Congruence properties of pk(n)
| Autor: | Edward Y. S. Liu, Julia Q. D. Du, Jack C. D. Zhao |
|---|---|
| Rok vydání: | 2019 |
| Předmět: |
Algebra and Number Theory
Computer Science::Information Retrieval 010102 general mathematics Astrophysics::Instrumentation and Methods for Astrophysics Computer Science::Computation and Language (Computational Linguistics and Natural Language and Speech Processing) Congruence relation 01 natural sciences Combinatorics symbols.namesake TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES Integer 0103 physical sciences ComputingMethodologies_DOCUMENTANDTEXTPROCESSING symbols Computer Science::General Literature Congruence (manifolds) 010307 mathematical physics 0101 mathematics ComputingMilieux_MISCELLANEOUS Euler product Mathematics |
| Zdroj: | International Journal of Number Theory. 15:1267-1290 |
| ISSN: | 1793-7310 1793-0421 |
| DOI: | 10.1142/s1793042119500714 |
| Popis: | We present a unified approach to establish infinite families of congruences for [Formula: see text] for arbitrary positive integer [Formula: see text], where [Formula: see text] is given by the [Formula: see text]th power of the Euler product [Formula: see text]. For [Formula: see text], define [Formula: see text] to be the least positive integer such that [Formula: see text] and [Formula: see text] the least non-negative integer satisfying [Formula: see text]. Using the Atkin [Formula: see text]-operator, we find that the generating function of [Formula: see text] (respectively, [Formula: see text]) can be expressed as the product of an integral linear combination of modular functions on [Formula: see text] and [Formula: see text] (respectively, [Formula: see text]) for any [Formula: see text] and [Formula: see text]. By investigating the properties of the modular equations of the [Formula: see text]th order under the Atkin [Formula: see text]-operator, we obtain that these generating functions are determined by some linear recurring sequences. Utilizing the periodicity of these linear recurring sequences modulo [Formula: see text], we are led to infinite families of congruences for [Formula: see text] modulo any [Formula: see text] with [Formula: see text] and periodic relations between the values of [Formula: see text] modulo powers of [Formula: see text]. As applications, infinite families of congruences for many partition functions such as [Formula: see text]-core partition functions, the partition function and Andrews’ spt-function are easily obtained. |
| Databáze: | OpenAIRE |
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