Harmonic Numbers and Cubed Binomial Coefficients

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)=^∞_𝑟=11𝑟^𝑧,Re(𝑧)>1.(1.1) The generalized harmonic numbers of order 𝛼 are given by 𝐻^𝑛(𝛼)=^𝑛_𝑟=11𝑟^𝛼for(𝛼,𝑛)∈ℕ×ℕ,ℕ∶={1,2,3,…},(1.2) and for 𝛼=1,𝐻^𝑛(1)=𝐻_𝑛=¹⁰1−𝑡^𝑛1−𝑡𝑑𝑡=^𝑛_𝑟=11𝑟=𝛾+𝜓(𝑛+1),(1.3) where 𝛾 denotes the Euler-Mascheroni constant defined by 𝛾=lim_𝑛→∞^𝑛_𝑟=11𝑟−log(𝑛)=−𝜓(1)≈0.5772156649…,(1.4) and where 𝜓(𝑧) denotes the Psi, or digamma function defined by 𝜓(𝑧)=𝑑𝑑𝑧logΓ(𝑧)=Γ^(𝑧)Γ(𝑧)=^∞_𝑛=0𝑛+11−𝑛+1𝑧−𝛾,(1.5) and the Gamma function Γ(𝑧)=∫^∞0𝑢^𝑧−1𝑒^−𝑢𝑑𝑢, for ℝ(𝑧)>0.

Variant Euler sums of the form ^∞_𝑛=1𝐻^(1)2𝑛−(1/2)𝐻^𝑛(1)𝑥^2𝑛𝑛^𝑝for𝑝=1and𝑝=2(1.6) have been considered by Chen [1], and recently Boyadzhiev [2] evaluated various binomial identities involving power sums with harmonic numbers 𝐻^𝑛(1). Other remarkable harmonic number identities known to Euler are^∞_𝑛=1𝐻^𝑛(1)(𝑛+1)²=𝜁(3),^∞_𝑛=1𝐻^𝑛(1)𝑛³=54𝜁(4),(1.7) there is also a recurrence formula (2𝑛+1)𝜁(2𝑛)=2^𝑛−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 2^∞_𝑛=1𝐻^𝑛(1)𝑛^𝑞=(𝑞+2)𝜁(𝑞+1)−^𝑞−2_𝑟=1𝜁(𝑟+1)𝜁(𝑞−𝑟).