan, Ajit Singh

A cubic partition consists of partition pairs $(λ, μ)$ such that $∣ λ ∣ + ∣ μ ∣ = n$ where $μ$ involves only even integers but no restriction is placed on $λ$ . This paper in…

math.NT2023

MacMahon's sums-of-divisors and allied $q$ -series

Tewodros Amdeberhan, Ken Ono, Ajit Singh

Here we investigate the $q$ -series \begin{align*} \mathcal{U}_a(q)&=\sum_{n=0}^{\infty} MO(a;n)q^n&:=\sum_{0< k_1<k_2<\cdots<k_a} \frac{q^{k_1+k_2+\cdots+k_a}}{(1-q^{k_1})^2(1-q^{k…

Derivatives of theta functions as Traces of Partition Eisenstein series

Generalized cubic partitions

MacMahon's sums-of-divisors and allied q-series

MacMahon's sums-of-divisors and allied $q$ -series