International Journal of Combinatorics

International Journal of Combinatorics / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 208260 | https://doi.org/10.1155/2011/208260

Anthony Sofo, "Harmonic Numbers and Cubed Binomial Coefficients", International Journal of Combinatorics, vol. 2011, Article ID 208260, 14 pages, 2011. https://doi.org/10.1155/2011/208260

Harmonic Numbers and Cubed Binomial Coefficients

Academic Editor: Toufik Mansour
Received18 Jan 2011
Accepted03 Apr 2011
Published07 Jun 2011

Abstract

Euler related results on the sum of the ratio of harmonic numbers and cubed binomial coefficients are investigated in this paper. Integral and closed-form representation of sums are developed in terms of zeta and polygamma functions. The given representations are new.

1. Introduction

The well-known Riemann zeta function is defined as 𝜁(𝑧)=âˆžî“ğ‘Ÿ=11𝑟𝑧,Re(𝑧)>1.(1.1) The generalized harmonic numbers of order 𝛼 are given by 𝐻𝑛(𝛼)=𝑛𝑟=11𝑟𝛼for(𝛼,𝑛)∈ℕ×ℕ,ℕ∶={1,2,3,…},(1.2) and for 𝛼=1,𝐻𝑛(1)=𝐻𝑛=101−𝑡𝑛1−𝑡𝑑𝑡=𝑛𝑟=11𝑟=𝛾+𝜓(𝑛+1),(1.3) where 𝛾 denotes the Euler-Mascheroni constant defined by 𝛾=limğ‘›â†’âˆžîƒ©ğ‘›î“ğ‘Ÿ=11𝑟−log(𝑛)=−𝜓(1)≈0.5772156649…,(1.4) and where 𝜓(𝑧) denotes the Psi, or digamma function defined by 𝑑𝜓(𝑧)=Γ𝑑𝑧logΓ(𝑧)=(𝑧)=Γ(𝑧)âˆžî“ğ‘›=01−1𝑛+1𝑛+𝑧−𝛾,(1.5) and the Gamma function ∫Γ(𝑧)=∞0𝑢𝑧−1𝑒−𝑢𝑑𝑢, for ℝ(𝑧)>0.

Variant Euler sums of the form âˆžî“ğ‘›=1𝐻(1)2𝑛−(1/2)𝐻𝑛(1)𝑥2𝑛𝑛𝑝for𝑝=1and𝑝=2(1.6) have been considered by Chen [1], and recently Boyadzhiev [2] evaluated various binomial identities involving power sums with harmonic numbers 𝐻𝑛(1). Other remarkable harmonic number identities known to Euler areâˆžî“ğ‘›=1𝐻𝑛(1)(𝑛+1)2=𝜁(3),âˆžî“ğ‘›=1𝐻𝑛(1)𝑛3=54𝜁(4),(1.7) there is also a recurrence formula (2𝑛+1)𝜁(2𝑛)=2𝑛−1𝑟=1𝜁(2𝑟)𝜁(2𝑛−2𝑟),(1.8) which shows that in particular, for 𝑛=2,5𝜁(4)=2(𝜁(2))2 and more generally that 𝜁(2𝑛) is a rational multiple of (𝜁(𝑛))2. Another elegant recursion known to Euler [3] is 2âˆžî“ğ‘›=1𝐻𝑛(1)ğ‘›ğ‘ž=(ğ‘ž+2)𝜁(ğ‘ž+1)âˆ’ğ‘žâˆ’2𝑟=1𝜁(𝑟+1)𝜁(ğ‘žâˆ’ğ‘Ÿ).(1.9) Further work in the summation of harmonic numbers and binomial coefficients has also been done by Flajolet and Salvy [4] and Basu [5]. In this paper it is intended to add, in a small way, some results related to (1.7) and to extend the result of Cloitre, as reported in [6], âˆ‘âˆžğ‘›=1(𝐻𝑛(1)/𝑘𝑛+𝑘)=(𝑘/(𝑘−1)2) for 𝑘>1. Specifically, we investigate integral representations and closed form representations for sums of harmonic numbers and cubed binomial coefficients. The works of [7–13] also investigate various representations of binomial sums and zeta functions in simpler form by the use of the Beta function and other techniques. Some of the material in this paper was inspired by the work of Mansour, [8], where he used, in part, the Beta function to obtain very general results for finite binomial sums.

2. Integral Representations and Identities

The following Lemma, given by Sofo [11], is stated without proof and deals with the derivative of a reciprocal binomial coefficient.

