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Pentagonal number recurrence relations for
Kevin Gomez, Ken Ono, Hasan Saad +1
We revisit Euler's partition function recurrence, which asserts, for integers that $$ p(n)=p(n-1)+p(n-2)-p(n-5)-p(n-7)+\dots = \sum_{k\in \mathbb{Z}\setminus \{0\}} (-1)…
Traces of partition Eisenstein series
Tewodros Amdeberhan, Michael Griffin, Ken Ono +1
We study "partition Eisenstein series", extensions of the Eisenstein series defined by $$λ=(1^{m_1}, 2^{m_2},\dots, k^{m_k}) \vdash k \ \ \ \ \ \longmapsto \ \ \ \ \ G…
Certain infinite products in terms of MacMahon type series
Seokho Jin, Badri Vishal Pandey, Ajit Singh
Recently, Ono and the third author discovered that the reciprocals of the theta series and have infinitely many closed formulas…
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 -series and that he claimed to be quas…
Arithmetic properties for generalized cubic partitions and overpartitions modulo a prime
Tewodros Amdeberhan, James A. Sellers, Ajit Singh
A cubic partition is an integer partition wherein the even parts can appear in two colors. In this paper, we introduce the notion of generalized cubic partitions and prove a number…
Distribution of hooks in self-conjugate partitions
William Craig, Ken Ono, Ajit Singh
We confirm the speculation that the distribution of -hooks among unrestricted integer partitions essentially descends to self-conjugate partitions. Namely, we prove that the num…
Generalized cubic partitions
Tewodros Amdeberhan, Ajit Singh
A cubic partition consists of partition pairs such that where involves only even integers but no restriction is placed on . This paper in…
Remarks on MacMahon's -series
Ken Ono, Ajit Singh
In his important 1920 paper on partitions, MacMahon defined the partition generating functions \begin{align*} A_k(q)=\sum_{n=1}^{\infty}\mathfrak{m}(k;n)q^n&:=\sum_{0< s_1<s_2<\cdo…