- Open Access
Padé approximant related to the Wallis formula
© The Author(s) 2017
- Received: 17 March 2017
- Accepted: 15 May 2017
- Published: 8 June 2017
- gamma function
- psi function
- Wallis ratio
The numerical values given in this paper have been calculated via the computer program MAPLE 13.
Euler’s gamma function \(\Gamma(x)\) is one of the most important functions in mathematical analysis and has applications in diverse areas. The logarithmic derivative of \(\Gamma(x)\), denoted by \(\psi(x)=\Gamma'(x)/\Gamma(x)\), is called the psi (or digamma) function.
The following lemmas are required in the sequel.
Noting that (3.5) holds, we see by (3.13) that the left-hand side of (3.15) holds for \(n\geq4\), while the right-hand side of (3.15) is valid for \(n\geq3\). Elementary calculations show that the left-hand side of (3.15) is also valid for \(n =1, 2\) and 3, and the right-hand side of (3.15) is valid for \(n =1\) and 2. The proof is complete. □
6.673798 × 10−3
3.789512 × 10−3
2.264856 × 10−13
9.947434 × 10−12
2.398663 × 10−24
1.051407 × 10−20
2.408054 × 10−35
1.056218 × 10−29
2.408948 × 10−46
1.056690 × 10−38
The authors thank the referees for helpful comments.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
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