Pure Mathematics
Vol. 09  No. 06 ( 2019 ), Article ID: 31735 , 6 pages
10.12677/PM.2019.96098

Series Expansion of Generalized Ramanujan Constant $R\left(a,c-a\right)$

Xiaoyu Wang1, Peigui Zhou2, Fei Wang1*

1Teaching Section of Mathematics, Zhejiang Institute of Mechanical and Electrical Engineering, Hangzhou Zhejiang

2Keyi College of Zhejiang Sci-Tech University, Hangzhou Zhejiang

Received: Jul. 21st, 2019; accepted: Jul. 31st, 2019; published: Aug. 16th, 2019

ABSTRACT

In this paper, the authors present several kinds of series expansion expressions of generalized Ramanujan constant $R\left(a,c-a\right)=-2\gamma -\psi \left(a\right)-\psi \left(c-a\right)$ by the nth order derivative of $\psi \left(x\right)$ . By these results, some known results about $R\left(a,c-a\right)$ can be easily improved.

Keywords:Generalized Ramanujan Constant, Psi Function, Series Expansion

1浙江机电职业技术学院数学教研室，浙江 杭州

2浙江理工大学科技与艺术学院，浙江 杭州

1. 引言

$\Gamma \left(x\right)={\int }_{0}^{\infty }{t}^{x-1}{\text{e}}^{-t}\text{d}t,B\left(x,y\right)=\frac{\Gamma \left(x\right)\Gamma \left(y\right)}{\Gamma \left(x+y\right)},\psi \left(x\right)=\frac{{\Gamma }^{\prime }\left(x\right)}{\Gamma \left(x\right)}$ (1.1)

$\gamma =\underset{n\to \infty }{\mathrm{lim}}\left[\sum _{k=1}^{n}\frac{1}{k}-\mathrm{log}n\right]=\text{0}\text{.57721566}\cdots$ ，是Euler-Mascheroni常数，则

(1.2)

(1.3)

,

(1.4)

(1.5)

(1.6)

(1.7)

(1.8)

(1.9)

(1.10)

.

2. 主要结果

(2.1)

,(2.2)

.

, (2.3)

3. 主要结果的证明

，则，且

,

,

(3.1)

(3.2)

(3.3)

.

(3.4)

,

,

(3.5)

.

(3.6)

(3.7)

(3.8)

.

(3.9)

Series Expansion of Generalized Ramanujan Constant R(a,c-a)[J]. 理论数学, 2019, 09(06): 749-754. https://doi.org/10.12677/PM.2019.96098

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9. NOTES

*通讯作者。