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

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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