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

Acta mathematica scientia,Series A ›› 2023, Vol. 43 ›› Issue (6): 1659-1666.

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$q$-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)

Abstract

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

CLC Number:

O174.6

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

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