(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], ∑^∞𝑛=1(𝐻^𝑛(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 𝑎 be a positive real number, 𝑧≥0,𝑛 is a positive integer and let 𝑄(𝑎𝑛,𝑧)=^{𝑧𝑎𝑛+𝑧}⁻¹ be an analytic function of 𝑧. Then, 𝑄^(𝑎𝑛,𝑧)=𝑑𝑄𝑑𝑧=⎧⎪⎨⎪⎩−−𝑄𝐻(𝑎^𝑛(1)𝑛,,𝑧)[𝜓(𝑧+1+𝑎𝑛)−𝜓(𝑧+1)]foforr𝑧𝑧=>00,and𝑎=1.(2.1)

Theorem 2.2. Let 𝑎,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≥1𝑡^𝑛^{𝑛+𝑝−1𝑛−1}[𝜓(𝑗+1+𝑎𝑛)−𝜓(𝑗+1)]=−𝑛𝑎𝑘𝑙𝑚^{𝑗𝑎𝑛+𝑗}𝑡^{𝑘𝑏𝑛+𝑘}₁₀^{𝑙𝑐𝑛+𝑙}^{𝑚𝑑𝑛+𝑚}(1−𝑥)^𝑗(1−𝑦)^𝑘−1(1−𝑧)^𝑙−1(1−𝑤)^𝑚−11−𝑡𝑥^𝑎𝑦^𝑏𝑧^𝑐𝑤^𝑑^𝑝+1×ln(1−𝑥)⋅𝑥^𝑎−1𝑦^𝑏𝑧^𝑐𝑤^𝑑𝑑𝑥𝑑𝑦𝑑𝑧𝑑𝑤.(2.2)

Proof. Expand _𝑛≥1𝑡^𝑛^{𝑛+𝑝−1𝑛−1}𝑛=^{𝑗𝑎𝑛+𝑗}_𝑛≥1^{𝑘𝑏𝑛+𝑘}^{𝑙𝑐𝑛+𝑙}^{𝑚𝑑𝑛+𝑚}×𝑡Γ^𝑛(𝑐𝑛𝑛++𝑛−𝑝1)−1𝑙⋅1Γ(𝑙𝑎)𝑛ΓΓ𝑛((Γ𝑑𝑎𝑛𝑛(𝑎)+𝑛Γ1+()𝑗𝑚𝑗++⋅1Γ1))Γ(Γ𝑚(𝑏)(𝑏𝑛𝑛++1𝑘)𝑘+⋅1Γ)(𝑘)=𝑎𝑘𝑙𝑚₁₀Γ(𝑐𝑛+𝑙+1)Γ(𝑑𝑛+𝑚+1)(1−𝑥)^𝑗𝑥_𝑛≥1𝑥^𝑎𝑛𝑡^𝑛𝑛+𝑛−𝑝−11×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−1𝑦^𝛽−1𝑑𝑦₁₀(1−𝑦)^𝛽−1𝑦^𝛼−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≥1𝑡^𝑛^{𝑛+𝑝−1𝑛−1}(𝜓(𝑗+1+𝑎𝑛)−𝜓(𝑗+1))=−𝑛𝑎𝑘𝑙𝑚^{𝑗𝑎𝑛+𝑗}₁₀^{𝑘𝑏𝑛+𝑘}^{𝑙𝑐𝑛+𝑙}^{𝑚𝑑𝑛+𝑚}(1−𝑥)^𝑗𝑥ln(1−𝑥)_𝑛≥1𝑥^𝑎𝑛𝑡^𝑛=×−𝐵𝑎𝑘(𝑙𝑏𝑚𝑛+1,𝑘₁₀)𝐵(𝑐𝑛+1,𝑙)𝐵(𝑑𝑛+1,𝑚𝑛)𝑑+𝑛𝑥−,𝑝−11(1−𝑥)^𝑗ln(1−𝑥)𝑥(1−𝑦)^𝑘−1(1−𝑧)^𝑙−1(1−𝑤)^𝑚−1×_𝑛≥1𝑛+𝑛−𝑝−11𝑡𝑥^𝑎=−𝑎𝑘𝑙𝑚𝑡₁₀𝑦^𝑏𝑧^𝑐𝑤^𝑑^𝑛𝑑𝑥𝑑𝑦𝑑𝑧𝑑𝑤(1−𝑥)^𝑗ln(1−𝑥)(1−𝑦)^𝑘−1(1−𝑧)^𝑙−1(1−𝑤)^𝑚−11−𝑡𝑥^𝑎𝑦^𝑏𝑧^𝑐𝑤^𝑑^𝑝+1×𝑥^𝑎−1𝑦^𝑏𝑧^𝑐𝑤^𝑑𝑑𝑥𝑑𝑦𝑑𝑧𝑑𝑤(2.5) for |𝑡𝑥^𝑎𝑦^𝑏𝑧^𝑐𝑤^𝑑|<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)(𝑧)=𝑑^𝛼+1𝑑𝑧^𝛼+1logΓ(𝑧)=𝑑𝑑𝑧^𝛼𝛼[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)^𝛼𝛼!