Ramanujan Journal最新文献

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A positivity conjecture for a quotient of q-binomial coefficients. 一个关于q-二项式系数商的正性猜想。

IF 0.7 3区数学 Q3 MATHEMATICS

Ramanujan Journal

Pub Date : 2026-01-01 Epub Date: 2025-12-17 DOI: 10.1007/s11139-025-01285-2

M Gatzweiler, C Krattenthaler

We conjecture that, if the quotient of two q-binomial coefficients with the same top argument is a polynomial, then it has non-negative coefficients. We summarise what is known about the conjecture and prove it in two non-trivial cases. Moreover, we move ahead to extend our conjecture to D. Stanton's fake Gaussian sequences. As a corollary of one of our results we obtain that a polynomial that is conjectured to be a cyclic sieving polynomial for Kreweras words [S. Hopkins and M. Rubey, Selecta Math. (N.S.) 28 (2022), Paper No. 10] is indeed a polynomial with non-negative integer coefficients.

我们推测，如果具有相同顶参数的两个q-二项式系数的商是一个多项式，则它具有非负系数。我们总结了关于这个猜想的已知情况，并用两个非平凡的例子证明了它。此外，我们继续将我们的猜想扩展到D. Stanton的假高斯序列。作为我们其中一个结果的推论，我们得到了一个被推测为Kreweras词的循环筛选多项式的多项式[S]。霍普金斯和鲁比先生，选择数学。(n。)28(2022)，第10号论文]确实是一个系数非负整数的多项式。

引用次数: 0

Arithmetic properties of character degrees and the generalised knutson index. 特征度的算术性质与广义knutson索引。

IF 0.7 3区数学 Q3 MATHEMATICS

Ramanujan Journal

Pub Date : 2026-01-01 Epub Date: 2026-01-23 DOI: 10.1007/s11139-025-01311-3

Diego Martín Duro

In this paper, we introduce the generalised Knutson Index and compute it for the special linear groups and projective special linear groups of degree two by computing the lowest common multiple of the degrees of their irreducible representations. We also classify all alternating and symmetric groups such that the lowest common multiple of the degrees of their irreducible representations equals the order groups, which yields a lower bound on the generalised Knutson Indices of these groups.

本文引入广义Knutson指数，并通过计算二阶特殊线性群和投影线性群的不可约表示的最小公倍数来计算其广义Knutson指数。我们还对所有交替对称群进行了分类，使其不可约表示度的最小公倍数等于序群，从而得到了这些群的广义Knutson指标的下界。

引用次数: 0

A remarkable basic hypergeometric identity. 一个了不起的超几何恒等式。

IF 0.6 3区数学 Q3 MATHEMATICS

Ramanujan Journal

Pub Date : 2025-01-01 Epub Date: 2025-01-31 DOI: 10.1007/s11139-024-00994-4

Christian Krattenthaler, Wadim Zudilin

We give a closed form for quotients of truncated basic hypergeometric series where the base q is evaluated at roots of unity.

给出了截断基超几何级数的商的一个封闭形式，其中基q在单位根处求值。

引用次数: 0

Flipping operators and locally harmonic Maass forms. 翻转算子和局部调和质量形式。

IF 0.7 3区数学 Q3 MATHEMATICS

Ramanujan Journal

Pub Date : 2025-01-01 Epub Date: 2025-08-08 DOI: 10.1007/s11139-025-01183-7

Kathrin Bringmann, Andreas Mono, Larry Rolen

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.

在可管理增长的积分权调和质量形式理论中，两个关键的微分算子，Bol算子和阴影算子，起着基本的作用。调和质量形式的可管理增长通常分为两个部分，每个算子控制其中一个部分。第三个操作符，称为翻转操作符，交换这两个部分的角色。抛物型的Maass- poincar级数形成了可管理增长的负权和质量形式的便利基础，翻转具有负指标的作用。最近，人们对第一作者Kane和Kohnen定义的局部谐波质量形式很感兴趣。这些升降机属于庞加莱系列双曲型升降机，与志村升降机和新谷升降机密切相关。在这篇笔记中，我们证明了一个类似的性质适用于这些庞卡罗级数的翻转算子。

引用次数: 0

Explaining unforeseen congruence relationships between PEND and POND partitions via an Atkin-Lehner involution. 通过Atkin-Lehner对合解释PEND和POND分区之间不可预见的同余关系。