Lemma 2.1. Let ğ‘Ž be a positive real number, 𝑧≥0,𝑛 is a positive integer and let 𝑄(ğ‘Žğ‘›,𝑧)=ğ‘§ğ‘Žğ‘›+𝑧−1 be an analytic function of 𝑧. Then, ğ‘„î…ž(ğ‘Žğ‘›,𝑧)=𝑑𝑄=⎧⎪⎨⎪⎩[]𝑑𝑧−𝑄(ğ‘Žğ‘›,𝑧)𝜓(𝑧+1+ğ‘Žğ‘›)−𝜓(𝑧+1)for𝑧>0,−𝐻𝑛(1),for𝑧=0andğ‘Ž=1.(2.1)

Theorem 2.2. Let ğ‘Ž,𝑏,𝑐,𝑑≥0 be real positive numbers, |𝑡|≤1,𝑝≥0 and let 𝑗,𝑘,𝑙,𝑚≥0 be real positive numbers. Then 𝑛≥1𝑡𝑛𝑛+𝑝−1𝑛−1[𝜓](𝑗+1+ğ‘Žğ‘›)−𝜓(𝑗+1)ğ‘›î€·ğ‘—ğ‘Žğ‘›+𝑗𝑘𝑏𝑛+𝑘𝑙𝑐𝑛+𝑙𝑚𝑑𝑛+𝑚=âˆ’ğ‘Žğ‘˜ğ‘™ğ‘šğ‘¡î€œî€ž10(1−𝑥)𝑗(1−𝑦)𝑘−1(1−𝑧)𝑙−1(1−𝑤)𝑚−11âˆ’ğ‘¡ğ‘¥ğ‘Žğ‘¦ğ‘ğ‘§ğ‘ğ‘¤ğ‘‘î€¸ğ‘+1×ln(1−𝑥)â‹…ğ‘¥ğ‘Žâˆ’1𝑦𝑏𝑧𝑐𝑤𝑑𝑑𝑥𝑑𝑦𝑑𝑧𝑑𝑤.(2.2)

Proof. Expand 𝑛≥1𝑡𝑛𝑛+𝑝−1𝑛−1î€¸ğ‘›î€·ğ‘—ğ‘Žğ‘›+𝑗𝑘𝑏𝑛+𝑘𝑙𝑐𝑛+𝑙𝑚𝑑𝑛+𝑚=𝑛≥1𝑡𝑛𝑛+𝑝−1𝑛−1ğ‘Žğ‘›Î“(ğ‘Žğ‘›)Γ(𝑗+1)Γ(𝑏𝑛+1)𝑘⋅Γ(𝑘)×𝑛Γ(ğ‘Žğ‘›+𝑗+1)Γ(𝑏𝑛+𝑘+1)Γ(𝑐𝑛+1)𝑙⋅Γ(𝑙)Γ(𝑑𝑛+1)𝑚⋅Γ(𝑚)Γ(𝑐𝑛+𝑙+1)Γ(𝑑𝑛+𝑚+1)=ğ‘Žğ‘˜ğ‘™ğ‘š10(1−𝑥)𝑗𝑥𝑛≥1ğ‘¥ğ‘Žğ‘›ğ‘¡ğ‘›îƒ©îƒªğ‘›+𝑝−1𝑛−1×𝐵(𝑏𝑛+1,𝑘)𝐵(𝑐𝑛+1,𝑙)𝐵(𝑑𝑛+1,𝑚)𝑑𝑥,(2.3) where 𝐵(𝛼,𝛽)=Γ(𝛼)Γ(𝛽)=Γ(𝛼+𝛽)10(1−𝑦)𝛼−1𝑦𝛽−1=𝑑𝑦10(1−𝑦)𝛽−1𝑦𝛼−1𝑑𝑦,for𝛼>0and𝛽>0(2.4) is the classical Beta function. Differentiating with respect to the parameter 𝑗, and utilizing Lemma 2.1 implies the resulting equation is as follows: 𝑛≥1𝑡𝑛𝑛+𝑝−1𝑛−1(𝜓(𝑗+1+ğ‘Žğ‘›)−𝜓(𝑗+1))ğ‘›î€·ğ‘—ğ‘Žğ‘›+𝑗𝑘𝑏𝑛+𝑘𝑙𝑐𝑛+𝑙𝑚𝑑𝑛+𝑚=âˆ’ğ‘Žğ‘˜ğ‘™ğ‘š10(1−𝑥)𝑗𝑥ln(1−𝑥)𝑛≥1ğ‘¥ğ‘Žğ‘›ğ‘¡ğ‘›îƒ©îƒªğ‘›+𝑝−1𝑛−1×𝐵(𝑏𝑛+1,𝑘)𝐵(𝑐𝑛+1,𝑙)𝐵(𝑑𝑛+1,𝑚)𝑑𝑥,=âˆ’ğ‘Žğ‘˜ğ‘™ğ‘šî€œî€ž10(1−𝑥)𝑗ln(1−𝑥)𝑥(1−𝑦)𝑘−1(1−𝑧)𝑙−1(1−𝑤)𝑚−1×𝑛≥1𝑛+𝑝−1𝑛−1ğ‘¡ğ‘¥ğ‘Žğ‘¦ğ‘ğ‘§ğ‘ğ‘¤ğ‘‘î€¸ğ‘›ğ‘‘ğ‘¥ğ‘‘ğ‘¦ğ‘‘ğ‘§ğ‘‘ğ‘¤=âˆ’ğ‘Žğ‘˜ğ‘™ğ‘šğ‘¡î€œî€ž10(1−𝑥)𝑗ln(1−𝑥)(1−𝑦)𝑘−1(1−𝑧)𝑙−1(1−𝑤)𝑚−11âˆ’ğ‘¡ğ‘¥ğ‘Žğ‘¦ğ‘ğ‘§ğ‘ğ‘¤ğ‘‘î€¸ğ‘+1Ã—ğ‘¥ğ‘Žâˆ’1𝑦𝑏𝑧𝑐𝑤𝑑𝑑𝑥𝑑𝑦𝑑𝑧𝑑𝑤(2.5) for |ğ‘¡ğ‘¥ğ‘Žğ‘¦ğ‘ğ‘§ğ‘ğ‘¤ğ‘‘|<1.

