{"title":"赫尔维茨型 q 多项式和椭圆多项式的生成函数","authors":"Masaki Kato","doi":"10.1007/s11040-024-09480-1","DOIUrl":null,"url":null,"abstract":"<div><p>Ohno and Zagier (Indag Math 12:483–487, 2001) found that a generating function of sums of multiple polylogarithms can be written in terms of the Gauss hypergeometric function <span>\\({}_2F_1\\)</span>. As a generalization of the Ohno and Zagier formula, Ihara et al. (Can J Math 76:1–17, 2022) showed that a generating function of sums of interpolated multiple polylogarithms of Hurwitz type can be expressed in terms of the generalized hypergeometric function <span>\\({}_{r+1}F_r\\)</span>. In this paper, we establish <i>q</i>- and elliptic analogues of this result. We introduce elliptic <i>q</i>-multiple polylogarithms of Hurwitz type and study a generating function of sums of them. By taking the trigonometric and classical limits in the main theorem, we can obtain <i>q</i>- and elliptic generalizations of the Ihara, Kusunoki, Nakamura and Saeki formula.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"27 2","pages":""},"PeriodicalIF":0.9000,"publicationDate":"2024-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11040-024-09480-1.pdf","citationCount":"0","resultStr":"{\"title\":\"Generating Function of q- and Elliptic Multiple Polylogarithms of Hurwitz Type\",\"authors\":\"Masaki Kato\",\"doi\":\"10.1007/s11040-024-09480-1\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Ohno and Zagier (Indag Math 12:483–487, 2001) found that a generating function of sums of multiple polylogarithms can be written in terms of the Gauss hypergeometric function <span>\\\\({}_2F_1\\\\)</span>. As a generalization of the Ohno and Zagier formula, Ihara et al. (Can J Math 76:1–17, 2022) showed that a generating function of sums of interpolated multiple polylogarithms of Hurwitz type can be expressed in terms of the generalized hypergeometric function <span>\\\\({}_{r+1}F_r\\\\)</span>. In this paper, we establish <i>q</i>- and elliptic analogues of this result. We introduce elliptic <i>q</i>-multiple polylogarithms of Hurwitz type and study a generating function of sums of them. By taking the trigonometric and classical limits in the main theorem, we can obtain <i>q</i>- and elliptic generalizations of the Ihara, Kusunoki, Nakamura and Saeki formula.</p></div>\",\"PeriodicalId\":694,\"journal\":{\"name\":\"Mathematical Physics, Analysis and Geometry\",\"volume\":\"27 2\",\"pages\":\"\"},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2024-05-21\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://link.springer.com/content/pdf/10.1007/s11040-024-09480-1.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematical Physics, Analysis and Geometry\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s11040-024-09480-1\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Physics, Analysis and Geometry","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s11040-024-09480-1","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Generating Function of q- and Elliptic Multiple Polylogarithms of Hurwitz Type
Ohno and Zagier (Indag Math 12:483–487, 2001) found that a generating function of sums of multiple polylogarithms can be written in terms of the Gauss hypergeometric function \({}_2F_1\). As a generalization of the Ohno and Zagier formula, Ihara et al. (Can J Math 76:1–17, 2022) showed that a generating function of sums of interpolated multiple polylogarithms of Hurwitz type can be expressed in terms of the generalized hypergeometric function \({}_{r+1}F_r\). In this paper, we establish q- and elliptic analogues of this result. We introduce elliptic q-multiple polylogarithms of Hurwitz type and study a generating function of sums of them. By taking the trigonometric and classical limits in the main theorem, we can obtain q- and elliptic generalizations of the Ihara, Kusunoki, Nakamura and Saeki formula.
期刊介绍：
MPAG is a peer-reviewed journal organized in sections. Each section is editorially independent and provides a high forum for research articles in the respective areas.
The entire editorial board commits itself to combine the requirements of an accurate and fast refereeing process.
The section on Probability and Statistical Physics focuses on probabilistic models and spatial stochastic processes arising in statistical physics. Examples include: interacting particle systems, non-equilibrium statistical mechanics, integrable probability, random graphs and percolation, critical phenomena and conformal theories. Applications of probability theory and statistical physics to other areas of mathematics, such as analysis (stochastic pde''s), random geometry, combinatorial aspects are also addressed.
The section on Quantum Theory publishes research papers on developments in geometry, probability and analysis that are relevant to quantum theory. Topics that are covered in this section include: classical and algebraic quantum field theories, deformation and geometric quantisation, index theory, Lie algebras and Hopf algebras, non-commutative geometry, spectral theory for quantum systems, disordered quantum systems (Anderson localization, quantum diffusion), many-body quantum physics with applications to condensed matter theory, partial differential equations emerging from quantum theory, quantum lattice systems, topological phases of matter, equilibrium and non-equilibrium quantum statistical mechanics, multiscale analysis, rigorous renormalisation group.
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