11 papers
Quasimodular forms that detect primes are Eisenstein
Jan-Willem van Ittersum, Lukas Mauth, Ken Ono +1
MacMahon's partition functions and their extensions provide equations that identify prime numbers as solutions. These results depend on the theory of (mixed weight) quasimodular fo…
Some topological genera and Jacobi forms
Tewodros Amdeberhan, Michael Griffin, Ken Ono
We revisit and elucidate the -genus, Hirzebruch's -genus and Witten's -genus, cobordism invariants of special classes of manifolds. After slight modification, in…
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…
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…
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…
Distribution of the Hessian values of Gaussian hypergeometric functions
Ken Ono, Sudhir Pujahari, Hasan Saad +1
We consider a special family of Gaussian hypergeometric functions whose entries are cubic and trivial characters over finite fields. The special values of these functions are known…
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…
A note on odd partition numbers
Michael Griffin, Ken Ono
Ramanujan's celebrated partition congruences modulo assert that where satisfies $24δ_{\ell}\eq…
Hook lengths in self-conjugate partitions
Tewodros Amdeberhan, George E. Andrews, Ken Ono +1
In 2010, G.-N. Han obtained the generating function for the number of size hooks among integer partitions. Here we obtain these generating functions for self-conjugate partitio…
MacMahon's sums-of-divisors and allied -series
Tewodros Amdeberhan, Ken Ono, Ajit Singh
Here we investigate the -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…