IF 0.6 3区数学 Q3 MATHEMATICS

Ramanujan Journal

Pub Date : 2025-01-01 Epub Date: 2025-05-23 DOI: 10.1007/s11139-025-01111-9

James A Sellers, Nicolas Allen Smoot

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 $α \geq 1$ and all $n \geq 0,$ $\begin{matrix} pend (3^{2 α + 1} n + \frac{17 \cdot 3^{2 α} - 1}{8}) & \equiv 0 (mod 3), and \\ pond (3^{2 α + 1} n + \frac{23 \cdot 3^{2 α} + 1}{8}) & \equiv 0 (mod 3) \end{matrix}$ where $pend (n)$ counts the number of PEND partitions of weight n and $pond (n)$ counts the number of POND partitions of weight n. In this work, we revisit these families of congruences, and we show a relationship between them via an Atkin-Lehner involution. From this relationship, we can show that, once one of the above families of congruences is known, the other follows immediately.

在过去的几年中，许多作者从不同的角度研究了POD和PED分区。这些是整数分区，其中奇数部分必须是不同的（在POD分区的情况下），或者偶数部分必须是不同的（在PED分区的情况下）。最近，Ballantine和Welch被引导考虑POND和PEND分区，它们是整数分区，其中奇数部分不能区分（在POND分区的情况下）或偶数部分不能区分（在PEND分区的情况下）。不久后，第一作者通过初等q级数恒等式和生成函数的操作，结合数学归纳法证明了以下结果：对于所有α≥1,所有n≥0 , 挂件3 2α+ 1 n + 17·3 2α- 1 8≡0 (mod 3),和池塘3 2α+ 1 n + 23·3 2α+ 1 8≡0 3 (mod)挂件(n)计数挂件分区的数量重量n和池塘(n)计数池分区的数量重量n。在这个工作中,我们重新审视这些家庭的刻画,我们通过Atkin-Lehner对合显示它们之间的关系。从这个关系中，我们可以证明，一旦上述同余族中的一个已知，另一个就会紧随其后。

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引用次数: 0

Kloosterman sums on orthogonal groups. 正交群上的Kloosterman和。

IF 0.6 3区数学 Q3 MATHEMATICS

Ramanujan Journal

Pub Date : 2025-01-01 Epub Date: 2025-06-23 DOI: 10.1007/s11139-025-01135-1

Catinca Mujdei

We study Kloosterman sums on the orthogonal groups $S O_{3, 3}$ and $S O_{4, 2}$ , 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.

研究了与各自Weyl群的短元素相关的正交群so3,3和so4,2上的Kloosterman和。用多维指数和的形式给出了这些和的显式描述。这些是由代数几何和p进分析的方法组合而成的。

引用次数: 0

Irrationality and transcendence questions in the 'poor man's adèle ring'. “穷人的ad<e:1>圈”中的非理性与超越问题。

IF 0.6 3区数学 Q3 MATHEMATICS

Ramanujan Journal

Pub Date : 2025-01-01 Epub Date: 2025-06-18 DOI: 10.1007/s11139-025-01132-4

Florian Luca, Wadim Zudilin

We discuss arithmetic questions related to the 'poor man's adèle ring' $A$ whose elements are encoded by sequences ${(t_{p})}_{p}$ indexed by prime numbers, with each $t_{p}$ viewed as a residue in $Z / p Z$ . Our main theorem is about the $A$ -transcendence of the element ${(F_{p} (q))}_{p}$ , where $F_{n} (q)$ (Schur's q-Fibonacci numbers) are the (1, 1)-entries of $2 \times 2$ -matrices $\begin{matrix} (\begin{matrix} 1 & 1 \\ 1 & 0 \end{matrix}) (\begin{matrix} 1 & 1 \\ q & 0 \end{matrix}) (\begin{matrix} 1 & 1 \\ q^{2} & 0 \end{matrix}) \dots (\begin{matrix} 1 & 1 \\ q^{n - 2} & 0 \end{matrix}) \end{matrix}$ and $q > 1$ is an integer. This result was previously known for $q > 1$ square free under the GRH.

