{"title":"Sums of Two-Parameter Deformations of Multiple Polylogarithms","authors":"Masaki Kato","doi":"10.1007/s11040-021-09407-0","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper, we introduce a generating function of sums of two-parameter deformations of multiple polylogarithms, denoted by Φ<sub>2</sub>(<i>a</i>;<i>p</i>,<i>q</i>), and study a <i>q</i>-difference equation satisfied by it. We show that this <i>q</i>-difference equation can be solved by expanding Φ<sub>2</sub>(<i>a</i>;<i>p</i>,<i>q</i>) into power series of the parameter <i>p</i> and then using the method of variation of constants. By letting <span>\\(p \\rightarrow 0\\)</span> in the main theorem, we find that the generating function of sums of <i>q</i>-interpolated multiple zeta values can be written in terms of the <i>q</i>-hypergeometric function <sub>3</sub><i>ϕ</i><sub>2</sub>, which is due to Li-Wakabayashi.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2021-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11040-021-09407-0.pdf","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Physics, Analysis and Geometry","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s11040-021-09407-0","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 1
Abstract
In this paper, we introduce a generating function of sums of two-parameter deformations of multiple polylogarithms, denoted by Φ2(a;p,q), and study a q-difference equation satisfied by it. We show that this q-difference equation can be solved by expanding Φ2(a;p,q) into power series of the parameter p and then using the method of variation of constants. By letting \(p \rightarrow 0\) in the main theorem, we find that the generating function of sums of q-interpolated multiple zeta values can be written in terms of the q-hypergeometric function 3ϕ2, which is due to Li-Wakabayashi.
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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