By means of the modified Abel lemma on summation by parts, we investigate infinite series involving quadratic and cubic harmonic numbers. Several infinite series identities are established for π 2 and ζ(3) as consequences

Chu, Wenchang

Wang, Xiaoyuan

By means of the modified Abel lemma on summation by parts, we investigate infinite series involving quadratic and cubic harmonic numbers. Several infinite series identities are established for π 2 and ζ(3) as consequences.By means of the modified Abel lemma on summation by parts, we investigate infinite series involving quadratic and cubic harmonic numbers. Several infinite series identities are established for π 2 and ζ(3) as consequences.By means of the modified Abel lemma on summation by parts, we investigate infinite series involving quadratic and cubic harmonic numbers. Several infinite series identities are established for π 2 and ζ(3) as consequences

Revistes Catalanes amb Accés Obert

Infinite series identities involving quadratic and cubic harmonic numbers

Xiaoyuan Wang

Wenchang Chu

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Publicacions Matemàtiques

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Publ. Mat. 62 (2018), 285–300DOI: 10.5565/PUBLMAT6211813INFINITE SERIES IDENTITIES INVOLVINGQUADRATIC AND CUBIC HARMONIC NUMBERSXiaoyuan Wang and Wenchang ChuAbstract: By means of the modified Abel lemma on summation by parts, we inves-tigate infinite series involving quadratic and cubic harmonic numbers. Several infiniteseries identities are established for pi2 and ζ(3) as consequences.2010 Mathematics Subject Classification: Primary: 11M06; Secondary: 40A25.Key words: Riemann zeta function, harmonic numbers, Abel’s lemma on summa-tion by parts, partial fraction decomposition, telescoping method.1. Introduction and preliminariesFor two positive real numbers a and b, define the following two se-quences byhn(a, b) :=n∑k=11ak − a+ b and h¯n(a, b) :=n∑k=1(−1)kak − a+ b .They contain the well-known classical harmonic-like numbers as partic-ular cases:Hn := hn(1, 1) =n∑k=11kand On := hn(2, 1) =n∑k=112k − 1 ;H¯n := h¯n(1, 1) =n∑k=1(−1)kkand O¯n := h¯n(2, 1) =n∑k=1(−1)k2k − 1 .During the last two decades there has been growing interest in findingclosed formulae of finite sums involving harmonic numbers. There areseveral approaches to evaluate these sums. By applying derivative op-erator to the known binomial identities and terminating hypergeometricformulae, Chu and De Donno [9, 11, 12] established numerous summa-tion formulae. Boyadzhiev [2] evaluated several finite sums by the Eulertransformation. Cheon and El-Mikkawy [4] studied these sums by ex-pressing harmonic numbers in terms of the Stirling numbers of the firstkind (cf. Comtet [13, §5.6]). Flajolet and Salvy [15] developed a very286 X. Wang, W. Chupowerful method, based on the residue technique, for the summation ofseries involving harmonic numbers. Wang [20] found further formulaeconcerning harmonic numbers by Riordan arrays. The most powerfulmethod to investigate finite harmonic sums, “partial fraction decompo-sitions”, was introduced by Chu [7, 8] and subsequently developed byGe–Liu–Luo [16] and Prodinger [18].Instead, evaluation of infinite series containing harmonic numbers canbe dated back to Euler, who discovered several surprising identities re-lated to the Riemann zeta function. Three beautiful examples can bereproduced as follows (cf. [1, 6, 14]):2ζ(3) =∑k≥1Hkk2,17pi4360=∑k≥1H2kk2,pi432=∑k≥1O2kk2.In 1991, De Doelder [14] found numerous formulae by evaluating thedigamma function. Borwein brothers [1] reviewed some identities bycombining generating functions with Parseval’s identity about theFourier series in 1995. In 1997, Chu [6] devised a hypergeometric