In the following three corollaries we encounter harmonic numbers at possible rational values of the argument, of the form 𝐻(𝛼)(𝑟/𝑏)−1 where 𝑟=1,2,3,…,𝑘,𝛼=1,2,3,…, and 𝑘∈ℕ. The polygamma function 𝜓(𝛼)(𝑧) is defined as 𝜓(𝛼)𝑑(𝑧)=𝛼+1𝑑𝑧𝛼+1=𝑑logΓ(𝑧)𝛼𝑑𝑧𝛼[]𝜓(𝑧),𝑧≠{0,−1,−2,−3,…}.(2.6) To evaluate 𝐻(𝛼)(𝑟/𝑏)−1 we have available a relation in terms of the polygamma function 𝜓(𝛼)(𝑧), for rational arguments 𝑧,𝐻(𝛼+1)(𝑟/𝑏)−1=𝜁(𝛼+1)+(−1)𝛼𝜓𝛼!(𝛼)𝑟𝑏,(2.7) where 𝜁(𝑧) is the Riemann zeta function. We also define𝐻(1)(𝑟/𝑏)−1𝑟=𝛾+𝜓𝑏,𝐻0(𝛼)=0.(2.8) The evaluation of the polygamma function 𝜓(𝛼)(𝑟/𝑏) at rational values of the argument can be explicitly done via a formula as given by Kölbig [14] (see also [15]) or Choi and Cvijović [16] in terms of the polylogarithmic or other special functions. Some specific values are given as 𝜓(𝑛)12=(−1)𝑛2𝑛!𝑛+1𝐻−1𝜁(𝑛+1)(5)(−2/3)1=𝜁(5)−𝜓24(4)13=−2𝜋5√39−120𝜁(5),𝐻(2)1/4=16−8𝐺−5𝜁(2),(2.9) and can be confirmed on a mathematical computer package, such as Mathematica [17].

Corollary 2.3. Let ğ‘Ž=1,𝑑=𝑐=𝑏>0,𝑡=1,𝑝=0,𝑗=0 and let 𝑙=𝑚=𝑘≥1 be a positive integer. Then 𝑛≥1𝐻𝑛(1)𝑛𝑘𝑏𝑛+𝑘3=−𝑘310[](1−𝑦)(1−𝑧)(1−𝑤)𝑘−11−𝑥(𝑦𝑧𝑤)𝑏ln(1−𝑥)⋅(𝑦𝑧𝑤)𝑏=𝑑𝑥𝑑𝑦𝑑𝑧𝑑𝑤(2.10)𝑘𝑟=1(−1)𝑟+1𝑘𝑟3×−𝑟24𝑏2𝑟𝜁(4)−2𝑏2𝐻(1)(𝑟/𝑏)−1+𝑟𝑏+𝑟2𝑏+𝑟𝑋𝑅(𝑘)𝜁(3)1−𝑏𝐻(1)(𝑟/𝑏)−1−𝑟2𝑏2𝐻(2)(𝑟/𝑏)−1𝑟+𝑟1−𝑏𝐻(1)(𝑟/𝑏)−1𝑋𝑅(𝑘)+𝑟2+1𝑌𝑅(𝑘)𝜁(2)2𝐻(1)(𝑟/𝑏)−12+𝐻(2)(𝑟/𝑏)−1+𝑟𝑏𝐻(1)(𝑟/𝑏)−1𝐻(2)(𝑟/𝑏)−1+𝐻(3)(𝑟/𝑏)−1+𝑟22𝑏2𝐻(2)(𝑟/𝑏)−12+2𝐻(1)(𝑟/𝑏)−1𝐻(3)(𝑟/𝑏)−1+3𝐻(4)(𝑟/𝑏)−1+𝑟2𝐻(1)(𝑟/𝑏)−12+𝐻(2)(𝑟/𝑏)−1+𝑟2𝑏𝐻(1)(𝑟/𝑏)−1𝐻(2)(𝑟/𝑏)−1+𝐻(3)(𝑟/𝑏)−1+𝑟𝑋𝑅(𝑘)22𝐻(1)(𝑟/𝑏)−12+𝐻(2)(𝑟/𝑏)−1,𝑌𝑅(𝑘)(2.11) where 𝐻𝑋𝑅(𝑘)∶=−3(1)𝑘−𝑟−𝐻(1)𝑟−13,(2.12)𝑌𝑅(𝑘)∶=2𝑋𝑅2(𝑘)3+𝐻(2)𝑘−𝑟+𝐻(2)𝑟−1.(2.13)

