Congruence families modulo powers of 7 for 4-colored generalized Frobenius partitions
Abstract
In 2012, Peter Paule and Cristian-Silviu Radu proved an infinite family of Ramanujan type congruences for 2-colored Frobenius partitions cφ2 introduced by George Andrews. Recently, Frank Garvan, James Sellers and Nicolas Smoot showed that this family of congruences is equivalent to the family of congruences for (2,0)-colored Frobenius partitions c2,0 introduced by Brian Drake and by Yuze Jiang, Larry Rolen and Michael Woodbury for the general case. Motivated by Garvan, Sellers and Smoot's work, Rong Chen and Xiao-Jie Zhu found modular transformations relating the ck,β for fixed k and varying β. As an example, they proved a family of congruences for c3,1/2 following Paule and Radu's work and then proved the equivalence between c3,1/2 and cφ3=c3,3/2. In the present paper, we give a new example of Chen and Zhu's framework for c4,β. Our proof is considerably simpler.
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