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q-Racah probability distribution q-Racah 概率分布
The Ramanujan Journal Pub Date : 2024-05-07 DOI: 10.1007/s11139-024-00859-w
Masahito Hayashi, Akihito Hora, Shintarou Yanagida
{"title":"q-Racah probability distribution","authors":"Masahito Hayashi, Akihito Hora, Shintarou Yanagida","doi":"10.1007/s11139-024-00859-w","DOIUrl":"https://doi.org/10.1007/s11139-024-00859-w","url":null,"abstract":"<p>We introduce a certain discrete probability distribution <span>(P_{n,m,k,l;q})</span> having non-negative integer parameters <i>n</i>, <i>m</i>, <i>k</i>, <i>l</i> and quantum parameter <i>q</i> which arises from a zonal spherical function of the Grassmannian over the finite field <span>(mathbb {F}_q)</span> with a distinguished spherical vector. Using representation theoretic arguments and hypergeometric summation technique, we derive the presentation of the probability mass function by a single <i>q</i>-Racah polynomial, and also the presentation of the cumulative distribution function in terms of a terminating <span>({}_4 phi _3)</span>-hypergeometric series.</p>","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140889602","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Congruences for class numbers of $$mathbb {Q}(sqrt{pm 2p})$$ when $$pequiv 3$$ $$(text {mod }4)$$ is prime 当 $$pequiv 3$$$(text {mod }4)$$ 是素数时，$$mathbb {Q}(sqrt{pm 2p})$$的类数的协整关系
The Ramanujan Journal Pub Date : 2024-05-06 DOI: 10.1007/s11139-024-00863-0
Jigu Kim, Yoshinori Mizuno
{"title":"Congruences for class numbers of $$mathbb {Q}(sqrt{pm 2p})$$ when $$pequiv 3$$ $$(text {mod }4)$$ is prime","authors":"Jigu Kim, Yoshinori Mizuno","doi":"10.1007/s11139-024-00863-0","DOIUrl":"https://doi.org/10.1007/s11139-024-00863-0","url":null,"abstract":"<p>For a prime <span>(pequiv 3)</span> <span>((text {mod }4))</span>, let <span>(h(-8p))</span> and <i>h</i>(8<i>p</i>) be the class numbers of <span>(mathbb {Q}(sqrt{-2p}))</span> and <span>(mathbb {Q}(sqrt{2p}))</span>, respectively. Let <span>(Psi (xi ))</span> be the Hirzebruch sum of a quadratic irrational <span>(xi )</span>. We show that <span>(h(-8p)equiv h(8p)Big (Psi (2sqrt{2p})/3-Psi (frac{1+sqrt{2p}}{2})/3Big ))</span> <span>((text {mod }16))</span>. Also, we show that <span>(h(-8p)equiv 2,h(8p)Psi (2sqrt{2p})/3)</span> <span>((text {mod }8))</span> if <span>(pequiv 3)</span> <span>((text {mod }8))</span>, and <span>(h(-8p)equiv big (2,h(8p)Psi (2sqrt{2p})/3big )+4)</span> <span>((text {mod }8))</span> if <span>(pequiv 7)</span> <span>((text {mod }8))</span>.</p>","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140889422","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Moments of Kummer sums weighted by L-functions 用 L 函数加权的库默和的矩
The Ramanujan Journal Pub Date : 2024-05-04 DOI: 10.1007/s11139-024-00862-1
Nilanjan Bag
{"title":"Moments of Kummer sums weighted by L-functions","authors":"Nilanjan Bag","doi":"10.1007/s11139-024-00862-1","DOIUrl":"https://doi.org/10.1007/s11139-024-00862-1","url":null,"abstract":"<p>The main purpose of this article is to study higher order moments of Kummer sums weighted by <i>L</i>-functions using estimates for character sums and analytic methods. The results of this article complement a conjecture of Zhang Wenpeng (2002). Also the results in this article give analogous results of Kummer’s conjecture (1846).</p>","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140889428","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the orthogonality of Atkin-like polynomials and orthogonal polynomial expansion of generalized Faber polynomials 论阿特金类多项式的正交性和广义法布尔多项式的正交多项式展开
The Ramanujan Journal Pub Date : 2024-05-04 DOI: 10.1007/s11139-024-00861-2
Tomoaki Nakaya