Proof. Let 𝑛≥1𝐻𝑛(1)𝑛𝑘𝑏𝑛+𝑘3=𝑛≥1𝐻𝑛(1)(𝑘!)3𝑛∏𝑘𝑟=1(𝑏𝑛+𝑟)3=𝑛≥1𝐻𝑛(1)(𝑘!)3𝑛𝑘𝑟=1𝐴𝑟+𝐵𝑏𝑛+𝑟𝑟(𝑏𝑛+𝑟)2+𝐶𝑟(𝑏𝑛+𝑟)3,(2.14) where 𝐶𝑟=lim𝑛→(−𝑟/𝑏)(𝑏𝑛+𝑟)3∏𝑘𝑟=1(𝑏𝑛+𝑟)3=(−1)𝑟+1𝑟𝑘𝑟𝑘!3,𝐵𝑟=lim𝑛→(−𝑟/𝑏)𝑑𝑑𝑛(𝑏𝑛+𝑟)3∏𝑘𝑟=1(𝑏𝑛+𝑟)3=3(−1)𝑟𝑟𝑘𝑟𝑘!3𝐻(1)𝑘−𝑟−𝐻(1)𝑟−1,𝐴𝑟=12lim𝑛→(−𝑟/𝑏)𝑑2𝑑𝑛2(𝑏𝑛+𝑟)3∏𝑘𝑟=1(𝑏𝑛+𝑟)3=32(−1)𝑟+1𝑟𝑘𝑟𝑘!33𝐻(1)𝑘−𝑟−𝐻(1)𝑟−12+𝐻(2)𝑘−𝑟+𝐻(2)𝑟−1.(2.15) Now, by interchanging sums, we have 𝑛≥1𝐻𝑛(1)𝑛𝑘𝑏𝑛+𝑘3=𝑘𝑟=1𝐴𝑟(𝑘!)3𝑛≥1𝐻𝑛(1)𝑛(𝑏𝑛+𝑟)+𝐵𝑟(𝑘!)3𝑛≥1𝐻𝑛(1)𝑛(𝑏𝑛+𝑟)2+𝐶𝑟(𝑘!)3𝑛≥1𝐻𝑛(1)𝑛(𝑏𝑛+𝑟)3.(2.16) We can evaluate 𝑛≥1𝐻𝑛(1)=𝑛(𝑏𝑛+𝑟)𝑛≥1𝜓(𝑛)+(1/𝑛)+𝛾𝑛(𝑏𝑛+𝑟)=𝑛(𝑏𝑛+𝑟)𝑛≥1𝜓(𝑛)+1𝑛(𝑏𝑛+𝑟)𝑛2+𝛾(𝑏𝑛+𝑟)=𝑛(𝑏𝑛+𝑟)𝑛≥1𝜓(𝑛)+1𝑛(𝑏𝑛+𝑟)𝑟𝑛2+(𝛾−(𝑏/𝑟))𝑟𝑛≥01−1𝑛+1+1𝑛+(𝑟/𝑏)−1𝑛+(𝑟/𝑏)𝑛+1+(𝑟/𝑏)=−𝑏𝛾𝑟2−𝛾2+2𝑟3𝜁(2)+2𝑟(𝜓(𝑟/𝑏))2−𝜓2ğ‘Ÿî…ž(𝑟/𝑏)+𝛾2𝑟𝑟𝜓𝑟𝑏,+𝛾+𝑏(2.17) here we have used the result from [18] 𝑛≥1𝜓(𝑛)=1𝑛(𝑏𝑛+𝑟)𝜓𝑟2𝑟𝑏+12−𝛾2+𝜁(2)âˆ’ğœ“î…žî‚€ğ‘Ÿğ‘î‚î‚¶.+1(2.18)
Now using (2.7) and (2.8), we may write 𝑛≥1𝐻𝑛(1)=𝑛(𝑏𝑛+𝑟)𝜁(2)𝑟+1𝐻2𝑟(1)(𝑟/𝑏)−12+𝐻(2)(𝑟/𝑏)−1.(2.19) Similarly 𝑛≥1𝐻𝑛(1)𝑛(𝑏𝑛+𝑟)2=1𝑟𝑛≥1𝐻𝑛(1)−𝑏𝑛(𝑏𝑛+𝑟)𝑟𝑛≥1𝐻𝑛(1)(𝑏𝑛+𝑟)2=−𝜁(3)+𝑏𝑟𝜁(2)𝑟2−𝜁(2)𝐻(1)(𝑟/𝑏)−1+1𝑏𝑟2𝑟2𝐻(1)(𝑟/𝑏)−12+𝐻(2)(𝑟/𝑏)−1+1𝐻𝑏𝑟(1)(𝑟/𝑏)−1𝐻(2)(𝑟/𝑏)−1+𝐻(3)(𝑟/𝑏)−1,𝑛≥1𝐻𝑛(1)𝑛(𝑏𝑛+𝑟)3=𝑛≥1𝐻𝑛(1)1𝑟2−𝑏𝑛(𝑏𝑛+𝑟)𝑟2(𝑏𝑛+𝑟)2−𝑏𝑟(𝑏𝑛+𝑟)3=−𝜁(4)4𝑏2𝑟−𝜁(3)𝑏𝑟2−𝜁(3)𝐻(1)(𝑟/𝑏)−1𝑏2𝑟+𝜁(2)𝑟3−𝜁(2)𝐻(1)(𝑟/𝑏)−1𝑏𝑟2−𝜁(2)𝐻(2)(𝑟/𝑏)−1𝑏2𝑟+12𝑟3𝐻(1)(𝑟/𝑏)−12+𝐻(2)(𝑟/𝑏)−1+1𝑏𝑟2𝐻(1)(𝑟/𝑏)−1𝐻(2)(𝑟/𝑏)−1+𝐻(3)(𝑟/𝑏)−1+12𝑏2𝑟𝐻(2)(𝑟/𝑏)−12+2𝐻(1)(𝑟/𝑏)−1𝐻(3)(𝑟/𝑏)−1+3𝐻(4)(𝑟/𝑏)−1.(2.20) Substituting (2.19), (2.20) into (2.16) where 𝑋𝑅(𝑘) and 𝑌𝑅(𝑘) are given by (2.12) and (2.13), respectively, on simplifying the identity (2.11) is realized.

