Gamela E. Heragy, Zeinab S. I. Mansour, Karima M. Oraby
{"title":"从使用 q-Riccati 方程的方法得出不定 q 积分","authors":"Gamela E. Heragy, Zeinab S. I. Mansour, Karima M. Oraby","doi":"10.1007/s11139-024-00855-0","DOIUrl":null,"url":null,"abstract":"<p>In an earlier work, a method was introduced for obtaining indefinite <i>q</i>-integrals of <i>q</i>-special functions from the second-order linear <i>q</i>-difference equations that define them. In this paper, we reformulate the method in terms of <i>q</i>-Riccati equations, which are nonlinear and first order. We derive <i>q</i>-integrals using fragments of these Riccati equations, and here only two specific fragment types are examined in detail. The results presented here are for the <i>q</i>-Airy function, the Ramanujan function, the discrete <i>q</i>-Hermite I and II polynomials, the <i>q</i>-hypergeometric functions, the <i>q</i>-Laguerre polynomials, the Stieltjes-Wigert polynomial, the little <i>q</i>-Legendre and the big <i>q</i>-Legendre polynomials.</p>","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":"19 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Indefinite q-integrals from a method using q-Riccati equations\",\"authors\":\"Gamela E. Heragy, Zeinab S. I. Mansour, Karima M. Oraby\",\"doi\":\"10.1007/s11139-024-00855-0\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>In an earlier work, a method was introduced for obtaining indefinite <i>q</i>-integrals of <i>q</i>-special functions from the second-order linear <i>q</i>-difference equations that define them. In this paper, we reformulate the method in terms of <i>q</i>-Riccati equations, which are nonlinear and first order. We derive <i>q</i>-integrals using fragments of these Riccati equations, and here only two specific fragment types are examined in detail. The results presented here are for the <i>q</i>-Airy function, the Ramanujan function, the discrete <i>q</i>-Hermite I and II polynomials, the <i>q</i>-hypergeometric functions, the <i>q</i>-Laguerre polynomials, the Stieltjes-Wigert polynomial, the little <i>q</i>-Legendre and the big <i>q</i>-Legendre polynomials.</p>\",\"PeriodicalId\":501430,\"journal\":{\"name\":\"The Ramanujan Journal\",\"volume\":\"19 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-05-02\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"The Ramanujan Journal\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1007/s11139-024-00855-0\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"The Ramanujan Journal","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s11139-024-00855-0","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Indefinite q-integrals from a method using q-Riccati equations
In an earlier work, a method was introduced for obtaining indefinite q-integrals of q-special functions from the second-order linear q-difference equations that define them. In this paper, we reformulate the method in terms of q-Riccati equations, which are nonlinear and first order. We derive q-integrals using fragments of these Riccati equations, and here only two specific fragment types are examined in detail. The results presented here are for the q-Airy function, the Ramanujan function, the discrete q-Hermite I and II polynomials, the q-hypergeometric functions, the q-Laguerre polynomials, the Stieltjes-Wigert polynomial, the little q-Legendre and the big q-Legendre polynomials.