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Open Access Original research

Euler-related sums

Anthony Sofo

Author Affiliations

Victoria University College, Victoria University, Melbourne City, PO Box 14428, Victoria, 8001, Australia

Mathematical Sciences 2012, 6:10  doi:10.1186/2251-7456-6-10


The electronic version of this article is the complete one and can be found online at: http://www.iaumath.com/content/6/1/10


Received:10 April 2012
Accepted:9 July 2012
Published:9 July 2012

© 2012 Sofo; licensee Springer.

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.

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 <a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M1">View MathML</a>, 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,

<a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M2">View MathML</a>

(1)

be the nth harmonic number, γ denotes the Euler-Mascheroni constant, <a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M3">View MathML</a> is the digamma function and <a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M4">View MathML</a> is the well-known gamma function. Let also, <a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M5">View MathML</a> and <a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M6">View MathML</a> denote, respectively, the sets of real, complex and natural numbers. A generalized binomial coefficient <a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M7">View MathML</a> may be defined by

<a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M8">View MathML</a>

(2)

and in the special case when <a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M9">View MathML</a> we have

<a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M10">View MathML</a>

(3)

where

<a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M11">View MathML</a>

(4)

with <a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M12">View MathML</a> is known as the Pochhammer symbol. Some well-known Euler sums are (see, e.g., [1])

<a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M13">View MathML</a>

(5)

recently, Chen [2] obtained

<a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M14">View MathML</a>

(6)

In [3], we have, for k≥1,

<a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M15">View MathML</a>

(7)

and in [4],

<a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M16">View MathML</a>

(8)

where <a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M17">View MathML</a> denotes the generalized nth harmonic number in power r defined by

<a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M18">View MathML</a>

(9)

We study, in this paper, <a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M19">View MathML</a> 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

<a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M20">View MathML</a>

(10)

and in [8],

<a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M21">View MathML</a>

(11)

Also, from the study of Prodinger [9],

<a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M22">View MathML</a>

(12)

Further work in the summation of harmonic numbers and binomial coefficients has also been done by Sofo [10]. 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.

Lemma 1

Let n and r be positive integers. Then we have

<a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M23">View MathML</a>

(13)

<a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M24">View MathML</a>

(14)

<a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M25">View MathML</a>

(15)

<a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M26">View MathML</a>

(16)

Proof

From the definition of harmonic numbers and the digamma function,

<a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M27">View MathML</a>

(17)

and Equation 3 follows. From the double argument identity of the digamma function

<a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M28">View MathML</a>

(18)

using Equation 3 and rearranging, we obtain Equation 4. For Equation 5, we first note that for an arbitrary sequence <a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M29">View MathML</a>, the following identity holds:

<a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M30">View MathML</a>

(19)

hence,

<a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M31">View MathML</a>

(20)

The interesting identity (Equation 6) follows from Equation 5 and substituting

<a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M32">View MathML</a>

(21)

so that

<a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M33">View MathML</a>

(22)

<a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M34">View MathML</a>

(23)

replacing the counter, we obtain Equation 6. □

Main results and discussion

We now prove the two following theorems:

Theorem 1

Let <a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M35">View MathML</a> Then we have

<a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M36">View MathML</a>

(24)

Proof

Let <a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M37">View MathML</a> and consider the following expansion:

<a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M38">View MathML</a>

(25)

Now,

<a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M39">View MathML</a>

(26)

where

<a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M40">View MathML</a>

(27)

For an arbitrary positive sequence <a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M41">View MathML</a>, the following identity holds:

<a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M42">View MathML</a>

(28)

hence, from Equations 4 and 9,

<a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M43">View MathML</a>

(29)

Since we notice that

<a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M44">View MathML</a>

(30)

we get

<a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M45">View MathML</a>

(31)

Now,

<a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M46">View MathML</a>

(32)

substituting Equation 7 and simplifying, we have

<a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M47">View MathML</a>

(33)

hence, the identity (Equation 8) follows. □

Corollary 1

From Equation 8 and using Equations 3 and 4, we obtain the results,

<a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M48">View MathML</a>

(34)

and

<a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M49">View MathML</a>

(35)

Proof

We can use Equations 3 and 4 and also note that

<a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M50">View MathML</a>

(36)

From the rearrangement of <a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M51">View MathML</a> and Equation 1, we can obtain Equation 11; and from the rearrangement of <a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M52">View MathML</a> and Equation 13, we can obtain Equation 12. □

Example 1

For k=3 and 5,

<a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M53">View MathML</a>

(37)

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

Theorem 2

Let k, <a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M54">View MathML</a> Then we have

<a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M55">View MathML</a>

(38)

Proof

To prove Equation 14, we may write

<a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M56">View MathML</a>

(39)

where Ar is given by Equation 10, and by a rearrangement of sums,

<a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M57">View MathML</a>

(40)

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

Corollary 2

Let <a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M58">View MathML</a> Then we obtain

<a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M59">View MathML</a>

(41)

and

<a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M60">View MathML</a>

(42)

Proof

It is straightforward to show that

<a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M61">View MathML</a>

(43)

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

<a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M62">View MathML</a>

(44)

<a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M63">View MathML</a>

(45)

and

<a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M64">View MathML</a>

(46)

Conclusions

The author has generalized some results on variant Euler sums and specifically obtained identities for <a onClick="popup('http://www.iaumath.com/content/6/1/10/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.iaumath.com/content/6/1/10/mathml/M65">View MathML</a>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.

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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