The purpose of this paper is to develop a set of identities for Euler type sums of products of harmonic numbers and reciprocal binomial coefficients.
We use analytical methods to obtain our results.
Identities are successfully achieved for the sums under investigation. Some published results have been successfully generalized.
Keywords:Harmonic numbers; Binomial coefficients and gamma function; Polygamma function; Combinatorial series identities and summation formulas; Partial fraction approach; MSC (2000); primary: 05A10; 05A19; 11B65; secondary: 11B83; 11M06; 33B15; 33D60; 33C20
Background and preliminaries
In the spirit of Euler, we shall investigate the summation of some variant Euler sums. In common terminology, let, as usual,
be the nth harmonic number, γ denotes the Euler-Mascheroni constant, is the digamma function and is the well-known gamma function. Let also, and denote, respectively, the sets of real, complex and natural numbers. A generalized binomial coefficient may be defined by
with is known as the Pochhammer symbol. Some well-known Euler sums are (see, e.g., )
recently, Chen  obtained
In , we have, for k≥1,
and in ,
We study, in this paper, and its finite counterpart. Analogous results of Euler type for infinite series have been developed by many authors, see for example [5,6] and references therein. Many finite versions of harmonic number sum identities also exist in the literature, for example in , we have
and in ,
Also, from the study of Prodinger ,
Further work in the summation of harmonic numbers and binomial coefficients has also been done by Sofo . The works of [11-17] and references therein also investigate various representations of binomial sums and zeta functions in a simpler form by the use of the beta function and by means of certain summation theorems for hypergeometric series.
Let n and r be positive integers. Then we have
From the definition of harmonic numbers and the digamma function,
and Equation 3 follows. From the double argument identity of the digamma function
The interesting identity (Equation 6) follows from Equation 5 and substituting
replacing the counter, we obtain Equation 6. □
Main results and discussion
We now prove the two following theorems:
hence, from Equations 4 and 9,
Since we notice that
substituting Equation 7 and simplifying, we have
hence, the identity (Equation 8) follows. □
From Equation 8 and using Equations 3 and 4, we obtain the results,
We can use Equations 3 and 4 and also note that
For k=3 and 5,
Now, we consider the following finite version of Theorem 1:
To prove Equation 14, we may write
where Ar is given by Equation 10, and by a rearrangement of sums,
Substituting Equation 7 into Equation 15 and after simplification, Equation 14 follows. □
It is straightforward to show that
then rearranging Equation 14 and using Equation 18, we obtain Equation 16. Rearranging Equation 14 and using Equation 2, we obtain Equation 17. □
Some examples are
Analytical techniques have been employed in the analysis of our results. We have used many relations of the polygamma functions together with results of reordering of double sums and partial fraction decomposition.
The author declares that he has no competing interests.
Professor Anthony Sofo is a Fellow of the Australian Mathematical Society.
The author is grateful to an anonymous referee for the careful reading of the manuscript.
Choi, J: Certain summation formulas involving harmonic numbers and generalized harmonic numbers. Appl. Math. Comput. 218, 734–740 (2011). Publisher Full Text
Srivastava, R: Some families of combinatorial and other series identities and their applications. Appl. Math. Comput. 218, 1077–1083 (2011). Publisher Full Text
Osburn, R, Schneider, C: Gaussian hypergeometric series and supercongruences. Math. Comp.. 78, 275–292 (2009). Publisher Full Text
Sofo, A: Summation formula involving harmonic numbers. Analysis Mathematica. 37, 51–64 (2011). Publisher Full Text
Cheon, GS, El-Mikkawy, MEA: Generalized harmonic numbers with Riordan arrays. J. Number Theory. 128, 413–425 (2008). Publisher Full Text
He, TX, Hu, LC, Yin, D: A pair of summation formulas and their applications. Comp, Math. Appl.. 58, 1340–1348 (2009). Publisher Full Text
Sofo, A, Srivastava, HM: Identities for the harmonic numbers and binomial coefficients. Ramanujan J.. 25, 93–113 (2011). Publisher Full Text
Sofo, A: Sums of derivatives of binomial coefficients. Advances Appl. Math.. 42, 123–134 (2009). Publisher Full Text
Zheng, DY: Further summation formulae related to generalized harmonic numbers. J. Math. Anal. Appl. 335, 692–706 (2007). Publisher Full Text