An algorithmic approach to Ramanujan–Kolberg identities

Let M be a given positive integer and r=(rδ)δ|M a sequence indexed by the positive divisors δ of M. In this paper we present an algorithm that takes as input a generating function of the form ∑n=0∞ar(n)qn:=∏δ|M∏n=1∞(1−qδn)rδ and positive integers m, N and t∈{0,…,m−1}. Given this data we compute a se...

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Vydané v:	Journal of symbolic computation Ročník 68; s. 225 - 253
Hlavný autor:	Radu, Cristian-Silviu
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Elsevier Ltd 01.05.2015
Predmet:	Modular forms Number theoretic algorithm Partition identities Number theoretic algorithm Partition identities Modular forms
ISSN:	0747-7171, 1095-855X
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Popis
Shrnutí:	Let M be a given positive integer and r=(rδ)δ\|M a sequence indexed by the positive divisors δ of M. In this paper we present an algorithm that takes as input a generating function of the form ∑n=0∞ar(n)qn:=∏δ\|M∏n=1∞(1−qδn)rδ and positive integers m, N and t∈{0,…,m−1}. Given this data we compute a set Pm,r(t) which contains t and is uniquely defined by m, r and t. Next we decide if there exists a sequence (sδ)δ\|N indexed by the positive divisors δ of N, and modular functions b1,…,bk on Γ0(N) (where each bj equals the product of finitely many terms from {qδ/24∏n=1∞(1−qδn):δ\|N}), such that:qα∏δ\|N∏n=1∞(1−qδn)sδ×∏t′∈Pm,r(t)∑n=0∞a(mn+t′)qn=c1b1+⋯+ckbk for some c1,⋯,ck∈Q and α:=∑δ\|Nδsδ24+∑t′∈Pm,r(t)24t′+∑δ\|Mδrδ24m. Our algorithm builds on work by Rademacher (1942), Newman (1959), and Kolberg (1957).
ISSN:	0747-7171 1095-855X
DOI:	10.1016/j.jsc.2014.09.018