Partitions and the Maximal Excludant

For each nonempty integer partition $\pi$, we define the maximal excludant of $\pi$ as the largest nonnegative integer smaller than the largest part of $\pi$ that is not itself a part. Let $\sigma\!\operatorname{maex}(n)$ be the sum of maximal excludants over all partitions of $n$. We show that the generating function of $\sigma\!\operatorname{maex}(n)$ is closely related to a mock theta function studied by Andrews, Dyson and Hickerson, and Cohen, respectively. Further, we show that, as $n\to \infty$, $\sigma\!\operatorname{maex}(n)$ is asymptotic to the sum of largest parts over all partitions of $n$. Finally, the expectation of the difference of the largest part and the maximal excludant over all partitions of $n$ is shown to converge to $1$ as $n\to \infty$.

Higher depth mock theta functions and q-hypergeometric series

In the theory of harmonic Maaß forms and mock modular forms, mock theta functions are distinguished examples which arose from q-hypergeometric examples of Ramanujan. Recently, there has been a body of work on higher depth mock modular forms. Here, we introduce distinguished examples of these forms, which we call higher depth mock theta functions, and develop q-hypergeometric expressions for them. We provide three examples of mock theta functions of depth two, each arising by multiplying a classical mock theta function with a certain specialization of a universal mock theta function. In addition, we give their modular completions, and relate each to a q-hypergeometric series.

Congruences modulo powers of 5 for the rank parity function

International audience It is well known that Ramanujan conjectured congruences modulo powers of 5, 7 and 11 for the partition function. These were subsequently proved by Watson (1938) and Atkin (1967). In 2009 Choi, Kang, and Lovejoy proved congruences modulo powers of 5 for the crank parity function. The generating function for the rank parity function is f (q), which is the first example of a mock theta function that Ramanujan mentioned in his last letter to Hardy. We prove congruences modulo powers of 5 for the rank parity function.

On second order mock theta function $ B(q) $

<abstract><p>In this paper, we present some arithmetic properties for the second order mock theta function $ B(q) $ given by McIntosh as:</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ B(q) = \sum\limits_{n = 0}^{\infty}\frac{q^n(-q;q^2)_n}{(q;q^2)_{n+1}}. $\end{document} </tex-math></disp-formula></p> </abstract>