For 𝑘=1 and 𝑏=1 the following identity is valid: 𝑛≥1𝐻𝑛(1)𝑛(𝑛+1)3=𝜁(2)−𝜁(3)−𝜁(4)4.(2.21)

Theorem 2.4. 𝑛≥1𝑡𝑛𝑛+𝑝−1𝑛−1(𝜓(𝑗+1+ğ‘Žğ‘›)−𝜓(𝑗+1))𝑛2î€·ğ‘—ğ‘Žğ‘›+𝑗𝑘𝑏𝑛+𝑘𝑙𝑐𝑛+𝑙𝑚𝑑𝑛+𝑚=âˆ’ğ‘Žğ‘ğ‘™ğ‘šğ‘¡î€œî€ž10(1−𝑥)𝑗(1−𝑦)𝑘(1−𝑧)𝑙−1(1−𝑤)𝑚−11âˆ’ğ‘¡ğ‘¥ğ‘Žğ‘¦ğ‘ğ‘§ğ‘ğ‘¤ğ‘‘î€¸ğ‘+1×ln(1−𝑥)â‹…ğ‘¥ğ‘Žâˆ’1𝑦𝑏−1𝑧𝑐𝑤𝑑𝑑𝑥𝑑𝑦𝑑𝑧𝑑𝑤.(2.22)

Proof. The proof of this theorem is very similar to that of Theorem 2.2 and will not be given here.

