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.
We obtain identities for variant Euler sums of the type , and its finite counterpart, which generalize some results obtained by other authors.
Identities are successfully achieved for the sums under investigation. Some published results have been successfully generalized.
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 n th 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
and in the special case when we have
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 ,
where denotes the generalized nth 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 , 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:
Let Then we have
Let and consider the following expansion:
For an arbitrary positive sequence , the following identity holds:
Since we notice that
substituting Equation 7 and simplifying, we have
hence, the identity (Equation 8) follows. □
For k=3 and 5,
Now, we consider the following finite version of Theorem 1:
Let k, Then we have
To prove Equation 14, we may write
where A r is given by Equation 10, and by a rearrangement of sums,
Let Then we obtain
It is straightforward to show that
Some examples are
The author has generalized some results on variant Euler sums and specifically obtained identities for and its finite counterpart.
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.
Professor Anthony Sofo is a Fellow of the Australian Mathematical Society.
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The author is grateful to an anonymous referee for the careful reading of the manuscript.
The author declares that he has no competing interests.
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Sofo, A. Euler-related sums. Math Sci 6, 10 (2012). https://doi.org/10.1186/2251-7456-6-10