$q$-Ramanujan 渐近公式及 $q$-Ramanujan $R$-函数

数学物理学报 ›› 2023, Vol. 43 ›› Issue (6): 1659-1666.

$q$-Ramanujan 渐近公式及 $q$-Ramanujan $R$-函数

鲍琪¹,王淼坤^2,^*(),褚玉明^2,³

¹华东师范大学数学科学学院上海 201100
²湖州师范学院理学院浙江湖州 313000
³杭州师范大学数学学院杭州 311121

收稿日期:2022-07-01 修回日期:2023-03-17 出版日期:2023-12-26 发布日期:2023-11-16
通讯作者: *王淼坤，E-mail: wmk000@126.com
基金资助:
国家自然科学基金(11701176);国家自然科学基金(11901061)

$q$-Ramanujan Asymptotic Formula and $q$-Ramanujan $R$-function

Bao Qi¹,Wang Miaokun^2,^*(),Chu Yuming^2,³

¹School of Mathematical Sciences, East China Normal University, Shanghai 201100
²Department of Mathematics, Huzhou University, Zhejiang Huzhou 313000
³School of Mathematics, Hangzhou Normal University, Hangzhou 311121

Received:2022-07-01 Revised:2023-03-17 Online:2023-12-26 Published:2023-11-16
Supported by:
NSFC(11701176);NSFC(11901061)

摘要/Abstract

摘要：

该文将 Gauss 超几何函数 $_{2}F_{1}$ 的 Ramanujan 渐近公式及其相关的 Ramanujan $R$-函数推广到了基本超几何级数 $_{2}\phi_{1}$ 的情形. 一方面, 给出了 $_{2}\phi_{1}$ 的 $q$-Ramanujan 渐近公式并定义了 $q$-Ramanujan $R$-函数; 另一方面, 着重研究了 $q$-Ramanujan $R$-函数, 证明了包括级数展开式, 完全单调性和参数 $q$ 的单调性在内的一些分析性质. 作为应用, 推导得到 $q$-Ramanujan $R$-函数的几个渐近不等式.

关键词: $q$-Ramanujan 渐近公式, $q$-Ramanujan $R$-函数, $q$-模拟

Abstract:

In this paper, the Ramanujan asymptotic formula of the Gaussian hypergeometric function $_{2}F_{1}$ and its related Ramanujan $R$-function will be generalized to the case of basic hypergeometric series $_{2}\phi_{1}$. On the one hand, we shall present the $q$-Ramanujan asymptotic formula of $_{2}\phi_{1}$ and introduce the $q$-Ramanujan $R$-function; on the other hand, we shall mainly study the $q$-Ramanujan $R$-function, and prove some analytical properties of the $q$-Ramanujan $R$-function including series expansions, complete monotonicity property and monotonicity property with respect to the parameter $q$. As applications, several sharp inequalities for the $q$-Ramanujan $R$-function will be derived.

Key words: $q$-Ramanujan asymptotic formula, $q$-Ramanujan constant, $q$-analogy

中图分类号:

O174.6

鲍琪, 王淼坤, 褚玉明. $q$-Ramanujan 渐近公式及 $q$-Ramanujan $R$-函数[J]. 数学物理学报, 2023, 43(6): 1659-1666.

Bao Qi, Wang Miaokun, Chu Yuming. $q$-Ramanujan Asymptotic Formula and $q$-Ramanujan $R$-function[J]. Acta mathematica scientia,Series A, 2023, 43(6): 1659-1666.

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