Abstract
The problem of finding closed forms for a summation involving harmonic numbers is considered. Solutions for ∑ ni =1P(i)H (k)i , where p(i) is a polynomial, and ∑ ni =1 Hi/(i+m), where m is an integer, are given. A method to automate these results is presented. This is achieved by using Moenck’s algorithm and by exploiting the relationship between polygamma functions and harmonic numbers.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
R. W. Gosper, “Indefinite hypergeometric sums in MACSYMA,”Proc. MACYSMA User’s Conference (1977), Berkeley CA, 237–252.
R. W. Gosper, “Decision procedures for indefinite hypergeometric summation,”Proc. Nat. Acad. Sciences, USA 75 (1978), 40–42.
M. B. Hayden, “Automated tools for the analysis of algorithms,” Masters Thesis, University of Rhode Island, Kingston RI (1986).
M. B. Hayden and E. A. Lamagna, “Summation involving binomial coefficients using hypergeometric functions,”Proc. ACM Symposium on Symbolic and Algebraic Computation (1986), 367–372.
C. Jordan,Calculus of Finite Differences. Chelsea, New York NY (1965).
M. Karr, “Summation in finite terms,”Journal of the ACM 28 (1981), 305–350.
M. Karr, “Theory of summation in finite terms,”Journal of Symbolic Computation 1 (1985), 303–315.
D. E. Knuth,The Art Of Computer Programming, Vol 1: Fundamental Algorithms (Second Edition). Addison-Wesley, Reading MA (1973).
R. L. Graham, D. E. Knuth and O. Patashnik,Concrete Mathematics: A Foundation for Computer Science. Addison-Wesley, Reading MA (1989).
S.-M. Liu, “Symbolic expressions for summation of harmonic numbers,” Masters Thesis, Brown University, Providence RI (1986).
R. Moenck, “On computing closed forms for summations,”Proc. MACYSMA User’s Conference (1977), Berkeley CA, 225–236.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1989 Springer-Verlag New York Inc.
About this paper
Cite this paper
Savio, D.Y., Lamagna, E.A., Liu, SM. (1989). Summation of Harmonic Numbers. In: Kaltofen, E., Watt, S.M. (eds) Computers and Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9647-5_2
Download citation
DOI: https://doi.org/10.1007/978-1-4613-9647-5_2
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-97019-6
Online ISBN: 978-1-4613-9647-5
eBook Packages: Springer Book Archive