A unified proof of Ramanujan-type congruences modulo 5 for 4-colored generalized Frobenius partitions (Q335442)

In [Mem. Am. Math. Soc. 301, 44 p. (1984; Zbl 0544.10010)], \textit{G. E. Andrews} introduced the family of \(k\)-colored generalized Frobenius partition function \(c\phi_k(n)\), which denotes the number of generalized Frobenius partitions of \(n\) with \(k\) colors. The Frobenius partitions are a natural generalization of the ordinary partitions and are defined in terms of the Frobenius symbol, which in turn comes from the Ferrers graph representation of an ordinary partition. In recent time, there has been quite a lot of work on Ramanujan-type congruences for colored generalized Frobenius partitions. The paper under review is an addition to the literature, where the authors give a unified proof of several congruence relations due to \textit{J. A. Sellers} [J. Indian Math. Soc., New Ser. 2013, 97--103 (2013; Zbl 1290.05015) (special volume to commemorate the 125th birth anniversary of Srinivasa Ramanujan); \textit{E. X. W. Xia}, Ramanujan J. 39, No. 3, 567--576 (2016; Zbl 1401.11138); \textit{M. D. Hirschhorn} and \textit{J. A. Sellers}, ibid. 40, No. 1, 193--200 (2016; Zbl 1335.05018)].NEWLINENEWLINEThe main result proved in the paper is the following:NEWLINENEWLINEFor all \(n\geq 0\), NEWLINE\[NEWLINEc\phi_4(5n+1) := \begin{cases} k+1 \pmod 5, &\text{if } n=2k(3k+1)~\text{for some integer } k,\\ 0, &\text{otherwise.} \end{cases}NEWLINE\]NEWLINE The authors use a method similar to the one used by Hirschhorn and Sellers [loc. cit.], they prove a few auxiliary results, using standard techniques, which then proves the main result.