𝜓^(𝛼)𝑏𝑟,(2.7) where 𝜁(𝑧) is the Riemann zeta function. We also define𝐻^{(1)(𝑟/𝑏)−1}=𝛾+𝜓𝑟𝑏,𝐻^0(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)^𝑛𝐻𝑛!2^𝑛+1−1𝜁(𝑛+1)_(5)(−2/3)=𝜁(5)−124𝜓⁽⁴⁾31=−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 𝑎=1,𝑑=𝑐=𝑏>0,𝑡=1,𝑝=0,𝑗=0 and let 𝑙=𝑚=𝑘≥1 be a positive integer. Then _𝑛≥1𝐻^𝑛(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×−𝑟²4𝑏²𝜁(4)−𝑟²𝑏²𝐻^{(1)(𝑟/𝑏)−1}+𝑟+1−𝑟𝑏+𝑟𝑏²𝑋𝑅(𝑘)𝜁(3)𝑏𝐻^{(1)(𝑟/𝑏)−1}−𝑟²𝑏²𝐻^{(2)(𝑟/𝑏)−1}+𝑟1−𝑟𝑏𝐻^{(1)(𝑟/𝑏)−1}𝑋𝑅(𝑘)+𝑟²+1𝑌𝑅(𝑘)𝜁(2)2𝐻^{(1)(𝑟/𝑏)−1}²+𝐻^{(2)(𝑟/𝑏)−1}+𝑏𝑟𝐻^{(1)(𝑟/𝑏)−1}+𝑟₂𝐻^{(2)(𝑟/𝑏)−1}+𝐻^{(3)(𝑟/𝑏)−1}2𝑏²𝐻^{(2)(𝑟/𝑏)−1}²+2𝐻^{(1)(𝑟/𝑏)−1}𝐻^{(3)(𝑟/𝑏)−1}+3𝐻^{(4)(𝑟/𝑏)−1}+𝑟2𝐻^{(1)(𝑟/𝑏)−1}²+𝐻^{(2)(𝑟/𝑏)−1}+𝑟𝑏²𝐻^{(1)(𝑟/𝑏)−1}+𝑟𝐻^{(2)(𝑟/𝑏)−1}+𝐻^{(3)(𝑟/𝑏)−1}𝑋𝑅(𝑘)₂2𝐻^{(1)(𝑟/𝑏)−1}²+𝐻^{(2)(𝑟/𝑏)−1}𝑌𝑅(𝑘),(2.11) where 𝑋𝑌𝑅𝑅((𝑘𝑘))∶∶==3−3𝐻^{(1)𝑘−𝑟}−𝐻^(1)𝑟−1,(2.12)2𝑋𝑅²(𝑘)3+𝐻^{(2)𝑘−𝑟}+𝐻^(2)𝑟−1.(2.13)

Proof. Let _𝑛≥1𝐻^𝑛(1)𝑛^{𝑘𝑏𝑛+𝑘}³=_𝑛≥1𝐻^𝑛(1)(𝑘!)³𝑛∏^𝑘𝑟=1(𝑏𝑛+𝑟)³=_𝑛≥1𝐻^𝑛(1)(𝑘!)³𝑛^𝑘_𝑟=1𝑏𝑛𝐴+^𝑟𝑟+(𝑏𝑛𝐵+^𝑟𝑟)²+𝐶^𝑟(𝑏𝑛+𝑟)³,(2.14) where 𝐶_𝑟=lim_{𝑛→(−𝑟/𝑏)}∏(𝑏^𝑘𝑟=1𝑛(𝑏+𝑛𝑟+)³𝑟)³=(−1)^𝑟+1𝑘𝑟!𝑘𝑟³𝐵_𝑟,=lim_{𝑛→(−𝑟/𝑏)}𝑑=3(−1)_𝑟𝑑𝑛∏(𝑏^𝑘𝑟=1𝑛(𝑏+𝑛𝑟+)³𝑟)³𝑘𝑟!𝑘𝑟³𝐻^{(1)𝑘−𝑟}𝐴_𝑟−𝐻^(1)𝑟−1,=12lim^{𝑛→(−𝑟/𝑏)}𝑑²=3𝑑𝑛²∏(𝑏^𝑘𝑟=1𝑛(𝑏+𝑛𝑟+)³𝑟)³2(−1)^𝑟+1𝑘𝑟!