Congruence relations for r-colored partitions

Let l >= 5 be prime. For the partition function p(n) and 5 <= l <= 31, Atkin found a number of examples of primes Q >= 5 such that there exist congruences of the form p(lQ(3)n + beta) equivalent to 0 (mod l). Recently, Ahlgren, Allen, and Tang proved that there are infinitely many such congruences for every l. In this paper, for a wide range of c is an element of F-l, we prove congruences of the form p(lQ(3)n + beta(0)) equivalent to c . p(lQn + beta(1))(mod l) for infinitely many primes Q. For a positive integer r, let p(r)(n) be the r-colored partition function. Our methods yield similar congruences for p(r)(n). In particular, if r is an odd positive integer for which l > 5r + 19 and 2(r+2) not equivalent to 2(+/- 1)(mod l), then we show that there are infinitely many congruences of the form p(r)(lQ(3)n + beta) equivalent to 0 (mod l). Our methods involve the theory of modular Galois representations. (c) 2022 Elsevier Inc. All rights reserved.