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Journal Articles Journal of Integer Sequences Year : 2018

Short addition sequences for theta functions

Abstract

The main step in numerical evaluation of classical Sl2 (Z) modular forms and elliptic functions is to compute the sum of the first N nonzero terms in the sparse q-series belonging to the Dedekind eta function or the Jacobi theta constants. We construct short addition sequences to perform this task using N + o(N) multiplications. Our constructions rely on the representability of specific quadratic progressions of integers as sums of smaller numbers of the same kind. For example, we show that every generalised pentagonal number c 5 can be written as c = 2a + b where a, b are smaller generalised pentagonal numbers. We also give a baby-step giant-step algorithm that uses O(N/ log r N) multiplications for any r > 0, beating the lower bound of N multiplications required when computing the terms explicitly. These results lead to speed-ups in practice.
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Dates and versions

hal-01355926 , version 1 (24-08-2016)
hal-01355926 , version 2 (07-03-2018)

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Andreas Enge, William Hart, Fredrik Johansson. Short addition sequences for theta functions. Journal of Integer Sequences, 2018, 18 (2), pp.1-34. ⟨hal-01355926v2⟩
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