Dancing samba with Ramanujan partition congruences. (January 2018)

• Record Type: Journal Article
• Authors: Hemmecke, Ralf
• Is Part Of: Journal of symbolic computation. Volume 84(2018)
• Page Start: 14
• Page End: 24
• Publication Date: 2018-01
• Subjects: Partition identities -- Number theoretic algorithm -- Subalgebra basis; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2017.02.001
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: The article presents an algorithm to compute a C [ t ] -module basis G for a given subalgebra A over a polynomial ring R = C [ x ] with a Euclidean domain C as the domain of coefficients and t a given element of A . The reduction modulo G allows a subalgebra membership test. The algorithm also works for more general rings R, in particular for a ring R ⊂ C ( ( q ) ) with the property that f ∈ R is zero if and only if the order of f is positive. As an application, we algorithmically derive an explicit identity (in terms of quotients of Dedekind η -functions and Klein's j -invariant) that shows that p ( 11 n + 6 ) is divisible by 11 for every natural number n where p ( n ) denotes the number of partitions of n .