We give a formula for the quantum SU(2) invariant at ζ = e4π i/r for Lens space L(p, q), and we prove that the asymptotic expansion is represented by a sum of contributions from SL2C flat connections whose coefficients are square roots of the Reidemeister torsions.
{"title":"ON THE ASYMPTOTIC EXPANSION OF THE QUANTUM SU(2) INVARIANT AT ζ = e4πi/r FOR LENS SPACE L (p, q)","authors":"T. Takata, Rika Tanaka","doi":"10.2206/kyushujm.74.265","DOIUrl":"https://doi.org/10.2206/kyushujm.74.265","url":null,"abstract":"We give a formula for the quantum SU(2) invariant at ζ = e4π i/r for Lens space L(p, q), and we prove that the asymptotic expansion is represented by a sum of contributions from SL2C flat connections whose coefficients are square roots of the Reidemeister torsions.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68557643","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the rationality of gamma factors associated to certain Hasse zeta functions. We show many explicit examples of rational gamma factors coming from products of GL(n).
{"title":"GAMMA FACTORS OF ZETA FUNCTIONS AS ABSOLUTE ZETA FUNCTIONS","authors":"Hidekazu Tanaka","doi":"10.2206/kyushujm.74.441","DOIUrl":"https://doi.org/10.2206/kyushujm.74.441","url":null,"abstract":"We study the rationality of gamma factors associated to certain Hasse zeta functions. We show many explicit examples of rational gamma factors coming from products of GL(n).","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68557517","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Some of the connection problems associated with the system of differential equations E2, which is satisfied by Appell’s F2 function, are solved by using integrals of Euler type. The present results give another proof of connection formulas related with Appell’s F2, Horn’s H2 and Olsson’s FP functions, which are obtained by Olsson. 0. Introduction Appell’s hypergeometric function F2 is the analytic continuation of F2(a, b1, b2, c1, c2; x, y)= ∑ m,n≥0 (a)m+n(b1)m(b2)n m!n!(c1)m(c2)n xm yn, |x | + |y|< 1, where (a)n = 0(a + n)/0(a), and satisfies the system E2 of rank four [AKdF, Er]: (E2) [ x(1− x) ∂2 ∂x2 − xy ∂2 ∂x∂y + {c1 − (a + b1 + 1)x} ∂ ∂x − b1 y ∂ ∂y − ab1 ] F = 0, [ y(1− y) ∂2 ∂y2 − xy ∂2 ∂x∂y + {c2 − (a + b2 + 1)y} ∂ ∂y − b2x ∂ ∂x − ab2 ] F = 0, which is defined on the space C2 { {x = 0} ∪ {x = 1} ∪ {y = 0} ∪ {y = 1} ∪ {x + y = 1} } ⊂ (P1)2. In [Ol], Olsson shows that a fundamental set of solutions of E2 around the point (0, 1) in the case |x/(1− y)|< 1 or that around the point (0,∞) is given by Horn’s hypergeometric function H2 and Olsson’s hypergeometric function FP , while that around the point (0, 0) is given by F2. Moreover, he also derives some connection formulas related with F2, H2 and 2010 Mathematics Subject Classification: Primary 33C60; Secondary 33C65, 33C70.