𝑘𝑟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≥1𝐻^𝑛(1)𝑛^{𝑘𝑏𝑛+𝑘}³=^𝑘_𝑟=1𝐴^𝑟(𝑘!)³_𝑛≥1𝐻^𝑛(1)𝑛(𝑏𝑛+𝑟)+𝐵^𝑟(𝑘!)³_𝑛≥1𝐻^𝑛(1)𝑛(𝑏𝑛+𝑟)²+𝐶_𝑟(𝑘!)³_𝑛≥1𝐻^𝑛(1)𝑛(𝑏𝑛+𝑟)³.(2.16) We can evaluate _𝑛≥1𝐻^𝑛(1)𝑛(𝑏𝑛+𝑟)==_𝑛≥1𝜓𝑛(𝑛(𝑏)𝑛++(1𝑟/)𝑛)+𝑛(𝑏𝑛𝛾+𝑟)=_𝑛≥1𝑛(𝜓𝑏𝑛(𝑛+)𝑟)+𝑛²(𝑏𝑛1+𝑟)+𝑛(𝑏𝑛𝛾+𝑟)_𝑛≥1+(𝛾−𝑛((𝑏𝜓𝑏/𝑛𝑟()𝑛+))𝑟)+𝑟1𝑛²=−𝑏𝛾𝑟_𝑛≥0𝑛+11−𝑛+1(𝑟/𝑏)+𝑛+1(𝑟/𝑏)−𝑛+11+(𝑟/𝑏)𝑟²−𝛾²2𝑟+3𝜁2(𝑟2)+(𝜓(2𝑟𝑟/𝑏))²−𝜓^(𝑟/𝑏)2𝑟+𝛾𝑟𝜓𝑏𝑟+𝛾+𝑏,(2.17) here we have used the result from [18] _𝑛≥1𝜓(𝑛)𝑛(𝑏𝑛+𝑟)=21𝑟𝜓𝑏𝑟+1²−𝛾²+𝜁(2)−𝜓^𝑟𝑏+1.(2.18)
Now using (2.7) and (2.8), we may write _𝑛≥1𝐻^𝑛(1)𝑛(𝑏𝑛+𝑟)=𝜁(𝑟2)+21𝑟𝐻^{(1)(𝑟/𝑏)−1}²+𝐻^{(2)(𝑟/𝑏)−1}.(2.19) Similarly _𝑛≥1𝐻^𝑛(1)𝑛(𝑏𝑛+𝑟)²=1𝑟_𝑛≥1𝐻^𝑛(1)𝑛(𝑏𝑛+𝑟)−𝑏𝑟_𝑛≥1𝐻^𝑛(1)(𝑏𝑛+𝑟)²=−𝜁(3)𝑏𝑟+𝜁𝑟(²2)−𝜁(2)𝐻^{(1)(𝑟/𝑏)−1}𝑏𝑟+21𝑟²𝐻^{(1)(𝑟/𝑏)−1}²+𝐻^{(2)(𝑟/𝑏)−1}+1𝑏𝑟𝐻^{(1)(𝑟/𝑏)−1}_𝑛≥1𝐻^{(2)(𝑟/𝑏)−1}+𝐻^{(3)(𝑟/𝑏)−1},𝐻^𝑛(1)𝑛(𝑏𝑛+𝑟)³=_𝑛≥1𝐻^𝑛(1)𝑟²𝑛(𝑏1𝑛+𝑟)−𝑟²(𝑏𝑛𝑏+𝑟)²−𝑏=−𝜁(4)𝑟(𝑏𝑛+𝑟)³4𝑏²𝑟−𝜁𝑏(𝑟3²)−𝜁(3)𝐻^{(1)(𝑟/𝑏)−1}𝑏²𝑟+𝜁𝑟(³2)−𝜁(2)𝐻^{(1)(𝑟/𝑏)−1}𝑏𝑟²−𝜁(2)𝐻^{(2)(𝑟/𝑏)−1}𝑏²𝑟+21𝑟³𝐻^{(1)(𝑟/𝑏)−1}²+𝐻^{(2)(𝑟/𝑏)−1}+1𝑏𝑟²𝐻^{(1)(𝑟/𝑏)−1}+1𝐻^{(2)(𝑟/𝑏)−1}+𝐻^{(3)(𝑟/𝑏)−1}2𝑏²𝑟𝐻^{(2)(𝑟/𝑏)−1}²+2𝐻^{(1)(𝑟/𝑏)−1}𝐻^{(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≥1𝐻^𝑛(1)𝑛(𝑛+1)³=𝜁(2)−𝜁(3)−𝜁(4)4.(2.21)

