Abstract
Purpose
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.
Method
We use analytical methods to obtain our results.
Results
We obtain identities for variant Euler sums of the type , and its finite counterpart, which generalize some results obtained by other authors.
Conclusions
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; 33C20Background 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 EulerMascheroni constant, is the digamma function and is the wellknown gamma function. Let also, and denote, respectively, the sets of real, complex and natural numbers. A generalized binomial coefficient may be defined by
and in the special case when we have
where
with is known as the Pochhammer symbol. Some wellknown Euler sums are (see, e.g., [1])
recently, Chen [2] obtained
In [3], we have, for k≥1,
and in [4],
where denotes the generalized n^{th} harmonic number in power r defined by
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 [7], we have
and in [8],
Also, from the study of Prodinger [9],
Further work in the summation of harmonic numbers and binomial coefficients has also been done by Sofo [10]. The works of [1117] 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.
Lemma 1
Let n and r be positive integers. Then we have
Proof
From the definition of harmonic numbers and the digamma function,
and Equation 3 follows. From the double argument identity of the digamma function
using Equation 3 and rearranging, we obtain Equation 4. For Equation 5, we first note that for an arbitrary sequence , the following identity holds:
hence,
The interesting identity (Equation 6) follows from Equation 5 and substituting
so that
replacing the counter, we obtain Equation 6. □
Main results and discussion
We now prove the two following theorems:
Theorem 1
Proof
Let and consider the following expansion:
Now,
where
For an arbitrary positive sequence , the following identity holds:
hence, from Equations 4 and 9,
Since we notice that
we get
Now,
substituting Equation 7 and simplifying, we have
hence, the identity (Equation 8) follows. □
Corollary 1
From Equation 8 and using Equations 3 and 4, we obtain the results,
and
Proof
We can use Equations 3 and 4 and also note that
From the rearrangement of and Equation 1, we can obtain Equation 11; and from the rearrangement of and Equation 13, we can obtain Equation 12. □
Example 1
For k=3 and 5,
Now, we consider the following finite version of Theorem 1:
Theorem 2
Proof
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. □
Corollary 2
and
Proof
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. □
Example 2
Some examples are
and
Conclusions
Methods
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.
Competing interests
The author declares that he has no competing interests.
Author’s information
Professor Anthony Sofo is a Fellow of the Australian Mathematical Society.
Acknowledgements
The author is grateful to an anonymous referee for the careful reading of the manuscript.
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