本文用欧拉型积分法解决了由apappell的F2函数满足的微分方程组E2的一些连接问题。本文的结果再次证明了由Olsson. 0得到的与Appell 's F2、Horn 's H2和Olsson 's FP函数相关的连接公式。Appell的超几何函数F2是F2(a, b1, b2, c1, c2;x, y) =∑m, n≥0 m (a) + n (b1) m (b2) n m ! n ! (c1) m (c2) n xm yn, x y | + | | | < 1, (a) n = 0 (a + n) / 0 (a),并满足系统E2的排名四(AKdF,呃):(E2)[x(1−x)∂2∂x2−xy∂2∂x∂y + {c1−(+ b1 + 1) x}∂∂x−b1 y∂∂y−有所)F = 0, [y(1−y)∂2∂y2−xy∂2∂x∂y + {c2−(+ b2 + 1) y}∂∂y−b2x∂∂x−ab2) F = 0,这是空间上定义c2 {{x = 0}∪{x = 1}∪{y = 0}∪{y = 1}∪{x + y = 1}}⊂(P1) 2。在[Ol]中，Olsson证明了当|x/(1−y)|< 1或(0，∞)时E2绕点(0,1)的基本解集由Horn的超几何函数H2和Olsson的超几何函数FP给出，而绕点(0,0)的基本解集由F2给出。并推导出F2、H2与2010数学学科分类相关的关联公式:Primary 33C60;次级33C65, 33C70。
{"title":"CONNECTION FORMULAS RELATED WITH APPELL'S F2, HORN'S H2 AND OLSSON'S FP FUNCTIONS","authors":"K. Mimachi","doi":"10.2206/kyushujm.74.15","DOIUrl":"https://doi.org/10.2206/kyushujm.74.15","url":null,"abstract":"Some of the connection problems associated with the system of differential equations E2, which is satisfied by Appell’s F2 function, are solved by using integrals of Euler type. The present results give another proof of connection formulas related with Appell’s F2, Horn’s H2 and Olsson’s FP functions, which are obtained by Olsson. 0. Introduction Appell’s hypergeometric function F2 is the analytic continuation of F2(a, b1, b2, c1, c2; x, y)= ∑ m,n≥0 (a)m+n(b1)m(b2)n m!n!(c1)m(c2)n xm yn, |x | + |y|< 1, where (a)n = 0(a + n)/0(a), and satisfies the system E2 of rank four [AKdF, Er]: (E2) [ x(1− x) ∂2 ∂x2 − xy ∂2 ∂x∂y + {c1 − (a + b1 + 1)x} ∂ ∂x − b1 y ∂ ∂y − ab1 ] F = 0, [ y(1− y) ∂2 ∂y2 − xy ∂2 ∂x∂y + {c2 − (a + b2 + 1)y} ∂ ∂y − b2x ∂ ∂x − ab2 ] F = 0, which is defined on the space C2 { {x = 0} ∪ {x = 1} ∪ {y = 0} ∪ {y = 1} ∪ {x + y = 1} } ⊂ (P1)2. In [Ol], Olsson shows that a fundamental set of solutions of E2 around the point (0, 1) in the case |x/(1− y)|< 1 or that around the point (0,∞) is given by Horn’s hypergeometric function H2 and Olsson’s hypergeometric function FP , while that around the point (0, 0) is given by F2. Moreover, he also derives some connection formulas related with F2, H2 and 2010 Mathematics Subject Classification: Primary 33C60; Secondary 33C65, 33C70.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68557274","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study Pfaffian systems of confluent hypergeometric functions of two variables with rank three, by using rational twisted cohomology groups associated with Euler-type integral representations of them. We give bases of the cohomology groups, whose intersection matrices depend only on parameters. Each connection matrix of our Pfaffian systems admits a decomposition into five parts, each of which is the product of a constant matrix and a rational 1-form on the space of variables.
{"title":"PFAFFIAN SYSTEMS OF CONFLUENT HYPERGEOMETRIC FUNCTIONS OF TWO VARIABLES","authors":"Shigeo Mukai","doi":"10.2206/kyushujm.74.63","DOIUrl":"https://doi.org/10.2206/kyushujm.74.63","url":null,"abstract":"We study Pfaffian systems of confluent hypergeometric functions of two variables with rank three, by using rational twisted cohomology groups associated with Euler-type integral representations of them. We give bases of the cohomology groups, whose intersection matrices depend only on parameters. Each connection matrix of our Pfaffian systems admits a decomposition into five parts, each of which is the product of a constant matrix and a rational 1-form on the space of variables.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68557556","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. We give integral representations of Euler type for Appell’s hypergeometric functions F 2 , F 3 , Horn’s hypergeometric function H 2 and Olsson’s hypergeometric function F P . Their integrands are the same (up to a constant factor), and only the regions of integration vary. Olsson Takayama ], and Koornwinder known that Appell’s hypergeometric function F 3 , Horn’s hypergeometric function H 2 and Olsson’s hypergeometric function P also appear as solutions of . Here Appell’s F 3 , Horn’s H 2 and Olsson’s F P are analytic
