{"title":"多个多对数的双参数变形和","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":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"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":"{\"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\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"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\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s11040-021-09407-0\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s11040-021-09407-0","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Sums of Two-Parameter Deformations of Multiple Polylogarithms
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.
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