Corollary 2.5. Let ğ‘Ž=1,𝑑=𝑐=𝑏>0,𝑡=1,𝑝=0,𝑗=0, and let 𝑙=𝑚=𝑘≥1 be a positive integer. Then 𝑛≥1𝐻𝑛(1)𝑛2𝑘𝑏𝑛+𝑘3=−𝑏𝑘210(1−𝑦)𝑘[](1−𝑧)(1−𝑤)𝑘−11−𝑥(𝑦𝑧𝑤)𝑏ln(1−𝑥)⋅𝑦𝑏−1(𝑧𝑤)𝑏=𝑑𝑥𝑑𝑦𝑑𝑧𝑑𝑤(2.23)𝑘𝑟=1(−1)𝑟+1𝑘𝑟3×𝑟𝑟4𝑏𝜁(4)+4+𝑏𝐻(1)(𝑟/𝑏)−1+3𝑟𝑋𝑅(𝑘)+2𝑟2+−𝑌𝑅(𝑘)𝜁(3)3𝑏𝑟+2𝐻(1)(𝑟/𝑏)−1+𝑟𝑏𝐻(2)(𝑟/𝑏)−1+𝑟𝐻(1)(𝑟/𝑏)−1−−2𝑏𝑋𝑅(𝑘)−𝑏𝑟𝑌𝑅(𝑘)𝜁(2)3𝑏𝐻2𝑟(1)(𝑟/𝑏)−12+𝐻(2)(𝑟/𝑏)−1𝐻−2(1)(𝑟/𝑏)−1𝐻(2)(𝑟/𝑏)−1+𝐻(3)(𝑟/𝑏)−1−𝑟𝐻2𝑏(2)(𝑟/𝑏)−12+2𝐻(1)(𝑟/𝑏)−1𝐻(3)(𝑟/𝑏)−1+3𝐻(4)(𝑟/𝑏)−1−𝑏𝐻(1)(𝑟/𝑏)−12+𝐻(2)(𝑟/𝑏)−1𝐻+𝑟(1)(𝑟/𝑏)−1𝐻(2)(𝑟/𝑏)−1+𝐻(3)(𝑟/𝑏)−1−𝑋𝑅(𝑘)𝑏𝑟2𝐻(1)(𝑟/𝑏)−12+𝐻(2)(𝑟/𝑏)−1,𝑌𝑅(𝑘)(2.24) where 𝑋𝑅(𝑘) is given by (2.12) and 𝑌𝑅(𝑘) is given by (2.13).

Proof. Following similar steps to Corollary 2.3, we may write 𝑛≥1𝐻𝑛(1)𝑛2𝑘𝑏𝑛+𝑘3=𝑘𝑟=1𝐴𝑟(𝑘!)3𝑛≥1𝐻𝑛(1)𝑛2(𝑏𝑛+𝑟)+𝐵𝑟(𝑘!)3𝑛≥1𝐻𝑛(1)𝑛2(𝑏𝑛+𝑟)2+𝐶𝑟(𝑘!)3𝑛≥1𝐻𝑛(1)𝑛2(𝑏𝑛+𝑟)3,(2.25) and evaluate 𝑛≥1𝐻𝑛(1)𝑛2=(𝑏𝑛+𝑟)𝑛≥1𝐻𝑛(1)1𝑟𝑛2−𝑏=𝑟𝑛(𝑏𝑛+𝑟)2𝜁(3)𝑟−𝑏𝜁(2)𝑟2−𝑏2𝑟2𝐻(1)(𝑟/𝑏)−12+𝐻(2)(𝑟/𝑏)−1,𝑛≥1𝐻𝑛(1)𝑛2(𝑏𝑛+𝑟)2=3𝜁(3)𝑟2−2𝑏𝜁(2)𝑟3+𝜁(2)𝐻(1)(𝑟/𝑏)−1𝑟2−𝑏𝑟3𝐻(1)(𝑟/𝑏)−12+𝐻(2)(𝑟/𝑏)−1−1𝑟2𝐻(1)(𝑟/𝑏)−1𝐻(2)(𝑟/𝑏)−1+𝐻(3)(𝑟/𝑏)−1,𝑛≥1𝐻𝑛(1)𝑛2(𝑏𝑛+𝑟)3=𝜁(4)4𝑏𝑟2−3𝑏𝜁(2)𝑟4+2𝜁(2)𝐻(1)(𝑟/𝑏)−1𝑟3+𝜁(2)𝐻(2)(𝑟/𝑏)−1𝑏𝑟2+4𝜁(3)𝑟3+𝜁(3)𝐻(1)(𝑟/𝑏)−1𝑏𝑟2−3𝑏2𝑟4𝐻(1)(𝑟/𝑏)−12+𝐻(2)(𝑟/𝑏)−1−2𝑟3𝐻(1)(𝑟/𝑏)−1𝐻(2)(𝑟/𝑏)−1+𝐻(3)(𝑟/𝑏)−1−12𝑏𝑟2𝐻(2)(𝑟/𝑏)−12+2𝐻(1)(𝑟/𝑏)−1𝐻(3)(𝑟/𝑏)−1+3𝐻(4)(𝑟/𝑏)−1.(2.26) By substituting (2.26) into (2.25) and collecting zeta functions, the identity (2.24) is obtained.

