Ramanujan-type congruences for three colored Frobenius partitions (Q5928010)

A generalized Frobenius partition of a positive integer \(n\) is a two-rowed array of nonnegative integers \(\left( \begin{smallmatrix} a_1 &a_2 &\dots &a_k\\ b_1 &b_2 &\dots &b_k\end{smallmatrix} \right)\), where the entries in each row are arranged in nonincreasing order and \(n= k+ a_1+\cdots+ a_k+ b_1+\cdots+ b_k\). A generalized Frobenius partition in 3 colours is an array of the above form where the entries are taken from 3 distinct copies of the nonnegative integers distinguished by colour and the rows are ordered first by size and then by colour with no two consecutive like entries in any row. Let \(c\varphi_3(n)\) denote the number of generalized Frobenius partitions of \(n\) in 3 colours. \textit{K. Ono} [J. Number Theory 57, 170-180 (1996; Zbl 0852.11057)] proved that \(c\varphi_3 (63n+ 5550)\equiv 0\bmod 7\) and investigated \(c\varphi_3 (5n+2)\bmod 5\). NEWLINENEWLINENEWLINEIn the paper under review the author proves the following congruences: NEWLINE\[NEWLINE\begin{aligned} &c\varphi_3 (45n+ 23)\equiv c\varphi_3 (45n+41) \equiv 0\bmod 5;\\ &c\varphi_3 (99n+95)\equiv 0\bmod 11;\\ &c\varphi_3 (171n+50)\equiv 0\bmod 19. \end{aligned}NEWLINE\]NEWLINE These congruences are proved by constructing modular forms whose Fourier coefficients are related to \(c\varphi_3(n)\) and whose congruence properties can be verified with a finite computation.