On an application of Binet’s second formula
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- by Ruiming Zhang
- Proc. Amer. Math. Soc. 145 (2017), 5267-5272
- DOI: https://doi.org/10.1090/proc/13711
- Published electronically: June 22, 2017
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Abstract:
In this work we apply the second Binet formula for Euler’s gamma function $\Gamma (x)$ and a Laplace transform formula to derive an infinite series expansion for the auxiliary function $f(x)$ in the computations of sine integral and cosine integral functions in terms of $\log \Gamma (x)$ and the Möbius function. Then we apply Möbius inversion to obtain a Kummer type series expansion for $\log \Gamma (x)$. Unlike the original Kummer formula, our formula is not a Fourier series anymore. By differentiating the series expansion for $f(x)$ we obtain an infinite series expansion for the auxiliary function $g(x)$ associated with sine integral and cosine integral functions as well.References
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Bibliographic Information
- Ruiming Zhang
- Affiliation: College of Science, Northwest A&F University, Yangling, Shaanxi 712100, People’s Republic of China
- MR Author ID: 257230
- Email: ruimingzhang@yahoo.com
- Received by editor(s): December 16, 2016
- Received by editor(s) in revised form: December 23, 2016, and January 12, 2017
- Published electronically: June 22, 2017
- Additional Notes: This research was partially supported by National Natural Science Foundation of China, Grant No. 11371294, and Northwest A&F University
- Communicated by: Mourad Ismail
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 5267-5272
- MSC (2010): Primary 33B15, 33E20
- DOI: https://doi.org/10.1090/proc/13711
- MathSciNet review: 3717955