{"title":"INTEGRAL REPRESENTATIONS OF APPELL'S F2, F3, HORN'S H2 AND OLSSON'S FP FUNCTIONS","authors":"K. Mimachi","doi":"10.2206/kyushujm.74.1","DOIUrl":"https://doi.org/10.2206/kyushujm.74.1","url":null,"abstract":". We give integral representations of Euler type for Appell’s hypergeometric functions F 2 , F 3 , Horn’s hypergeometric function H 2 and Olsson’s hypergeometric function F P . Their integrands are the same (up to a constant factor), and only the regions of integration vary. Olsson Takayama ], and Koornwinder known that Appell’s hypergeometric function F 3 , Horn’s hypergeometric function H 2 and Olsson’s hypergeometric function P also appear as solutions of . Here Appell’s F 3 , Horn’s H 2 and Olsson’s F P are analytic","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68556993","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce a new class of domains Dn,m(μ, p), called FBH-type domains, in Cn × Cm , where 0< μ ∈ R and p ∈ N. In the special case of p = 1, these domains are just the Fock–Bargmann–Hartogs domains Dn,m(μ) in Cn × Cm introduced by Yamamori. In this paper we obtain a complete description of an arbitrarily given proper holomorphic mapping between two equidimensional FBH-type domains. In particular, we prove that the holomorphic automorphism group Aut(Dn,m(μ, p)) of any FBH-type domain Dn,m(μ, p) with p 6= 1 is a Lie group isomorphic to the compact connected Lie group U (n)×U (m). This tells us that the structure of Aut(Dn,m(μ, p)) with p 6= 1 is essentially different from that of Aut(Dn,m(μ)).
{"title":"ON PROPER HOLOMORPHIC MAPPINGS BETWEEN TWO EQUIDIMENSIONAL FBH-TYPE DOMAINS","authors":"A. Kodama","doi":"10.2206/kyushujm.74.149","DOIUrl":"https://doi.org/10.2206/kyushujm.74.149","url":null,"abstract":"We introduce a new class of domains Dn,m(μ, p), called FBH-type domains, in Cn × Cm , where 0< μ ∈ R and p ∈ N. In the special case of p = 1, these domains are just the Fock–Bargmann–Hartogs domains Dn,m(μ) in Cn × Cm introduced by Yamamori. In this paper we obtain a complete description of an arbitrarily given proper holomorphic mapping between two equidimensional FBH-type domains. In particular, we prove that the holomorphic automorphism group Aut(Dn,m(μ, p)) of any FBH-type domain Dn,m(μ, p) with p 6= 1 is a Lie group isomorphic to the compact connected Lie group U (n)×U (m). This tells us that the structure of Aut(Dn,m(μ, p)) with p 6= 1 is essentially different from that of Aut(Dn,m(μ)).","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68557211","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Kaneko and Koike introduced the notion of extremal quasi-modular form and proposed conjectures on their arithmetic properties. The aim of this note is to prove a rather sharp multiplicity estimate for these quasi-modular forms. The note ends with discussions and partial answers around these conjectures and an appendix by G. Nebe containing the proof of the integrality of the Fourier coefficients of the normalised extremal quasimodular form of weight 14 and depth 1.
{"title":"ON EXTREMAL QUASI-MODULAR FORMS AFTER KANEKO AND KOIKE","authors":"F. Pellarin, G. Nebe","doi":"10.2206/kyushujm.74.401","DOIUrl":"https://doi.org/10.2206/kyushujm.74.401","url":null,"abstract":"Kaneko and Koike introduced the notion of extremal quasi-modular form and proposed conjectures on their arithmetic properties. The aim of this note is to prove a rather sharp multiplicity estimate for these quasi-modular forms. The note ends with discussions and partial answers around these conjectures and an appendix by G. Nebe containing the proof of the integrality of the Fourier coefficients of the normalised extremal quasimodular form of weight 14 and depth 1.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2019-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47812635","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we discuss the parity result for multiple Dirichlet series which contains some special values of multiple zeta functions as special cases, Mordell--Tornheim type of multiple zeta values, zeta values of the root systems and so on. Moreover, we can give explicit expression in terms of lower series by using main theorem.
