Euler-related sums

Mathematical Sciences20126:10

DOI: 10.1186/2251-7456-6-10

Received: 10 April 2012

Accepted: 9 July 2012

Published: 9 July 2012

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 http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_IEq1_HTML.gif , 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 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,
http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_Equ1_HTML.gif
be the nth harmonic number, γ denotes the Euler-Mascheroni constant, http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_IEq2_HTML.gif is the digamma function and http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_IEq3_HTML.gif is the well-known gamma function. Let also, http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_IEq4_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_IEq5_HTML.gif denote, respectively, the sets of real, complex and natural numbers. A generalized binomial coefficient http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_IEq6_HTML.gif may be defined by
http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_Equ2_HTML.gif
and in the special case when http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_IEq7_HTML.gif we have
http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_Equ3_HTML.gif
where
http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_Equ4_HTML.gif
with http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_IEq8_HTML.gif is known as the Pochhammer symbol. Some well-known Euler sums are (see, e.g., [1])
http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_Equ5_HTML.gif
recently, Chen [2] obtained
http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_Equ6_HTML.gif
In [3], we have, for k≥1,
http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_Equ7_HTML.gif
(1)
and in [4],
http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_Equ8_HTML.gif
(2)
where http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_IEq9_HTML.gif denotes the generalized n th harmonic number in power r defined by
http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_Equ9_HTML.gif
We study, in this paper, http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_IEq10_HTML.gif 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
http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_Equ10_HTML.gif
and in [8],
http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_Equ11_HTML.gif
Also, from the study of Prodinger [9],
http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_Equ12_HTML.gif

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
http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_Equ13_HTML.gif
(3)
http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_Equ14_HTML.gif
(4)
http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_Equ15_HTML.gif
(5)
http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_Equ16_HTML.gif
(6)

Proof

From the definition of harmonic numbers and the digamma function,
http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_Equ17_HTML.gif
and Equation 3 follows. From the double argument identity of the digamma function
http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_Equ18_HTML.gif
using Equation 3 and rearranging, we obtain Equation 4. For Equation 5, we first note that for an arbitrary sequence http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_IEq11_HTML.gif , the following identity holds:
http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_Equ19_HTML.gif
hence,
http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_Equ20_HTML.gif
The interesting identity (Equation 6) follows from Equation 5 and substituting
http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_Equ21_HTML.gif
(7)
so that
http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_Equ22_HTML.gif
http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_Equ23_HTML.gif

replacing the counter, we obtain Equation 6. □

Main results and discussion

We now prove the two following theorems:

Theorem 1

Let http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_IEq12_HTML.gif Then we have
http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_Equ24_HTML.gif
(8)

Proof

Let http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_IEq13_HTML.gif and consider the following expansion:
http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_Equ25_HTML.gif
Now,
http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_Equ26_HTML.gif
(9)
where
http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_Equ27_HTML.gif
(10)
For an arbitrary positive sequence http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_IEq14_HTML.gif , the following identity holds:
http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_Equ28_HTML.gif
hence, from Equations 4 and 9,
http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_Equ29_HTML.gif
Since we notice that
http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_Equ30_HTML.gif
we get
http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_Equ31_HTML.gif
Now,
http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_Equ32_HTML.gif
substituting Equation 7 and simplifying, we have
http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_Equ33_HTML.gif

hence, the identity (Equation 8) follows. □

Corollary 1

From Equation 8 and using Equations 3 and 4, we obtain the results,
http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_Equ34_HTML.gif
(11)
and
http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_Equ35_HTML.gif
(12)

Proof

We can use Equations 3 and 4 and also note that
http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_Equ36_HTML.gif
(13)

From the rearrangement of http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_IEq15_HTML.gif and Equation 1, we can obtain Equation 11; and from the rearrangement of http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_IEq16_HTML.gif and Equation 13, we can obtain Equation 12. □

Example 1

For k=3 and 5,
http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_Equ37_HTML.gif

Now, we consider the following finite version of Theorem 1:

Theorem 2

Let k, http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_IEq17_HTML.gif Then we have
http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_Equ38_HTML.gif
(14)

Proof

To prove Equation 14, we may write
http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_Equ39_HTML.gif
where A r is given by Equation 10, and by a rearrangement of sums,
http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_Equ40_HTML.gif
(15)

Substituting Equation 7 into Equation 15 and after simplification, Equation 14 follows. □

Corollary 2

Let http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_IEq18_HTML.gif Then we obtain
http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_Equ41_HTML.gif
(16)
and
http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_Equ42_HTML.gif
(17)

Proof

It is straightforward to show that
http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_Equ43_HTML.gif
(18)

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
http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_Equ44_HTML.gif
http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_Equ45_HTML.gif
and
http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_Equ46_HTML.gif

Conclusions

The author has generalized some results on variant Euler sums and specifically obtained identities for http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-10/MediaObjects/40096_2012_9_IEq19_HTML.gif and its finite counterpart.

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.

Author’s information

Professor Anthony Sofo is a Fellow of the Australian Mathematical Society.

Declarations

Acknowledgements

The author is grateful to an anonymous referee for the careful reading of the manuscript.

Authors’ Affiliations

(1)
Victoria University College, Victoria University

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Copyright

© Sofo; licensee Springer. 2012

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://​creativecommons.​org/​licenses/​by/​2.​0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.