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 \(\overline{p}(5n+2)\) and \(\overline{p}(5n+3)\) and Andrews–Paule’s broken 2-diamond partition functions \(\triangle _{2}(25n+14)\) and \(\triangle _{2}(25n+24)\). 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 \(\overline{Q}_{3,1}(9n+3)\) and \( \overline{Q}_{3,1}(9n+6)\) due to Shen, the 2-dissection formulas of Ramanujan, and the 8-dissection formulas due to Hirschhorn.

中文翻译：

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查找Ramanujan类型身份的模块化函数

本文涉及由Radu引入并根据eta商定义的一类分区函数a（n）。通过利用Newman，Schoeneberg和Robins的变换定律以及Radu的算法，我们提出了一种算法来找到\（a（mn + t）\）的Ramanujan型恒等式。虽然不能保证此算法成功，但它适用于许多情况。例如，我们用整数系数推导\（p（11n + 6）\）的证人身份。我们的算法还导致了超分区函数\（\ overline {p}（5n + 2）\）和\（\ overline {p}（5n + 3）\）和Andrews–Paule破损的2钻石的Ramanujan型身份分区函数\（\ triangle _ {2}（25n + 14）\）和\（\ triangle _ {2}（25n + 24）\）。它也可以扩展为在更通用的分区函数类上派生Ramanujan类型的标识。例如，它在安德鲁斯的奇异过分函数\（\ overline {Q} _ {3,1}（9n + 3）\）和\（\ overline {Q} _ {{3,1}}}中产生Ramanujan型恒等式（9n + 6）\）归因于Shen，Ramanujan的2分解公式和Hirschhorn归因于8分解公式。