我们讨论了与“穷人ad环”A有关的算术问题，其元素由素数索引的序列（t p） p编码，每个t p被视为Z / p Z中的残数。我们的主要定理是关于元素（F p (q)） p的A -超越，其中F n (q)（舒尔的q-斐波那契数）是2 × 2矩阵1 1 1 1 1 1 1 q 0 1 1 q 2 0⋯1 1 q n - 20的（1,1）项，q > 1是一个整数。这个结果之前在GRH下是已知的qbbb101平方自由度。

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引用次数: 0

Normal distribution of bad reduction. 不良还原率正态分布。

IF 0.6 3区数学 Q3 MATHEMATICS

Ramanujan Journal

Pub Date : 2025-01-01 Epub Date: 2025-05-22 DOI: 10.1007/s11139-025-01108-4

Robert J Lemke Oliver, Daniel Loughran, Ari Shnidman

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, $100 %$ of curves in these families have, in a precise sense, many such primes.

证明了曲线族中不良半稳约素数的正态分布规律。因此，我们推断，当按高度排序时，这些科中100%的曲线在精确意义上有许多这样的素数。

引用次数: 0

Ramanujan's partition generating functions modulo ℓ. 拉马努金分割生成函数模为l。

IF 0.7 3区数学 Q3 MATHEMATICS

Ramanujan Journal

Pub Date : 2025-01-01 Epub Date: 2025-11-04 DOI: 10.1007/s11139-025-01241-0

Kathrin Bringmann, William Craig, Ken Ono

For the partition function p(n), Ramanujan proved the striking identities <dispformula> <math> <mrow> <mtable> <mtr> <mtd> <mrow> <mtable> <mtr> <mtd> <mrow><msub><mi>P</mi> <mn>5</mn></msub> <mrow><mo>(</mo> <mi>q</mi> <mo>)</mo></mrow> <mo>:</mo> <mo>=</mo> <munder><mo>∑</mo> <mrow><mi>n</mi> <mo>≥</mo> <mn>0</mn></mrow> </munder> <mi>p</mi> <mrow><mo>(</mo> <mn>5</mn> <mi>n</mi> <mo>+</mo> <mn>4</mn> <mo>)</mo></mrow> <msup><mi>q</mi> <mi>n</mi></msup> </mrow> </mtd> <mtd><mrow><mo>=</mo> <mn>5</mn> <munder><mo>∏</mo> <mrow><mi>n</mi> <mo>≥</mo> <mn>1</mn></mrow> </munder> <mfrac> <msubsup> <mfenced><msup><mi>q</mi> <mn>5</mn></msup> <mo>;</mo> <msup><mi>q</mi> <mn>5</mn></msup> </mfenced> <mrow><mi>∞</mi></mrow> <mn>5</mn></msubsup> <msubsup><mrow><mo>(</mo> <mi>q</mi> <mo>;</mo> <mi>q</mi> <mo>)</mo></mrow> <mrow><mi>∞</mi></mrow> <mn>6</mn></msubsup> </mfrac> <mo>,</mo></mrow> </mtd> </mtr> <mtr> <mtd><mrow><mrow></mrow> <msub><mi>P</mi> <mn>7</mn></msub> <mrow><mo>(</mo> <mi>q</mi> <mo>)</mo></mrow> <mo>:</mo> <mo>=</mo> <munder><mo>∑</mo> <mrow><mi>n</mi> <mo>≥</mo> <mn>0</mn></mrow> </munder> <mi>p</mi> <mrow><mo>(</mo> <mn>7</mn> <mi>n</mi> <mo>+</mo> <mn>5</mn> <mo>)</mo></mrow> <msup><mi>q</mi> <mi>n</mi></msup> </mrow> </mtd> <mtd><mrow><mo>=</mo> <mn>7</mn> <munder><mo>∏</mo> <mrow><mi>n</mi> <mo>≥</mo> <mn>1</mn></mrow> </munder> <mfrac> <msubsup> <mfenced><msup><mi>q</mi> <mn>7</mn></msup> <mo>;</mo> <msup><mi>q</mi> <mn>7</mn></msup> </mfenced> <mrow><mi>∞</mi></mrow> <mn>3</mn></msubsup> <msubsup><mrow><mo>(</mo> <mi>q</mi> <mo>;</mo> <mi>q</mi> <mo>)</mo></mrow> <mrow><mi>∞</mi></mrow> <mn>4</mn></msubsup> </mfrac> <mo>+</mo> <mn>49</mn> <mi>q</mi> <munder><mo>∏</mo> <mrow><mi>n</mi> <mo>≥</mo> <mn>1</mn></mrow> </munder> <mfrac> <msubsup> <mfenced><msup><mi>q</mi> <mn>7</mn></msup> <mo>;</mo> <msup><mi>q</mi> <mn>7</mn></msup> </mfenced> <mrow><mi>∞</mi></mrow> <mn>7</mn></msubsup> <msubsup><mrow><mo>(</mo> <mi>q</mi> <mo>;</mo> <mi>q</mi> <mo>)</mo></mrow> <mrow><mi>∞</mi></mrow> <mn>8</mn></msubsup> </mfrac> <mo>,</mo></mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> </mrow> </math> </dispformula> where <math> <mrow> <msub><mrow><mo>(</mo> <mi>q</mi> <mo>;</mo> <mi>q</mi> <mo>)</mo></mrow> <mi>∞</mi></msub> <mo>:</mo> <mo>=</mo> <msub><mo>∏</mo> <mrow><mi>n</mi> <mo>≥</mo> <mn>1</mn></mrow> </msub> <mrow><mo>(</mo> <mn>1</mn> <mo>-</mo> <msup><mi>q</mi> <mi>n</mi></msup> <mo>)</mo></mrow> <mo>.</mo></mrow> </math> As these identities imply his celebrated congruences modulo 5 and 7, it is natural to seek, for primes <math><mrow><mi>ℓ</mi> <mo>≥</mo> <mn>5</mn> <mo>,</mo></mrow> </math> closed form expressions of the power series <dispformula> <math> <mrow><msub><mi>P</mi> <mi>ℓ</mi></msub> <mrow><mo>(</mo> <mi>q</mi> <mo>)</mo></mrow> <mo>:</mo> <mo>=</mo> <munder><mo>∑</mo> <mrow><mi>n</mi> <mo>≥</mo> <mn>0</mn></mrow> </munder> <mi>p</mi> <mrow><mo>(</mo> <mi>ℓ</mi> <mi>n</mi> <mo>-</mo> <m

