Infinitely many congruences modulo 5 for 4-colored Frobenius partitions

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• 作者：Michael D. Hirschhorn ; James A. Sellers
• 刊名：The Ramanujan Journal
• 出版年：2016
• 出版时间：May 2016
• 年：2016
• 卷：40
• 期：1
• 页码：193-200
• 全文大小：352 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Number Theory
  Field Theory and Polynomials
  Combinatorics
  Fourier Analysis
  Functions of a Complex Variable
  
• 出版者：Springer U.S.
• ISSN：1572-9303
• 卷排序：40

文摘

In his 1984 AMS Memoir, Andrews introduced the family of functions \(c\phi _k(n),\) which denotes the number of generalized Frobenius partitions of \(n\) into \(k\) colors. Recently, Baruah and Sarmah, Lin, Sellers, and Xia established several Ramanujan-like congruences for \(c\phi _4(n)\) relative to different moduli. In this paper, employing classical results in \(q\)-series, the well-known theta functions of Ramanujan, and elementary generating function manipulations, we prove a characterization of \(c\phi _4(10n+1)\) modulo 5 which leads to an infinite set of Ramanujan-like congruences modulo 5 satisfied by \(c\phi _4.\) This work greatly extends the recent work of Xia on \(c\phi _4\) modulo 5.KeywordsCongruencesGeneralized Frobenius partitionsTheta functionsGenerating functions