{"title":"非均匀格上的分数阶和与分数阶差分及欧拉和柯西公式的模拟","authors":"Jin-fa Cheng","doi":"10.1007/s11766-021-4013-1","DOIUrl":null,"url":null,"abstract":"<div><p>As is well known, the definitions of fractional sum and fractional difference of <i>f</i> (<i>z</i>) on non-uniform lattices <i>x</i>(<i>z</i>) = <i>c</i><sub>1</sub><i>z</i><sup>2</sup> + <i>c</i><sub>2</sub><i>z</i> + <i>c</i><sub>3</sub> or <i>x</i>(<i>z</i>) = <i>c</i><sub>1</sub><i>q</i><sup><i>z</i></sup> + <i>c</i><sub>2</sub><i>q</i><sup>−<i>z</i></sup> + <i>c</i><sub>3</sub> are more difficult and complicated. In this article, for the first time we propose the definitions of the fractional sum and fractional difference on non-uniform lattices by two different ways. The analogue of Euler’s Beta formula, Cauchy’ Beta formula on non-uniform lattices are established, and some fundamental theorems of fractional calculas, the solution of the generalized Abel equation on non-uniform lattices are obtained etc.</p></div>","PeriodicalId":55568,"journal":{"name":"Applied Mathematics-A Journal of Chinese Universities Series B","volume":"36 3","pages":"420 - 442"},"PeriodicalIF":1.0000,"publicationDate":"2021-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Fractional sum and fractional difference on non-uniform lattices and analogue of Euler and Cauchy Beta formulas\",\"authors\":\"Jin-fa Cheng\",\"doi\":\"10.1007/s11766-021-4013-1\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>As is well known, the definitions of fractional sum and fractional difference of <i>f</i> (<i>z</i>) on non-uniform lattices <i>x</i>(<i>z</i>) = <i>c</i><sub>1</sub><i>z</i><sup>2</sup> + <i>c</i><sub>2</sub><i>z</i> + <i>c</i><sub>3</sub> or <i>x</i>(<i>z</i>) = <i>c</i><sub>1</sub><i>q</i><sup><i>z</i></sup> + <i>c</i><sub>2</sub><i>q</i><sup>−<i>z</i></sup> + <i>c</i><sub>3</sub> are more difficult and complicated. In this article, for the first time we propose the definitions of the fractional sum and fractional difference on non-uniform lattices by two different ways. The analogue of Euler’s Beta formula, Cauchy’ Beta formula on non-uniform lattices are established, and some fundamental theorems of fractional calculas, the solution of the generalized Abel equation on non-uniform lattices are obtained etc.</p></div>\",\"PeriodicalId\":55568,\"journal\":{\"name\":\"Applied Mathematics-A Journal of Chinese Universities Series B\",\"volume\":\"36 3\",\"pages\":\"420 - 442\"},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2021-09-20\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Applied Mathematics-A Journal of Chinese Universities Series B\",\"FirstCategoryId\":\"1089\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s11766-021-4013-1\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Applied Mathematics-A Journal of Chinese Universities Series B","FirstCategoryId":"1089","ListUrlMain":"https://link.springer.com/article/10.1007/s11766-021-4013-1","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Fractional sum and fractional difference on non-uniform lattices and analogue of Euler and Cauchy Beta formulas
As is well known, the definitions of fractional sum and fractional difference of f (z) on non-uniform lattices x(z) = c1z2 + c2z + c3 or x(z) = c1qz + c2q−z + c3 are more difficult and complicated. In this article, for the first time we propose the definitions of the fractional sum and fractional difference on non-uniform lattices by two different ways. The analogue of Euler’s Beta formula, Cauchy’ Beta formula on non-uniform lattices are established, and some fundamental theorems of fractional calculas, the solution of the generalized Abel equation on non-uniform lattices are obtained etc.
期刊介绍:
Applied Mathematics promotes the integration of mathematics with other scientific disciplines, expanding its fields of study and promoting the development of relevant interdisciplinary subjects.
The journal mainly publishes original research papers that apply mathematical concepts, theories and methods to other subjects such as physics, chemistry, biology, information science, energy, environmental science, economics, and finance. In addition, it also reports the latest developments and trends in which mathematics interacts with other disciplines. Readers include professors and students, professionals in applied mathematics, and engineers at research institutes and in industry.
Applied Mathematics - A Journal of Chinese Universities has been an English-language quarterly since 1993. The English edition, abbreviated as Series B, has different contents than this Chinese edition, Series A.