We present a unified approach to establish infinite families of congruences for [Formula: see text] for arbitrary positive integer [Formula: see text], where [Formula: see text] is given by the [Formula: see text]th power of the Euler product [Formula: see text]. For [Formula: see text], define [Formula: see text] to be the least positive integer such that [Formula: see text] and [Formula: see text] the least non-negative integer satisfying [Formula: see text]. Using the Atkin [Formula: see text]-operator, we find that the generating function of [Formula: see text] (respectively, [Formula: see text]) can be expressed as the product of an integral linear combination of modular functions on [Formula: see text] and [Formula: see text] (respectively, [Formula: see text]) for any [Formula: see text] and [Formula: see text]. By investigating the properties of the modular equations of the [Formula: see text]th order under the Atkin [Formula: see text]-operator, we obtain that these generating functions are determined by some linear recurring sequences. Utilizing the periodicity of these linear recurring sequences modulo [Formula: see text], we are led to infinite families of congruences for [Formula: see text] modulo any [Formula: see text] with [Formula: see text] and periodic relations between the values of [Formula: see text] modulo powers of [Formula: see text]. As applications, infinite families of congruences for many partition functions such as [Formula: see text]-core partition functions, the partition function and Andrews’ spt-function are easily obtained.
This paper is concerned with a class of partition functions a(n) introduced by Radu and defined in terms of eta-quotients. By utilizing the transformation laws of Newman, Schoeneberg and Robins, and Radu's algorithms, we present an algorithm to find Ramanujan-type identities for a(mn + t). While this algorithm is not guaranteed to succeed, it applies to many cases. For example, we deduce a witness identity for p(11n + 6) with integer coefficients. Our algorithm also leads to Ramanujan-type identities for the overpartition functions p(5n + 2) and p(5n + 3) and Andrews-Paule's broken 2-diamond partition functions △ 2 (25n + 14) and △ 2 (25n + 24). It can also be extended to derive Ramanujan-type identities on a more general class of partition functions. For example, it yields the Ramanujan-type identities on Andrews' singular overpartition functions Q 3,1 (9n + 3) and Q 3,1 (9n + 6) due to Shen, the 2-dissection formulas of Ramanujan and the 8-dissection formulas due to Hirschhorn.Atkin and Swinnerton-Dyer [5] have shown that g t can always be expressed by certain infinite products for m > 3. Then the left hand side of (1.3) can be expressed in terms of certain infinite products. Kolberg pointed out that when m > 5, this becomes much more complicated. For m = 11, 13, Bilgici and Ekin [7,8] used the method of Kolberg to compute the generating function ∞ n=0 p(mn + t)q mn+t 1. M |N .
The study of Andrews–Beck type congruences for partitions has its origin in the work by Andrews, who proved two congruences on the total number of parts in the partitions of [Formula: see text] with the Dyson rank, conjectured by George Beck. Recently, Lin, Peng and Toh proved many Andrews–Beck type congruences for [Formula: see text]-colored partitions. Moreover, they posed eight conjectural congruences. In this paper, we confirm two congruences modulo [Formula: see text] by utilizing some [Formula: see text]-series techniques and the theory of modular forms.
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