On an application of Binet's second formula

Author:
Ruiming Zhang

Journal:
Proc. Amer. Math. Soc. **145** (2017), 5267-5272

MSC (2010):
Primary 33B15, 33E20

DOI:
https://doi.org/10.1090/proc/13711

Published electronically:
June 22, 2017

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this work we apply the second Binet formula for Euler's gamma function and a Laplace transform formula to derive an infinite series expansion for the auxiliary function in the computations of sine integral and cosine integral functions in terms of and the Möbius function. Then we apply Möbius inversion to obtain a Kummer type series expansion for . Unlike the original Kummer formula, our formula is not a Fourier series anymore. By differentiating the series expansion for we obtain an infinite series expansion for the auxiliary function associated with sine integral and cosine integral functions as well.

**[1]**George E. Andrews, Richard Askey, and Ranjan Roy,*Special functions*, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR**1688958****[2]**Tom M. Apostol,*Introduction to analytic number theory*, Springer-Verlag, New York-Heidelberg, 1976. Undergraduate Texts in Mathematics. MR**0434929****[3]**Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi,*Higher transcendental functions. Vol. I*, Robert E. Krieger Publishing Co., Inc., Melbourne, Fla., 1981. Based on notes left by Harry Bateman; With a preface by Mina Rees; With a foreword by E. C. Watson; Reprint of the 1953 original. MR**698779**

Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi,*Higher transcendental functions. Vol. II*, Robert E. Krieger Publishing Co., Inc., Melbourne, Fla., 1981. Based on notes left by Harry Bateman; Reprint of the 1953 original. MR**698780**

Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi,*Higher transcendental functions. Vol. III*, Robert E. Krieger Publishing Co., Inc., Melbourne, Fla., 1981. Based on notes left by Harry Bateman; Reprint of the 1955 original. MR**698781****[4]**C. Krattenthaler and H. M. Srivastava,*Summations for basic hypergeometric series involving a 𝑞-analogue of the digamma function*, Comput. Math. Appl.**32**(1996), no. 3, 73–91. MR**1398550**, https://doi.org/10.1016/0898-1221(96)00114-9**[5]**Fritz Oberhettinger and Larry Badii,*Tables of Laplace transforms*, Springer-Verlag, New York-Heidelberg, 1973. MR**0352889****[6]**Frank W. J. Olver, Daniel W. Lozier, Ronald F. Boisvert, and Charles W. Clark (eds.),*NIST handbook of mathematical functions*, U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge, 2010. With 1 CD-ROM (Windows, Macintosh and UNIX). MR**2723248****[7]**George Pólya and Gabor Szegő,*Problems and theorems in analysis. I*, Classics in Mathematics, Springer-Verlag, Berlin, 1998. Series, integral calculus, theory of functions; Translated from the German by Dorothee Aeppli; Reprint of the 1978 English translation. MR**1492447**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2010):
33B15,
33E20

Retrieve articles in all journals with MSC (2010): 33B15, 33E20

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