Congruences for the coefficients of the powers of the Euler Product

Let p k ( n ) be given by the series expansion of the k -th power of the Euler Product ∏ n = 1 ∞ ( 1 - q n ) k = ∑ n = 0 ∞ p k ( n ) q n . By investigating the properties of the modular equations of the second and the third order under the Atkin U -operator, we determine the generating functions of p 8 k ( 2 2 α n + k ( 2 2 α - 1 ) 3 ) ( 1 ≤ k ≤ 3 ) and p 3 k ( 3 2 β n + k ( 3 2 β - 1 ) 8 ) ( 1 ≤ k ≤ 8 ) in terms of some linear recurring sequences. Combining with a result of Engstrom about the periodicity of linear recurring sequences modulo m , we obtain infinite families of congruences for p k ( n ) modulo any m ≥ 2 , where 1 ≤ k ≤ 24 and 3| k or 8| k . Based on these congruences for p k ( n ) , infinite families of congruences for many partition functions such as the overpartition function, t -core partition functions and ℓ -regular partition functions are easily obtained.