Theorem 2.4. _𝑛≥1𝑡^𝑛^{𝑛+𝑝−1𝑛−1}(𝜓(𝑗+1+𝑎𝑛)−𝜓(𝑗+1))=𝑛−²𝑎𝑏𝑙𝑚𝑡^{𝑗𝑎𝑛+𝑗}^{𝑘𝑏𝑛+𝑘}₁₀^{𝑙𝑐𝑛+𝑙}^{𝑚𝑑𝑛+𝑚}(1−𝑥)^𝑗(1−𝑦)^𝑘(1−𝑧)^𝑙−1(1−𝑤)^𝑚−11−𝑡𝑥^𝑎𝑦^𝑏𝑧^𝑐𝑤^𝑑^𝑝+1×ln(1−𝑥)⋅𝑥^𝑎−1𝑦^𝑏−1𝑧^𝑐𝑤^𝑑𝑑𝑥𝑑𝑦𝑑𝑧𝑑𝑤.(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 𝑎=1,𝑑=𝑐=𝑏>0,𝑡=1,𝑝=0,𝑗=0, and let 𝑙=𝑚=𝑘≥1 be a positive integer. Then _𝑛≥1𝐻^𝑛(1)𝑛²^{𝑘𝑏𝑛+𝑘}³=−𝑏𝑘²¹⁰(1−𝑦)^𝑘[(1−𝑧)(1−𝑤)]^𝑘−11−𝑥(𝑦𝑧𝑤)^𝑏ln(1−𝑥)⋅𝑦^𝑏−1=(𝑧𝑤)^𝑏𝑑𝑥𝑑𝑦𝑑𝑧𝑑𝑤(2.23)_𝑘_𝑟=1(−1)^𝑟+1𝑘𝑟³×𝑟4𝑏𝜁(4)+4+𝑏𝑟𝐻^{(1)(𝑟/𝑏)−1}+−3𝑏+3𝑟𝑋𝑅(𝑘)+2𝑟²𝑌𝑅(𝑘)𝜁(3)𝑟+2𝐻^{(1)(𝑟/𝑏)−1}+𝑟𝑏𝐻^{(2)(𝑟/𝑏)−1}−3𝑏+𝑟𝐻^{(1)(𝑟/𝑏)−1}−2𝑏𝑋𝑅(𝑘)−𝑏𝑟𝑌𝑅(𝑘)𝜁(2)2𝑟𝐻^{(1)(𝑟/𝑏)−1}²+𝐻^{(2)(𝑟/𝑏)−1}−2𝐻^{(1)(𝑟/𝑏)−1}−𝑟𝐻^{(2)(𝑟/𝑏)−1}+𝐻^{(3)(𝑟/𝑏)−1}2𝑏𝐻^{(2)(𝑟/𝑏)−1}²+2𝐻^{(1)(𝑟/𝑏)−1}𝐻^{(3)(𝑟/𝑏)−1}+3𝐻^{(4)(𝑟/𝑏)−1}−𝑏𝐻_{(1)(𝑟/𝑏)−1}−𝑏𝑟²+𝐻^{(2)(𝑟/𝑏)−1}+𝑟𝐻^{(1)(𝑟/𝑏)−1}𝐻^{(2)(𝑟/𝑏)−1}+𝐻^{(3)(𝑟/𝑏)−1}𝑋𝑅(𝑘)2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≥1𝐻^𝑛(1)𝑛²^{𝑘𝑏𝑛+𝑘}³=^𝑘_𝑟=1𝐴^𝑟(𝑘!)³_𝑛≥1𝐻^𝑛(1)𝑛²(𝑏𝑛+𝑟)+𝐵^𝑟(𝑘!)³_𝑛≥1𝐻^𝑛(1)𝑛²(𝑏𝑛+𝑟)²+𝐶_𝑟(𝑘!)³_𝑛≥1𝐻^𝑛(1)𝑛²(𝑏𝑛+𝑟)³,(2.25) and evaluate _𝑛≥1𝐻^𝑛(1)𝑛²(𝑏𝑛+𝑟)=_𝑛≥1𝐻^𝑛(1)𝑟1𝑛²−𝑏=2𝜁(3)𝑟𝑛(𝑏𝑛+𝑟)𝑟−𝑏𝜁𝑟(²2)−𝑏2𝑟²𝐻^{(1)(𝑟/𝑏)−1}²+𝐻^{(2)(𝑟/𝑏)−1}_𝑛≥1,𝐻^𝑛(1)𝑛²(𝑏𝑛+𝑟)²=3𝜁(3)𝑟²−2𝑏𝜁(2)𝑟³+𝜁(2)𝐻^{(1)(𝑟/𝑏)−1}𝑟²−𝑏𝑟³𝐻^{(1)(𝑟/𝑏)−1}²+𝐻^{(2)(𝑟/𝑏)−1}−𝑟1²𝐻^{(1)(𝑟/𝑏)−1}_𝑛≥1𝐻^{(2)(𝑟/𝑏)−1}+𝐻^{(3)(𝑟/𝑏)−1},𝐻^𝑛(1)𝑛²(𝑏𝑛+𝑟)³=𝜁(4)4𝑏𝑟²−3𝑏𝜁(2)𝑟⁴+2𝜁(2)𝐻^{(1)(𝑟/𝑏)−1}𝑟³+𝜁(2)𝐻^{(2)(𝑟/𝑏)−1}𝑏𝑟²+4𝜁(3)𝑟³+𝜁(3)𝐻^{(1)(𝑟/