We give a closed form for quotients of truncated basic hypergeometric series where the base q is evaluated at roots of unity.
We give a closed form for quotients of truncated basic hypergeometric series where the base q is evaluated at roots of unity.
In the theory of integral weight harmonic Maass forms of manageable growth, two key differential operators, the Bol operator and the shadow operator, play a fundamental role. Harmonic Maass forms of manageable growth canonically split into two parts, and each operator controls one of these parts. A third operator, called the flipping operator, exchanges the role of these two parts. Maass-Poincaré series (of parabolic type) form a convenient basis of negative weight harmonic Maass forms of manageable growth, and flipping has the effect of negating an index. Recently, there has been much interest in locally harmonic Maass forms defined by the first author, Kane, and Kohnen. These are lifts of Poincaré series of hyperbolic type, and are intimately related to the Shimura and Shintani lifts. In this note, we prove that a similar property holds for the flipping operator applied to these Poincaré series.
For the past several years, numerous authors have studied POD and PED partitions from a variety of perspectives. These are integer partitions wherein the odd parts must be distinct (in the case of POD partitions) or the even parts must be distinct (in the case of PED partitions). More recently, Ballantine and Welch were led to consider POND and PEND partitions, which are integer partitions wherein the odd parts cannot be distinct (in the case of POND partitions) or the even parts cannot be distinct (in the case of PEND partitions). Soon after, the first author proved the following results via elementary q-series identities and generating function manipulations, along with mathematical induction: For all and all
We study Kloosterman sums on the orthogonal groups and , associated to short elements of their respective Weyl groups. An explicit description for these sums is obtained in terms of multi-dimensional exponential sums. These are bounded by a combination of methods from algebraic geometry and p-adic analysis.
We discuss arithmetic questions related to the 'poor man's adèle ring' whose elements are encoded by sequences indexed by prime numbers, with each viewed as a residue in . Our main theorem is about the -transcendence of the element , where (Schur's q-Fibonacci numbers) are the (1, 1)-entries of -matrices
We prove normal distribution laws for primes of bad semistable reduction in families of curves. As a consequence, we deduce that when ordered by height, of curves in these families have, in a precise sense, many such primes.
We study cylindric partitions with two-element profiles using MacMahon's partition analysis. We find explicit formulas for the generating functions of the number of cylindric partitions by first finding the recurrences using partition analysis and then solving them. We also note some q-series identities related to these objects that show the manifestly positive nature of some alternating series. We generalize the proven identities and conjecture new polynomial refinements of Andrews-Gordon and Bressoud identities, which are companion to Foda-Quano's refinements. Finally, using a variant of the Bailey lemma, we present many new infinite hierarchies of polynomial identities.
Supplementary information: The online version contains supplementary material available at 10.1007/s11139-025-01225-0.
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