Arithmetic properties of 4-tuple -regular partitions

For any positive integers ℓ , T, k(n) T ℓ , k ( n ) denotes the number of k -tuple ℓ -regular partitions of an integer n . By employing q -identities, modular equations of degree 7 and continued fraction identities, we establish new congruence modulo 2, 3, 7. Our analysis is based on the continued fraction and a collection of associated q -identities and in particular we employ the modular and transformation properties of the continued fraction to derive explicit generating function dissections and establish the desired congruence relation.