For 𝑘=1 and 𝑏=1 the following identity is valid: 𝑛≥1𝐻𝑛(1)𝑛2(𝑛+1)3=𝜁(4)4+4𝜁(3)−3𝜁(2).(2.27)

Theorem 2.6. 𝑛≥1𝑡𝑛𝑛+𝑝−1𝑛−1(𝜓(𝑗+1+ğ‘Žğ‘›)−𝜓(𝑗+1))𝑛3î€·ğ‘—ğ‘Žğ‘›+𝑗𝑘𝑏𝑛+𝑘𝑙𝑐𝑛+𝑙𝑚𝑑𝑛+𝑚=âˆ’ğ‘Žğ‘ğ‘ğ‘šğ‘¡î€œî€ž10(1−𝑥)𝑗(1−𝑦)𝑘(1−𝑧)𝑙(1−𝑤)𝑚−11âˆ’ğ‘¡ğ‘¥ğ‘Žğ‘¦ğ‘ğ‘§ğ‘ğ‘¤ğ‘‘î€¸ğ‘+1×ln(1−𝑥)â‹…ğ‘¥ğ‘Žâˆ’1𝑦𝑏−1𝑧𝑐−1𝑤𝑑𝑑𝑥𝑑𝑦𝑑𝑧𝑑𝑤.(2.28)

Proof. The proof of this theorem is very similar to that of Theorem 2.2 and will not be given here.

Corollary 2.7. Let ğ‘Ž=1,𝑑=𝑐=𝑏>0,𝑡=1,𝑗=0, and let 𝑙=𝑚=𝑘≥1 be a positive integer. Then 𝑛≥1𝐻𝑛(1)𝑛3𝑘𝑏𝑛+𝑘3=−𝑏2ğ‘˜î€œî€ž10((1−𝑦)(1−𝑧))𝑘(1−𝑤)𝑘−11−𝑥(𝑦𝑧𝑤)𝑏ln(1−𝑥)⋅(𝑦𝑧)𝑏−1𝑤𝑏=𝑑𝑥𝑑𝑦𝑑𝑧𝑑𝑤(2.29)𝑘𝑟=1(−1)𝑟+1𝑘𝑟3×51+45𝑋𝑅(𝑘)+4𝑟2−𝑌𝑅(𝑘)𝜁(4)9𝑏𝑟+𝐻(1)(𝑟/𝑏)−1++5𝑏𝑋𝑅(𝑘)+2𝑏𝑟𝑌𝑅(𝑘)𝜁(3)6𝑏2𝑟2−3𝑏𝐻(1)(𝑟/𝑏)−1𝑟−𝐻(2)(𝑟/𝑏)−1−3𝑏2𝑟+𝑏𝐻(1)(𝑟/𝑏)−1𝑋𝑅(𝑘)+𝑏2+𝑌𝑅(𝑘)𝜁(2)3𝑏2𝑟2𝐻(1)(𝑟/𝑏)−12+𝐻(2)(𝑟/𝑏)−1+3𝑏𝑟𝐻(1)(𝑟/𝑏)−1𝐻(2)(𝑟/𝑏)−1+𝐻(3)(𝑟/𝑏)−1+12𝐻(2)(𝑟/𝑏)−12+2𝐻(1)(𝑟/𝑏)−1𝐻(3)(𝑟/𝑏)−1+3𝐻(4)(𝑟/𝑏)−1+3𝑏2𝐻2𝑟(1)(𝑟/𝑏)−12+𝐻(2)(𝑟/𝑏)−1𝐻+𝑏(1)(𝑟/𝑏)−1𝐻(2)(𝑟/𝑏)−1+𝐻(3)(𝑟/𝑏)−1+𝑏𝑋𝑅(𝑘)22𝐻(1)(𝑟/𝑏)−12+𝐻(2)(𝑟/𝑏)−1,𝑌𝑅(𝑘)(2.30) where 𝑋𝑅(𝑘) is given by (2.12) and 𝑌𝑅(𝑘) is given by (2.13).

