$q$-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.

Trendmd

References 15

[1]	Abramowitz M, Stegun I A. Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables. New York: Dover, 1965
[2]	Anderson G D, Vamanamurthy M K, Vuorinen M. Conformal Invariants, Inequalities, and Quasiconformal Maps. New York: John Wiley & Sons, 1997
[3]	Anderson G D, Qiu S L, Vamanamurthy M K, Vuorinen M. Generalized elliptic integrals and modular equations. Pacific J Math, 2000, 192(1): 1-37 doi: 10.2140/pjm
[4]	Ahlfors L V. Lectures on Quasiconformal Mappings. Providence, RI: Amer Math Soc, 2006
[5]	Berndt B C. Ramanujan's Notebook IV. New York: Springer-Verlag, 1993
[6]	Qiu S L, Ma X Y, Huang T R. Some properties of the difference between the Ramanujan constant and beta function. J Math Anal Appl, 2017, 446(1): 114-129 doi: 10.1016/j.jmaa.2016.08.043
[7]	Chu H H, Yang Z H, Zhang W, Chu Y M. Improvements of the bounds for Ramanujan constant function. J Inequal Appl, 2016, Article number: 196(2016): 1-9 (Paper No. 196)
[8]	Qiu S L, Ma X Y, Huang T R. Sharp approximations for the Ramanujan constant. Constr Approx, 2020, 51(2): 303-330 doi: 10.1007/s00365-019-09464-3
[9]	Wang M K, Chu Y M, Qiu S L. Sharp bounds for generalized elliptic integrals of the first kind. J Math Anal Appl, 2015, 429(2): 744-757 doi: 10.1016/j.jmaa.2015.04.035
[10]	Gasper G, Rahman M. Basic Hypergeometric Series. Cambridge: Cambridge University Press, 1990
[11]	Tian J F, Yang Z H. Logarithmically complete monotonicity of ratios of $q$ -gamma functions. J Math Anal Appl, 2022, 508(1): 125868 doi: 10.1016/j.jmaa.2021.125868
[12]	Heine E. Untersuchungenüber die Reihe. J Reine Angew Math, 1847, 34: 285-328
[13]	Evans R J, Stanton D. Asymptotic formulas for zero-balanced hypergeometric series. SIAM J Math Anal, 1984, 15: 1010-1020 doi: 10.1137/0515078
[14]	Batir N. $q$ -extensions of some estimates associated with the digamma function. J Approx Theory, 2013, 174: 54-64 doi: 10.1016/j.jat.2013.06.002
[15]	Krattenthaler C, Srivastava H M. Summations for basic hypergeometric series involving a $q$ -analogue of the digamma function. Comput Math Appl, 1996, 32(3): 73-91

Metrics

Viewed

Full text

240

HTML			PDF

Just accepted	Online first	Issue	Just accepted	Online first	Issue
0	0	3	0	0	237

	From	local

	Times	240
	Rate	100%

Abstract

100

Just accepted	Online first	Issue

0	0	100

From	Others	local

Times	99	1
Rate	99%	1%

Cited

Web of Science	Crossref	ScienceDirect	Search for Citations in Google Scholar >>


This page requires you have already subscribed to WoS.

Shared

Discussed

Comments

Recommended 10

[1]	Li Xiaodong, Gao Hongliang, Xu Jing. Exact Multiplicity of Positive Solutions for a Semipositone Mean Curvature Problem with Concave Nonlinearity[J]. Acta mathematica scientia,Series A, 2023, 43(5): 1341 -1349 .
[2]	Li Wenxin,Lian Pan,Liang Yuxia. The Schödinger Uncertainty Relation in the Fock-Type Spaces[J]. Acta mathematica scientia,Series A, 2023, 43(5): 1321 -1332 .
[3]	Sheng Weimin, Wang Yinhang. An Extension of Minkowski Formulae for Free Boundary Hypersurfaces in a Ball[J]. Acta mathematica scientia,Series A, 2023, 43(6): 1641 -1648 .
[4]	Lan Chao, Fan Xingya. The Discrete Series of Affine Symmetric Space ${SO^\ast(6)/SO(3,\mathbb{C})}$ [J]. Acta mathematica scientia,Series A, 2023, 43(6): 1649 -1658 .
[5]	Yang Jihua, Ma Liang. Limit Cycle Bifurcations of a Non-smooth Differential System with a Cuspidal Loop[J]. Acta mathematica scientia,Series A, 2023, 43(6): 1667 -1680 .
[6]	Wang Wenxia. The Method of Sum Operator and Unique Positive Solution for Fractional Nonlinear Integral Boundary Value Problems with $p$ -Laplacian Operator[J]. Acta mathematica scientia,Series A, 2023, 43(6): 1731 -1743 .
[7]	Han Zhongjie, He Yiheng, Zhao Zhixue. Indirect Stabilization and Optimal Decay Rates of Weakly Coupled Plates with Various Types of Damping[J]. Acta mathematica scientia,Series A, 2023, 43(6): 1681 -1698 .
[8]	Jian Weigang, Long Wei. Tauberian Theorem for Asymptotically Periodic Functions and Its Application to Abstract Cauchy Problems[J]. Acta mathematica scientia,Series A, 2023, 43(6): 1699 -1709 .
[9]	Xu Fei, Zhang Yong. Uniqueness and Asymptotic Stability of Time-Periodic Solutions for the Fractional Burgers Equation[J]. Acta mathematica scientia,Series A, 2023, 43(6): 1710 -1722 .
[10]	Li Renhua, Wang Zhengping. Normalized Solution of Fractional Schrödinger-Poisson Equations with Coercive Potential[J]. Acta mathematica scientia,Series A, 2023, 43(6): 1723 -1730 .

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

RichHTML

PDF (PC)

Abstract

Cite this article

share this article

References 15

Related Articles 3

Metrics

Comments

Recommended 10

[1]	Miaokun Wang,Yuming Chu,Songliang Qiu. The Simple Proof and Generalization of a Conjecture Concerning Generalized Legendre Identity [J]. Acta mathematica scientia,Series A, 2020, 40(3): 662-666.
[2]	Jinfa Cheng,Lukun Jia. Hypergeometric Type Difference Equations on Nonuniform Lattices: Rodrigues Type Representation for the Second Kind Solution [J]. Acta mathematica scientia,Series A, 2019, 39(4): 875-893.
[3]	Liu Hongmei, Zhou Wenshu, Li Yang. Transformation and Reduction Formulae for Bivariate Hypergeometric Series [J]. Acta mathematica scientia,Series A, 2016, 36(1): 150-156.

qq-Ramanujan Asymptotic Formula and qq-Ramanujan RR-function

RichHTML

PDF (PC)

Abstract

Cite this article

share this article

References 15

Related Articles 3

Metrics

Comments

Recommended 10

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