{"title":"On the orthogonality of Atkin-like polynomials and orthogonal polynomial expansion of generalized Faber polynomials","authors":"Tomoaki Nakaya","doi":"10.1007/s11139-024-00861-2","DOIUrl":"https://doi.org/10.1007/s11139-024-00861-2","url":null,"abstract":"<p>In this paper, we consider the Atkin-like polynomials that appeared in the study of normalized extremal quasimodular forms of depth 1 on <span>(SL_{2}(mathbb {Z}))</span> by Kaneko and Koike as orthogonal polynomials and clarify their properties. Using them, we show that the normalized extremal quasimodular forms have a certain expression by the linear functional corresponding to the Atkin inner product and prove an unexpected connection between generalized Faber polynomials, which are closely related to certain bases of the vector space of weakly holomorphic modular forms, and normalized extremal quasimodular forms. In particular, we reveal that the orthogonal polynomial expansion coefficients of the generalized Faber polynomials by the Atkin-like polynomials appear in the Fourier coefficients of normalized extremal quasimodular forms multiplied by certain (weakly) holomorphic modular forms.</p>","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140889279","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Summation formulas for a class of terminating $$_4phi _3$$ -series 一类终止 $$_4phi _3$$ 序列的求和公式
The Ramanujan Journal Pub Date : 2024-05-04 DOI: 10.1007/s11139-024-00857-y
Xu Yong, Wenlong Zhang
{"title":"Summation formulas for a class of terminating $$_4phi _3$$ -series","authors":"Xu Yong, Wenlong Zhang","doi":"10.1007/s11139-024-00857-y","DOIUrl":"https://doi.org/10.1007/s11139-024-00857-y","url":null,"abstract":"<p>In this paper we investigate a class of terminating <span>(_4phi _3)</span>-series. By utilizing the Carlitz inversion and four contiguous relations for the <span>(Omega _{lambda ,mu })</span>-series, many new summation formulas for terminating <span>(_4phi _3)</span>-series are obtained.</p>","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140889362","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
An asymptotic expansion for a Lambert series associated with Siegel cusp forms 与西格尔顶点形式相关的朗伯数列的渐近展开
The Ramanujan Journal Pub Date : 2024-05-04 DOI: 10.1007/s11139-024-00864-z
Babita, Abhash Kumar Jha, Abhishek Juyal, Bibekananda Maji
{"title":"An asymptotic expansion for a Lambert series associated with Siegel cusp forms","authors":"Babita, Abhash Kumar Jha, Abhishek Juyal, Bibekananda Maji","doi":"10.1007/s11139-024-00864-z","DOIUrl":"https://doi.org/10.1007/s11139-024-00864-z","url":null,"abstract":"<p>In 2000, Hafner and Stopple proved a conjecture of Zagier which states that the constant term of the automorphic function <span>(|Delta (x+iy)|^2)</span>, i.e., the Lambert series <span>(sum _{n=1}^infty tau (n)^2 e^{-4 pi n y})</span>, can be expressed in terms of the non-trivial zeros of the Riemann zeta function. In this article, we study an asymptotic expansion of a generalized version of the aforementioned Lambert series associated with Siegel cusp forms.</p>","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140889424","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Lie algebras of differential operators for matrix valued Laguerre type polynomials 矩阵值拉盖尔型多项式的微分算子的李代数
The Ramanujan Journal Pub Date : 2024-05-04 DOI: 10.1007/s11139-024-00858-x
Andrea L. Gallo, Pablo Román
{"title":"Lie algebras of differential operators for matrix valued Laguerre type polynomials","authors":"Andrea L. Gallo, Pablo Román","doi":"10.1007/s11139-024-00858-x","DOIUrl":"https://doi.org/10.1007/s11139-024-00858-x","url":null,"abstract":"<p>We study algebras of differential and difference operators acting on matrix valued orthogonal polynomials (MVOPs) with respect to a weight matrix of the form <span>(W^{(nu )}_{phi }(x) = x^{nu }e^{-phi (x)} W^{(nu )}_textrm{pol}(x))</span>, where <span>(nu &gt;0)</span>, <span>(W^{(nu )}_textrm{pol}(x))</span> is a certain matrix valued polynomial and <span>(phi )</span> is an analytic function. We introduce differential operators <span>({mathcal {D}})</span>, <span>({mathcal {D}}^{dagger })</span> which are mutually adjoint with respect to the matrix inner product induced by <span>(W^{(nu )}_{phi }(x))</span>. We prove that the Lie algebra generated by <span>({mathcal {D}})</span> and <span>({mathcal {D}}^{dagger })</span> is finite dimensional if and only if <span>(phi )</span> is a polynomial. For a polynomial <span>(phi )</span>, we describe the structure of this Lie algebra. As a byproduct, we give a partial answer to a problem by Ismail about finite dimensional Lie algebras related to scalar Laguerre type polynomials. The case <span>(phi (x)=x)</span> is discussed in detail.</p>","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140889430","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Intersective polynomials along the irreducibles in function fields 沿函数域中的不还原性的相交多项式
