Closed-form formulas of two Gauss hypergeometric functions of specific parameters

IF 1.2 3区 数学 Q1 MATHEMATICS
Gradimir V. Milovanović , Feng Qi
{"title":"Closed-form formulas of two Gauss hypergeometric functions of specific parameters","authors":"Gradimir V. Milovanović ,&nbsp;Feng Qi","doi":"10.1016/j.jmaa.2024.129024","DOIUrl":null,"url":null,"abstract":"<div><div>Using the Faà di Bruno formula, along with three identities of the partial Bell polynomials, and leveraging two differentiation formulas for the Gauss hypergeometric functions, the authors present several closed-form formulas for the Gauss hypergeometric functions<span><span><span><math><mmultiscripts><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow><none></none><mprescripts></mprescripts><mrow><mn>2</mn></mrow><none></none></mmultiscripts><mo>(</mo><mi>n</mi><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>,</mo><mi>n</mi><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>;</mo><mi>n</mi><mo>+</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>;</mo><mo>−</mo><msup><mrow><mi>z</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mspace></mspace><mtext>and</mtext><mspace></mspace><mmultiscripts><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow><none></none><mprescripts></mprescripts><mrow><mn>2</mn></mrow><none></none></mmultiscripts><mo>(</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>;</mo><mi>n</mi><mo>+</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>;</mo><msup><mrow><mi>z</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span></span></span> for <span><math><mi>n</mi><mo>∈</mo><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>}</mo></math></span> and <span><math><mo>|</mo><mi>z</mi><mo>|</mo><mo>&lt;</mo><mn>1</mn></math></span>. These formulas are analyzed in light of three Gauss relations for contiguous functions, with the aid of a relation between the Gauss hypergeometric functions and the Lerch transcendent. Additionally, the authors determine the location and distribution of the zeros of two polynomials involved in these representations, which contain generalized binomial coefficients. By comparing these formulas, they also derive several combinatorial identities.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"543 2","pages":"Article 129024"},"PeriodicalIF":1.2000,"publicationDate":"2024-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Mathematical Analysis and Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022247X24009466","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0

Abstract

Using the Faà di Bruno formula, along with three identities of the partial Bell polynomials, and leveraging two differentiation formulas for the Gauss hypergeometric functions, the authors present several closed-form formulas for the Gauss hypergeometric functionsF12(n+12,n+12;n+32;z2)andF12(1,n+12;n+32;z2) for n{0,1,2,} and |z|<1. These formulas are analyzed in light of three Gauss relations for contiguous functions, with the aid of a relation between the Gauss hypergeometric functions and the Lerch transcendent. Additionally, the authors determine the location and distribution of the zeros of two polynomials involved in these representations, which contain generalized binomial coefficients. By comparing these formulas, they also derive several combinatorial identities.
特定参数的两个高斯超几何函数的闭式公式
作者使用 Faà di Bruno 公式以及贝尔局部多项式的三个同余式,并利用高斯超几何函数的两个微分公式,提出了 n∈{0,1,2,...} 和 |z|<1 时高斯超几何函数 F12(n+12,n+12;n+32;-z2)andF12(1,n+12;n+32;z2) 的几个闭式公式。作者借助高斯超几何函数与勒奇超越之间的关系,根据连续函数的三个高斯关系对这些公式进行了分析。此外,作者还确定了这些表示中涉及的两个多项式的零点位置和分布,其中包含广义二项式系数。通过比较这些公式,他们还推导出了几个组合等式。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 求助全文
来源期刊
CiteScore
2.50
自引率
7.70%
发文量
790
审稿时长
6 months
期刊介绍: The Journal of Mathematical Analysis and Applications presents papers that treat mathematical analysis and its numerous applications. The journal emphasizes articles devoted to the mathematical treatment of questions arising in physics, chemistry, biology, and engineering, particularly those that stress analytical aspects and novel problems and their solutions. Papers are sought which employ one or more of the following areas of classical analysis: • Analytic number theory • Functional analysis and operator theory • Real and harmonic analysis • Complex analysis • Numerical analysis • Applied mathematics • Partial differential equations • Dynamical systems • Control and Optimization • Probability • Mathematical biology • Combinatorics • Mathematical physics.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信