Lacunary recurrences for Eisenstein series

Abstract

Using results from the theory of modular forms, we reprove and extend a result of Romik about lacunary recurrence relations for Eisenstein series.

On the number of $n$-dimensional representations of $\operatorname{SU}(3)$, the Bernoulli numbers, and the Witten zeta function

“…It turns out that the identity j α n,j = 1 is still correct in that case, and this is both easier and more natural to prove. That is, we now wish to show that S(n) := n k=0 F (n, k) = 1 (32) for all integer n ≥ 0, where we define…”

“…Update added in revision: following our description of this problem in the original version of this paper, a proof of (23) based on known results about modular forms was found recently by Mertens and Rolen[32].…”

A family of lacunary recurrences for Fibonacci numbers

“…The speed of the computation increases with the size of the gap. Lacunary recurrences have been studied for many types of sequences including Bernoulli numbers [1,8,9,11], Euler numbers [8], k-Fibonacci numbers (which are Fibonacci polynomials at positive integer values) [4], Eisenstein series [10], Tribonacci numbers [6], more general sequences that include Bernoulli, Euler, Fibonacci and Genocchi numbers [5], and sequences satisfying an arbitrary linear recurrence [3,12,13].…”