{"title":"ON THE PARITY RESULT FOR MULTIPLE DIRICHLET SERIES","authors":"Shin-ya Kadota","doi":"10.2206/kyushujm.76.1","DOIUrl":"https://doi.org/10.2206/kyushujm.76.1","url":null,"abstract":"In this paper, we discuss the parity result for multiple Dirichlet series which contains some special values of multiple zeta functions as special cases, Mordell--Tornheim type of multiple zeta values, zeta values of the root systems and so on. Moreover, we can give explicit expression in terms of lower series by using main theorem.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2019-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46420503","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
S. Aouissi, A. Azizi, M. C. Ismaili, D. C. Mayer, M. Talbi
Let $mathit{k}=mathbb{Q}(sqrt[3]{d},zeta_3)$, where $d>1$ is a cube-free positive integer, $mathit{k}_0=mathbb{Q}(zeta_3)$ be the cyclotomic field containing a primitive cube root of unity $zeta_3$, and $G=operatorname{Gal}(mathit{k}/mathit{k}_0)$. The possible prime factorizations of $d$ in our main result [2, Thm. 1.1] give rise to new phenomena concerning the chain $Theta=(theta_i)_{iinmathbb{Z}}$ of textit{lattice minima} in the underlying pure cubic subfield $L=mathbb{Q}(sqrt[3]{d})$ of $mathit{k}$. The aims of the present work are to give criteria for the occurrence of generators of primitive ambiguous principal ideals $(alpha)inmathcal{P}_{mathit{k}}^G/mathcal{P}_{mathit{k}_0}$ among the lattice minima $Theta=(theta_i)_{iinmathbb{Z}}$ of the underlying pure cubic field $L=mathbb{Q}(sqrt[3]{d})$, and to explain exceptional behavior of the chain $Theta$ for certain radicands $d$ with impact on determining the principal factorization type of $L$ and $mathit{k}$ by means of Voronoi's algorithm.
{"title":"PRINCIPAL FACTORS AND LATTICE MINIMA IN CUBIC FIELDS","authors":"S. Aouissi, A. Azizi, M. C. Ismaili, D. C. Mayer, M. Talbi","doi":"10.2206/kyushujm.76.101","DOIUrl":"https://doi.org/10.2206/kyushujm.76.101","url":null,"abstract":"Let $mathit{k}=mathbb{Q}(sqrt[3]{d},zeta_3)$, where $d>1$ is a cube-free positive integer, $mathit{k}_0=mathbb{Q}(zeta_3)$ be the cyclotomic field containing a primitive cube root of unity $zeta_3$, and $G=operatorname{Gal}(mathit{k}/mathit{k}_0)$. The possible prime factorizations of $d$ in our main result [2, Thm. 1.1] give rise to new phenomena concerning the chain $Theta=(theta_i)_{iinmathbb{Z}}$ of textit{lattice minima} in the underlying pure cubic subfield $L=mathbb{Q}(sqrt[3]{d})$ of $mathit{k}$. The aims of the present work are to give criteria for the occurrence of generators of primitive ambiguous principal ideals $(alpha)inmathcal{P}_{mathit{k}}^G/mathcal{P}_{mathit{k}_0}$ among the lattice minima $Theta=(theta_i)_{iinmathbb{Z}}$ of the underlying pure cubic field $L=mathbb{Q}(sqrt[3]{d})$, and to explain exceptional behavior of the chain $Theta$ for certain radicands $d$ with impact on determining the principal factorization type of $L$ and $mathit{k}$ by means of Voronoi's algorithm.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2019-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47412242","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce an iterated integral version of (generalized) log-sine integrals (iterated log-sine integrals) and prove a relation between a multiple polylogarithm and iterated log-sine integrals. We also give a new method for obtaining relations among multiple zeta values, which uses iterated log-sine integrals, and give alternative proofs of several known results related to multiple zeta values and log-sine integrals.
{"title":"MULTIPLE ZETA VALUES AND ITERATED LOG-SINE INTEGRALS","authors":"Ryo Umezawa","doi":"10.2206/kyushujm.74.233","DOIUrl":"https://doi.org/10.2206/kyushujm.74.233","url":null,"abstract":"We introduce an iterated integral version of (generalized) log-sine integrals (iterated log-sine integrals) and prove a relation between a multiple polylogarithm and iterated log-sine integrals. We also give a new method for obtaining relations among multiple zeta values, which uses iterated log-sine integrals, and give alternative proofs of several known results related to multiple zeta values and log-sine integrals.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2019-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47339766","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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