The Ramanujan Journal Pub Date : 2024-05-04 DOI: 10.1007/s11139-024-00865-y
Guoquan Li
{"title":"Intersective polynomials along the irreducibles in function fields","authors":"Guoquan Li","doi":"10.1007/s11139-024-00865-y","DOIUrl":"https://doi.org/10.1007/s11139-024-00865-y","url":null,"abstract":"<p>Let <span>(mathbb {F}_q[t])</span> be the polynomial ring over the finite field <span>(mathbb {F}_q)</span> of <i>q</i> elements. For a natural number <span>(Nge 1,)</span> let <span>(mathbb {G}_N)</span> be the subset of <span>(mathbb {F}_q[t])</span> containing all polynomials of degree less than <i>N</i>. Let <span>(hin mathbb {F}_q[t][x])</span> be a polynomial of degree <span>(2le k&lt;p,)</span> the characteristic of <span>(mathbb {F}_q.)</span> Suppose that for every <span>(din mathbb {F}_q[t]setminus {0},)</span> there exists <span>(min mathbb {F}_q[t])</span> such that <span>(dmid h(m))</span> and <span>((d,m)=1.)</span> Let <span>(Asubseteq mathbb {G}_N)</span> with <span>(|A|=delta q^N.)</span> Suppose further that <span>((A-A)cap left( h(Omega )setminus {0}right) =emptyset ,)</span> where <span>(A-A)</span> is the difference set of <i>A</i> and <span>(Omega )</span> denotes the set of all monic irreducible polynomials in <span>(mathbb {F}_q[t])</span>. It is proved that <span>(delta ll N^{-mu })</span> for any <span>(0&lt;mu &lt;1/(2k-2),)</span> where the implied constant depends only on <span>(q, h)</span> and <span>(mu .)</span></p>","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140889603","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
An explicit version of Chen’s theorem assuming the Generalized Riemann Hypothesis 假设广义黎曼假设的陈氏定理的明确版本
The Ramanujan Journal Pub Date : 2024-05-03 DOI: 10.1007/s11139-024-00866-x
Matteo Bordignon, Valeriia Starichkova
{"title":"An explicit version of Chen’s theorem assuming the Generalized Riemann Hypothesis","authors":"Matteo Bordignon, Valeriia Starichkova","doi":"10.1007/s11139-024-00866-x","DOIUrl":"https://doi.org/10.1007/s11139-024-00866-x","url":null,"abstract":"<p>We prove that assuming the Generalized Riemann Hypothesis every even integer larger than <span>(exp (exp (14)))</span> can be written as the sum of a prime number and a number that has at most two prime factors.</p>","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140889426","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Some determinantal representations of derangement numbers and polynomials 出差数和多项式的一些行列式表示法
The Ramanujan Journal Pub Date : 2024-05-03 DOI: 10.1007/s11139-024-00867-w
Chak-On Chow
{"title":"Some determinantal representations of derangement numbers and polynomials","authors":"Chak-On Chow","doi":"10.1007/s11139-024-00867-w","DOIUrl":"https://doi.org/10.1007/s11139-024-00867-w","url":null,"abstract":"<p>Munarini (J Integer Seq 23: Article 20.3.8, 2020) recently showed that the derangement polynomial <span>(d_n(q)=sum _{sigma in {mathcal {D}}_n}q^{{{,textrm{maj},}}(sigma )})</span> is expressible as the determinant of either an <span>(ntimes n)</span> tridiagonal matrix or an <span>(ntimes n)</span> lower Hessenberg matrix. Qi et al. (Cogent Math 3:1232878, 2016) showed that the classical derangement number <span>(d_n=n!sum _{k=0}^nfrac{(-1)^k}{k!})</span> is expressible as a tridiagonal determinant of order <span>(n+1)</span>. We show in this work that similar determinantal expressions exist for the type <i>B</i> derangement polynomial <span>(d_n^B(q)=sum _{sigma in {mathcal {D}}_n^B}q^{{{,textrm{fmaj},}}(sigma )})</span> studied previously by Chow (Sém Lothar Combin 55:B55b, 2006), and the type <i>D</i> derangement polynomial <span>(d_n^D(q)=sum _{sigma in {mathcal {D}}_n^D}q^{{{,textrm{maj},}}(sigma )})</span> studied recently by Chow (Taiwanese J Math 27(4):629–646, 2023). Representations of the types <i>B</i> and <i>D</i> derangement numbers <span>(d_n^B)</span> and <span>(d_n^D)</span> as determinants of order <span>(n+1)</span> are also presented.</p>","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140889281","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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