The Taylor coefficients of the Jacobi theta constant $$\theta _3$$

Abstract

We study the Taylor expansion around the point x = 1 of a classical modular form, the Jacobi theta constant θ 3 . This leads naturally to a new sequence (d(n)) ∞ n=0 = 1, 1, −1, 51, 849, −26199, . . . of integers, which arise as the Taylor coefficients in the expansion of a related "centered" version of θ 3 . We prove several results about the numbers d(n) and conjecture that they satisfy the congruence d(n) ≡ (−1) n−1 (mod 5) and other similar congruence relations.

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

“…(i) d(n) ≡ 1 (mod 2) for all n ≥ 0, (ii) d(n) ≡ (−1) n+1 (mod 5) for all n ≥ 1, and (iii) if p is prime and p ≡ 3 (mod 4), then d(n) ≡ 0 (mod p) for all n > p 2 −1 2 . This proves half of Conjecture 13 (b) in [8], where the sequence (d(n)) was first introduced. (The half of the statement that we don't prove is that for primes p = 4k + 1, the sequence (d(n)) ∞ n=0 mod p is periodic, although Theorem 1 is a specific example of this phenomenon in the case p = 5.…”

“…where M is the Möbius transformation given by M (w) = z−zw 1−w that maps the unit disk to H, with M (0) = z. In the case of the modular form ϑ(τ ) = θ 3 (−iτ ), the Taylor coefficients in the above expansion, about the CM point z = i, are the sequence (d(n)) (after a proper normalization) introduced in [8] and studied here.…”

“…

Recently, Romik determined in [8] the Taylor expansion of the Jacobi theta constant θ 3 , around the point x = 1. He discovered a new integer sequence, (d(n)) ∞ n=0 = 1, 1, −1, 51, 849, −26199, .

…”

“…and define the sequence (d(n)) ∞ n=0 by Thus, the numbers (d(n)) ∞ n=0 are the Taylor coefficients, modulo trivial factors, of σ at 0. It's not at all clear from the definition that the numbers d(n) are integers, but this is shown to be true in [8]. Furthermore, the connection of the sequence (d(n)) to the derivatives of θ 3 at 1 can be made explicit:…”

Taylor coefficients of the Jacobi θ3(q) function

“…A combinatorial interpretation of the cumulants. In the concluding Open Problems section of his recent article [16], Dan Romik asked for a combinatorial interpretation of the sequence d (n) . We were not able to find such an interpretation, but we can provide one for the sequence of cumulants as follows.…”