Congruences modulo primes of the Romik sequence related to the Taylor expansion of the Jacobi theta constant $$\theta _3$$

Abstract

Recently, Romik determined (in: Ramanujan J, 2019, https://doi.org/10.1007/s11139-018-0109-5 ) the Taylor expansion of the Jacobi theta constant $$\theta _3$$ , around the point $$x=1$$ . He discovered a new integer sequence, $$(d(n))_{n=0}^\infty =1,1,-1,51,849,-26199,\ldots $$ , from which the Taylor coefficients are built, and conjectured that the numbers d(n) satisfy certain congruences. In this paper, we prove some of these conjectures, for example that $$d(n)\equiv (-1)^{n+1}$$ (mod 5) for all $$n\ge 1$$ , and that for any prime $$p\equiv 3$$ (mod 4), d(n) vanishes modulo p for all large enough n.