CLOSED FORM REPRESENTATIONS OF HARMONIC SUMS

We develop a master theorem from which we are able to represent infinite sums of harmonic numbers and binomial coefficients in both integral and closed form. The new results extend known existing results in the published literature.

Keywords:

Harmonic numbers, Zeta functions, Binomial coefficients, Integral representations 2000 AMS Classification: Primary: 05 A 10; Secondary: 33 C 20,

PDF

___

Alzer, H. and Koumandos, S. Series representations for γ and other mathematical constants, Analysis Mathematica 34, 1–8, 2008.
Alzer, H. and Koumandos, S. Series and product representations for some mathematical constants, Period. Math. Hungar 58 (1), 71–82, 2009.
Alzer, H., Karayannakis, D. and Srivastava, H. M. Series representations of some mathe- matical constants, J. Math. Anal. Appl. 320, 145–162, 2006.
Alzer, H. Sharp inequalities for harmonic numbers, Expo. Math. 24, 385–388, 2006.
Basu, A. A new method in the study of Euler sums, Ramanujan J. 16, 7–24, 2008.
Chu, W. and Fu, A. M. Dougall-Dixon formula and harmonic number identities, Ramanujan J. 18, 11–31, 2009.
Chu, W. and DeDonno, L. Identita’ binomiali e numeri armonici, Boll. Della Unione Ital- iana, Sez B., 213, 2007.
Euler, L. Opera Omnia, Ser. 1, Vol XV (Teubner, Berlin,1917).
Flajolet, P. and Salvy, B. Euler sums and contour integral representations, Expo. Math. 7, –35, 1998.
Georghiou, C. and Philippou, A. N. Harmonic sums and the Zeta function, Fibonacci Quart. , 29–36, 1983.
Krattenthaler, C. and Rao, K. S. Automatic generation of hypergeometric identities by the beta integral method, J. Comput. Math. Appl. 160, 159–173, 2003.
Sofo, A. Integral forms of sums associated with harmonic numbers, Appl. Math. Comput. , 365–372, 2009.
Sofo, A. Computational Techniques for the Summation of Series (Kluwer Academic/Plenum Publishers, New York, 2003).
Sofo, A. Sums of derivatives of binomial coefficients, Advances Appl. Math. 42, 123–134, Sofo, A. Harmonic numbers and double binomial coefficients, Integral Transforms and Spec. Funct. 20 (11), 847–857, 2009.
Sondow, J. and Weisstein, E. W. Harmonic Number, From MathWorld–A Wolfram Web Resource, availabe online at (http://mathworld.wolfram.com/HarmonicNumber.html). Wolfram Research Inc. Mathematica (Wolfram Research Inc., Champaign, IL).