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On an application of Binet's second formula


Author: Ruiming Zhang
Journal: Proc. Amer. Math. Soc.
MSC (2010): Primary 33B15, 33E20
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.


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Additional Information

Ruiming Zhang
Affiliation: College of Science, Northwest A&F University, Yangling, Shaanxi 712100, People’s Republic of China
Email: ruimingzhang@yahoo.com

DOI: https://doi.org/10.1090/proc/13711
Keywords: Gamma function, Binet's second formula, Kummer's Fourier series expansion, M\"obius inversion
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
Article copyright: © Copyright 2017 American Mathematical Society