Eichler integrals of Eisenstein series as $q$-brackets of weighted $t$-hook functions on partitions

Abstract

We consider the t-hook functions on partitions f a,t : P → C defined bywhere H t (λ) is the multiset of partition hook numbers that are multiples of t. The Bloch-Okounkov q-brackets f a,t q include Eichler integrals of the classical Eisenstein series. For even a ≥ 2, we show that these q-brackets are natural pieces of weight 2 − a sesquiharmonic and harmonic Maass forms, while for odd a ≤ −1, we show that they are holomorphic quantum modular forms. We use these results to obtain new formulas of Chowla-Selberg type, and asymptotic expansions involving values of the Riemann zeta-function and Bernoulli numbers. We make use of work of Berndt, Han and Ji, and Zagier.f q := λ∈P f (λ)q |λ| λ∈P q |λ| ∈ C[[q]],

Horizontal and Vertical Log-Concavity

“…In the case of g(n) = σ(n) and T = q := e 2πiτ , where τ is in the upper complex half-plane, E g (T ) is the Eichler integral of 1−E 2 24 of the weight 2 quasi-modular Eisenstein series E 2 (τ ). We refer to [BOW20] for recent work on Eichler integrals.…”

The Bloch–Okounkov theorem for congruence subgroups and Taylor coefficients of quasi-Jacobi forms

“…It should be noted that the above theorems are true in greater generality. For example, the so-called hook-length moments introduced in [6], and studied in the context of harmonic Maass forms for a congruence subgroup in [3], also have natural generalisations obtained by studying their corresponding n-point functions. Similarly, the moment functions and their generalisations in [22] can equally well be generalized to congruence subgroups.…”

“…which (up to a constant) also occurs in [3]. Denote by H (N ) the algebra generated by the H t k for which k is even and t | N .…”

Semi-modular forms from Fibonacci-Eisenstein series

“…At this stage, seeing how the proofs above follow easily from the recursion (3), a natural next question is: are other Fibonacci-like sequences such as Lucas sequences subject to the same treatment, leading to further families of semi-modular functions? Recall for a, b ∈ Z that a classical Lucas sequence 3 {L n (a, b)} is defined for n ≥ 2 by the recursion (12) L n (a, b)…”