对于配分函数p(n)， Ramanujan证明了显著恒等式p5 (q): =∑n≥0 p(5n + 4) q n = 5∏n≥1 q 5；q 5∞5 (q; q)∞6,p7 (q): =∑n≥0 P (7n + 5) q n = 7∏n≥1 q 7；Q 7∞3 (Q； Q)∞4 + 49 Q∏n≥1 Q 7；Q 7∞7 (Q； Q)∞8，其中（Q； Q）∞:=∏n≥1 （1 - Q n）。由于这些恒等式包含了他著名的模5和模7同余，所以对于≥5的素数，我们很自然地寻求幂级数P (q): =∑n≥0 P (n - δ l) q n （mod l）的封闭形式表达式，其中δ l: = 2 - 1 24。在本文中，我们证明了P r (q)≡c r T r (q) q r；q r∞(mod r)，其中c r∈Z是显式的，T r (q)是SL 2 (Z)上权值为r - 1的特殊Dirichlet级数的分支值的Hecke迹的生成函数。这是拉马努金同余模5、7和11的一个新证明，因为不存在权4、6和10的非平凡尖角形式。

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A MacMahon analysis view of cylindric partitions. 圆柱形分区的麦克马洪分析视图。

IF 0.7 3区数学 Q3 MATHEMATICS

Ramanujan Journal

Pub Date : 2025-01-01 Epub Date: 2025-09-22 DOI: 10.1007/s11139-025-01225-0

Runqiao Li, Ali K Uncu

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.

本文用MacMahon划分分析方法研究了两元剖面的圆柱分区。首先利用分格分析找到递归式，然后求解，得到了圆柱分格数生成函数的显式公式。我们还注意到与这些对象相关的一些q级数恒等式，它们显示出某些交替级数的明显的正性质。我们推广了已证明的恒等式，并推测了与Foda-Quano的改进相伴随的Andrews-Gordon恒等式和Bressoud恒等式的新的多项式改进。最后，利用贝利引理的一个变体，我们给出了多项式恒等式的许多新的无限层次。补充信息：在线版本包含补充资料，提供地址为10.1007/s11139-025-01225-0。

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