New congruences modulo powers of 2 and 3 for 9-regular partitions

Let b(9)(n) denote the number of 9-regular partitions of n. Recently, employing the theory of modular forms, Keith established several congruences modulo 2 and 3 for b(9)(n). He also presented four conjectures on b(9)(n) and two of them have been proved by Lin, and Xia and Yao. The remaining two conjectures are b(9)(32n + 13) equivalent to 0 (mod 12) and b(9)(64n 13) equivalent to 0 (mod 24) for n >= 0. In this paper, employing 2-dissection formulas for certain quotients of theta functions, we prove that b(9)(32n + 13) equivalent to 0 (mod 4) and b(9)(64n + 13) equivalent to 0 (mod 8) for n >= 0. Combining these two congruences and the congruence b(9)(16n + 13) equivalent to 0 (mod 3) proved by Keith, we confirm the remaining two conjectures of Keith. We also establish two infinite families of congruences modulo 9 for b(9)(n). For example, we prove that for all integers n >= 0 and k >= 1, b(9)(2(6k)n + 5x2(6 kappa-1)-1/3) equivalent to 0 (mod 9). (C) 2014 Elsevier Inc. All rights reserved.