congruence properties of Romik's sequence of Taylor
coefficients of Jacobi's theta function θ3
In [Ramanujan J. 52 (2020), 275-290], Romik considered the
Taylor expansion of Jacobi's theta function θ3(q)$
at q = e-π and encoded it
in an integer sequence (d(n))n≥0
for which he provided a recursive procedure
to compute the terms of the sequence.
He observed intriguing behaviour of d(n) modulo primes and prime powers.
Here we prove (1) that d(n) eventually vanishes modulo any prime power
with p≡3 (mod 4), (2) that d(n) is eventually periodic
modulo any prime power pe
with p≡1 (mod 4), and (3) that d(n) is purely periodic
modulo any 2-power 2e.
Our results also provide more detailed information on
period length, respectively from when on the sequence vanishes or becomes periodic.
The corresponding bounds may not be optimal though, as computer data suggest.
Our approach shows that the above congruence properties
hold at a much finer, polynomial level.
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