𝑏)−1}𝑏𝑟²−3𝑏2𝑟⁴𝐻^{(1)(𝑟/𝑏)−1}²+𝐻^{(2)(𝑟/𝑏)−1}−𝑟2³𝐻^{(1)(𝑟/𝑏)−1}−1𝐻^{(2)(𝑟/𝑏)−1}+𝐻^{(3)(𝑟/𝑏)−1}2𝑏𝑟²𝐻^{(2)(𝑟/𝑏)−1}²+2𝐻^{(1)(𝑟/𝑏)−1}𝐻^{(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≥1𝐻^𝑛(1)𝑛²(𝑛+1)³=𝜁(4)4+4𝜁(3)−3𝜁(2).(2.27)

Theorem 2.6. _𝑛≥1𝑡^𝑛^{𝑛+𝑝−1𝑛−1}(𝜓(𝑗+1+𝑎𝑛)−𝜓(𝑗+1))=𝑛−³𝑎𝑏𝑐𝑚^{𝑗𝑎𝑛+𝑗}𝑡^{𝑘𝑏𝑛+𝑘}₁₀^{𝑙𝑐𝑛+𝑙}^{𝑚𝑑𝑛+𝑚}(1−𝑥)^𝑗(1−𝑦)^𝑘(1−𝑧)^𝑙(1−𝑤)^𝑚−11−𝑡𝑥^𝑎𝑦^𝑏𝑧^𝑐𝑤^𝑑^𝑝+1×ln(1−𝑥)⋅𝑥^𝑎−1𝑦^𝑏−1𝑧^𝑐−1𝑤^𝑑𝑑𝑥𝑑𝑦𝑑𝑧𝑑𝑤.(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 𝑎=1,𝑑=𝑐=𝑏>0,𝑡=1,𝑗=0, and let 𝑙=𝑚=𝑘≥1 be a positive integer. Then _𝑛≥1𝐻^𝑛(1)𝑛³^{𝑘𝑏𝑛+𝑘}³=−𝑏²𝑘¹⁰((1−𝑦)(1−𝑧))^𝑘(1−𝑤)^𝑘−11−𝑥(𝑦𝑧𝑤)^𝑏ln(1−𝑥)⋅(𝑦𝑧)^𝑏−1=𝑤^𝑏𝑑𝑥𝑑𝑦𝑑𝑧𝑑𝑤(2.29)_𝑘_𝑟=1(−1)^𝑟+1𝑘𝑟³×1+54𝑋𝑅(𝑘)+54𝑟²−9𝑏𝑌𝑅(𝑘)𝜁(4)𝑟+𝐻^{(1)(𝑟/𝑏)−1}+6𝑏₂+5𝑏𝑋𝑅(𝑘)+2𝑏𝑟𝑌𝑅(𝑘)𝜁(3)𝑟²−3𝑏𝐻^{(1)(𝑟/𝑏)−1}𝑟−𝐻^{(2)(𝑟/𝑏)−1}−3𝑏²𝑟+𝑏𝐻^{(1)(𝑟/𝑏)−1}𝑋𝑅(𝑘)+𝑏²+3𝑏₂𝑌𝑅(𝑘)𝜁(2)𝑟²𝐻^{(1)(𝑟/𝑏)−1}²+𝐻^{(2)(𝑟/𝑏)−1}+3𝑟𝑏𝐻^{(1)(𝑟/𝑏)−1}+1𝐻^{(2)(𝑟/𝑏)−1}+𝐻^{(3)(𝑟/𝑏)−1}2𝐻^{(2)(𝑟/𝑏)−1}²+2𝐻^{(1)(𝑟/𝑏)−1}𝐻^{(3)(𝑟/𝑏)−1}+3𝐻^{(4)(𝑟/𝑏)−1}+3𝑏₂2𝑟𝐻^{(1)(𝑟/𝑏)−1}²+𝐻^{(2)(𝑟/𝑏)−1}+𝑏𝐻^{(1)(𝑟/𝑏)−1}+𝑏𝐻^{(2)(𝑟/𝑏)−1}+𝐻^{(3)(𝑟/𝑏)−1}𝑋𝑅(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≥1𝐻^𝑛(1)𝑛³^{𝑘𝑏𝑛+𝑘}³=^𝑘_𝑟=1𝐴^𝑟(𝑘!)³_𝑛≥1𝐻^𝑛(1)𝑛³(𝑏𝑛+𝑟)+𝐵^𝑟(𝑘!)³_𝑛≥1𝐻^𝑛(1)𝑛³(𝑏𝑛+𝑟)²+𝐶_𝑟(𝑘!)³_𝑛≥1𝐻^𝑛(1)𝑛³(𝑏𝑛+𝑟)³.