Proof. We follow similar steps as the previous corollary so that 𝑛≥1𝐻𝑛(1)𝑛3𝑘𝑏𝑛+𝑘3=𝑘𝑟=1𝐴𝑟(𝑘!)3𝑛≥1𝐻𝑛(1)𝑛3(𝑏𝑛+𝑟)+𝐵𝑟(𝑘!)3𝑛≥1𝐻𝑛(1)𝑛3(𝑏𝑛+𝑟)2+𝐶𝑟(𝑘!)3𝑛≥1𝐻𝑛(1)𝑛3(𝑏𝑛+𝑟)3.(2.31) After much algebraic simplification, the following identity is obtained: 𝑛≥1𝐻𝑛(1)𝑛3(𝑏𝑛+𝑟)3=𝑛≥1𝐻𝑛(1)1𝑟3𝑛3−3𝑏𝑟4𝑛2−𝑏3𝑟3(𝑏𝑛+𝑟)3−3𝑏3𝑟4(𝑏𝑛+𝑟)2+6𝑏2𝑟4𝑛=(𝑏𝑛+𝑟)𝜁(4)𝑟3−9𝑏𝜁(3)𝑟4−𝜁(3)𝐻(1)(𝑟/𝑏)−1𝑟3+6𝑏2𝜁(2)𝑟5−3𝑏𝜁(2)𝐻(1)(𝑟/𝑏)−1𝑟4−𝜁(2)𝐻(2)(𝑟/𝑏)−1𝑟3+3𝑏2𝑟5𝐻(1)(𝑟/𝑏)−12+𝐻(2)(𝑟/𝑏)−1+3𝑏𝑟4𝐻(1)(𝑟/𝑏)−1𝐻(2)(𝑟/𝑏)−1+𝐻(3)(𝑟/𝑏)−1+12𝑟3𝐻(2)(𝑟/𝑏)−12+2𝐻(1)(𝑟/𝑏)−1𝐻(3)(𝑟/𝑏)−1+3𝐻(4)(𝑟/𝑏)−1,𝑛≥1𝐻𝑛(1)𝑛3(𝑏𝑛+𝑟)2=5𝜁(4)4𝑟2−5𝑏𝜁(3)𝑟3+3𝑏2𝜁(2)𝑟4−𝑏𝜁(2)𝐻(1)(𝑟/𝑏)−1𝑟3+3𝑏22𝑟4𝐻(1)(𝑟/𝑏)−12+𝐻(2)(𝑟/𝑏)−1+𝑏𝑟3𝐻(1)(𝑟/𝑏)−1𝐻(2)(𝑟/𝑏)−1+𝐻(3)(𝑟/𝑏)−1,𝑛≥1𝐻𝑛(1)𝑛3=(𝑏𝑛+𝑟)5𝜁(4)+𝑏4𝑟2𝜁(2)𝑟3−2𝑏𝜁(3)𝑟2+𝑏22𝑟3𝐻(1)(𝑟/𝑏)−12+𝐻(2)(𝑟/𝑏)−1.(2.32) Now we can substitute (2.32) into (2.31), collecting zeta functions and using (2.12) and (2.13) for 𝑋𝑅(𝑘) and 𝑌𝑅(𝑘), respectively, the identity (2.30) is obtained.

Some specific examples of Corollary 2.7 are as follows.

For 𝑘=1 and 𝑏=1 the following identity is valid, 𝑛≥1𝐻𝑛(1)𝑛3(𝑛+1)31=𝜁(4)−9𝜁(3)+6𝜁(2),𝑛≥1𝐻𝑛(1)𝑛384𝑛+83=12729073000−27264101740083211025ln2+4403800186883675(ln2)2−16604790784315𝐺+11272192𝐺2+1576747008−35𝐺ln21390679293952𝜋−33075172233728𝜋3−105203992775989−2450𝜁(2)4069970159210𝜁(3)+28974848ln2𝜁(3)−622467372𝜁(4),(2.33) where 𝐺 is Catalan's constant, defined by 1𝐺=210𝐾(𝑠)𝑑𝑠=âˆžî“ğ‘Ÿ=1(−1)𝑟(2𝑟+1)2≈0.915965…,(2.34) and 𝐾(𝑠) is the complete elliptic integral of the first kind. The degenerate case 𝑘=0, gives the well-known result 𝑛≥1𝐻𝑛(1)𝑛3=54𝜁(4).(2.35)

Remark 2.8. Corollaries 2.3, 2.5, and 2.7 are important and can be evaluated as demonstrated independently of their integral representations. Similarly the proofs of Corollaries 2.3, 2.5, and 2.7 are not obvious therefore their explicit representations is desired.

Remark 2.9. Theoretically it should be possible to obtain an integral representation for the general sum 𝑛≥1𝑡𝑛𝑛+𝑝−1𝑛−1(𝜓(𝑗+1+ğ‘Žğ‘›)−𝜓(𝑗+1))ğ‘›ğ‘žî€·ğ‘—ğ‘Žğ‘›+𝑗𝑘𝑏𝑛+𝑘𝑙𝑐𝑛+𝑙𝑚𝑑𝑛+𝑚,forğ‘ž=1,2,3,…,(2.36) with its associated corollaries. This work will be investigated in a forthcoming paper.

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Copyright © 2011 Anthony Sofo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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