Transformation properties of Andrews-Beck NT functions and generalized Appell-Lerch series
Abstract
In 2021, Andrews mentioned that George Beck introduced a partition statistic NT(r,m,n) which is related to Dyson's rank statistic. Motivated by Andrews's work, scholars have established a number of congruences and identities involving NT(r,m,n). In this paper, we strengthen and extend a recent work of Mao on the transformation properties of the NT function and provide an analogy of Hickerson and Mortenson's work on the rank function. As an application, we demonstrate how one can deduce from our results many identities involving NT(r,m,n) and another crank-analog statistic Mω(r,m,n). As a related result, some new properties of generalized Appell-Lerch series are given.
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