(2.31) After much algebraic simplification, the following identity is obtained: _𝑛≥1𝐻^𝑛(1)𝑛³(𝑏𝑛+𝑟)³=_𝑛≥1𝐻^𝑛(1)𝑟³1𝑛³−3𝑏𝑟⁴𝑛²−𝑏³𝑟³(𝑏𝑛+𝑟)³−3𝑏³𝑟⁴(𝑏𝑛+𝑟)²+6𝑏²=𝜁(4)𝑟⁴𝑛(𝑏𝑛+𝑟)𝑟³−9𝑏𝜁(3)𝑟⁴−𝜁(3)𝐻^{(1)(𝑟/𝑏)−1}𝑟³+6𝑏²𝜁(2)𝑟⁵−3𝑏𝜁(2)𝐻^{(1)(𝑟/𝑏)−1}𝑟⁴−𝜁(2)𝐻^{(2)(𝑟/𝑏)−1}𝑟³+3𝑏²𝑟⁵𝐻^{(1)(𝑟/𝑏)−1}²+𝐻^{(2)(𝑟/𝑏)−1}+3𝑟⁴𝑏𝐻^{(1)(𝑟/𝑏)−1}+1𝐻^{(2)(𝑟/𝑏)−1}+𝐻^{(3)(𝑟/𝑏)−1}2𝑟³𝐻^{(2)(𝑟/𝑏)−1}²+2𝐻^{(1)(𝑟/𝑏)−1}𝐻^{(3)(𝑟/𝑏)−1}+3𝐻^{(4)(𝑟/𝑏)−1}_𝑛≥1,𝐻^𝑛(1)𝑛³(𝑏𝑛+𝑟)²=5𝜁(4)4𝑟²−5𝑏𝜁(3)𝑟³+3𝑏²𝜁(2)𝑟⁴−𝑏𝜁(2)𝐻^{(1)(𝑟/𝑏)−1}𝑟³+3𝑏²2𝑟⁴𝐻^{(1)(𝑟/𝑏)−1}²+𝐻^{(2)(𝑟/𝑏)−1}+𝑟𝑏³𝐻^{(1)(𝑟/𝑏)−1}_𝑛≥1𝐻^{(2)(𝑟/𝑏)−1}+𝐻^{(3)(𝑟/𝑏)−1},𝐻^𝑛(1)𝑛³(𝑏𝑛+𝑟)=5𝜁4(𝑟4)+𝑏²𝜁(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≥1𝐻^𝑛(1)𝑛³(𝑛+1)³=𝜁(4)−9𝜁(3)+16𝜁(2),_𝑛≥1𝐻^𝑛(1)𝑛³^84𝑛+8³=1272907300027−264101117042050832ln2+4403830607158688(ln2)²−16604790784315𝐺+11272192𝐺²[phone]−1390679293952𝜋35𝐺ln2−406993730017559−172213035728𝜋³−2039922475705989𝜁(2)210𝜁(3)[phone]ln2𝜁(3)−622426737𝜁(4),(2.33) where 𝐺 is Catalan's constant, defined by 𝐺=12¹⁰𝐾(𝑠)𝑑𝑠=^∞_𝑟=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≥1𝐻^𝑛(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≥1𝑡^𝑛^{𝑛+𝑝−1𝑛−1}(𝜓(𝑗+1+𝑎𝑛)−𝜓(𝑗+1))𝑛^𝑞^{𝑗𝑎𝑛+𝑗}^{𝑘𝑏𝑛+𝑘}^{𝑙𝑐𝑛+𝑙}^{𝑚𝑑𝑛+𝑚},for𝑞=1,2,3,…,(2.36) with its associated corollaries. This work will be investigated in a forthcoming paper.