Congruences for the coefficients of a pair of third and sixth order mock theta functions.

Recently, da Silva and Sellers [11] studied the arithmetical properties of the coefficients p ξ (n) of the third order mock theta function ξ (q) , which was introduced by Gordon and McIntosh and is defined by ξ (q) = 1 + 2 ∑ n = 1 ∞ q 6 n 2 - 6 n + 1 q ; q 6 n q 5 ; q 6 n = : ∑ n = 0 ∞ p ξ (n) q n. Da Silva and Sellers discovered several congruences modulo 3, 4, 5, 8, and 9 for p ξ (n) . In this paper, we find unexpected congruences modulo some multiples of 5 and 7 for the coefficients of ξ (q) , namely, for all n ≥ 0 , p ξ 1800 n + r ≡ 0 (m o d 24 · 5) for r ∈ { 435 , 795 , 1155 , 1515 } , p ξ 504 n + r ≡ 0 (m o d 576 · 7) for r ∈ { 219 , 291 , 435 }. We also confirm all the conjectural congruences modulo 25 and 125 posed by Zhang and Shi [23] for the coefficients of the sixth order mock theta function β (q) , which is defined by β (q) = ∑ n = 0 ∞ q 3 n 2 + 3 n + 1 q ; q 3 n + 1 q 2 ; q 3 n + 1 . [ABSTRACT FROM AUTHOR]