methodby examining the coefficient of a monomial across known hypergeometricseries equalities, that leads to a systematic investigation of the infiniteseries expressible in the Riemann zeta function (cf. [3, 5, 17, 21]).Sofo and Hassani [19] derived further formulae by logarithmic integrals.Recently, Abel’s lemma on summation by parts has been modified byChu [10] to derive several infinite series identities involving the classicalharmonic numbers and their variants. The objective of the present workis to explore its further applications to infinite series involving quadraticand cubic harmonic numbers. Several remarkable formulae will be es-tablished. Here we highlight two infinite series expressions for pi2 (seeEquations (9) and (11))pi224− 316=∑k≥1H2k(k + 1)(k + 2)(k + 3),pi2256− 196=∑k≥1O2k(2k + 1)(2k + 3)(2k + 5);and two other ones for ζ(3) (see Equations (32) and (34))6ζ(3) + 1912=∑k≥2(2k + 1)H3kk(k + 1)(k + 2)(k − 1) ,7ζ(3)− 4512=∑k≥1kO3k(2k − 3)(2k − 1)(2k + 1)(2k + 3) .Infinite Series Involving Harmonic Numbers 287For subsequent applications, we reproduce Abel’s lemma on summa-tion by parts as follows. With an arbitrary complex sequence {τk}, definethe backward and forward difference operators ∇ and ·∆, respectively,by(1) ∇τk = τk − τk−1 and ·∆τk = τk − τk+1,where ·∆ is adopted for convenience in the present paper, which differsfrom the usual operator ∆ only in the minus sign. Then Abel’s lemmaon summation by parts (see [10] for a proof) may be reformulated as(2) Ω :=∑k≥1Bk∇Ak = [AB]+ −A0B1 +∑k≥1Ak ·∆Bkprovided that one of both series is convergent and the following limitexists:(3) [AB]+ = limn→∞AnBn+1.Under certain convergence conditions, this lemma can successively beapplied. In the next section, we shall prove, by employing (2) twice,two main transformation theorems concerning the quadratic harmonicnumbers h2k(a, b) and h¯2k(a, b), that will yield several interesting identitiesfor pi2, ln 2, and the Catalan constant G. Then in Section 3, we shallestablish, by applying (2) thrice, yet another general transformationtheorem on the cubic harmonic numbers h3k(a, b), from which we shallderive further two infinite series expressions for pi2 and two for ζ(3). Inorder to ensure the accuracy, all the formulae displayed in this paperhave been checked numerically by appropriately devised Mathematicacommands.2. Infinite series involving quadratic harmonic numbersFor the Ω-sum in (2), if there exist another difference pair {A′k, B′k}and a constant β such thatΩ′ :=∑k≥1Ak ·∆Bk = β +∑k≥1B′k∇A′k,then we can reformulate the Ω′-sum further by (2) as follows:Ω′ =∑k≥1Ak ·∆Bk = β + [A′B′]+ −A′0B′1 +∑k≥1A′k ·∆B′k.Substituting this Ω′ into (2), we find, after having applied twice the mod-ified Abel lemma on summation by parts, the following transformationformula.288 X. Wang, W. ChuLemma 1. For the two difference pairs {Ak, Bk} and {A′k, B′k} satisfy-ing(4)∑k≥1Ak ·∆Bk = β +∑k≥1B′k∇A′k,the following infinite series formula holds(5) Ω :=∑k≥1Bk∇Ak = β+[AB]++[A′B′]+−A0B1−A′0B′1+∑k≥1A′k ·∆B′kprovided that the both limits [AB]+ and [A′B′]+ exist and one of the twoseries displayed in (5) is convergent.This lemma is more general than what we actually need. In fact, allthe difference pairs {Ak, Bk} and {A′k, B′k} in this section satisfy, insteadof (4), the following simpler condition Ak ·∆Bk = B′k∇A′k. Now we arein position to prove two main theorems, that will lead, as particularcases, to several infinite series identities containing quadratic harmonicnumbers.Theorem 2 (Two infinite series identities).∑k≥1a2h2k(a, b)(ak + b)(ak + a+ b)=1ab+∑k≥01(ak + b)2,(6)∑k≥14a3h2k(a, b)(ak+b)(ak+a+b)(ak+2a+b)=∑k≥01(ak + b)2− a+ 2b2ab(a+b).(7)Proof: Let {Bk, B′k} be the two sequencesBk := h2k(a, b) and B′k := hk(a, b) + hk+1(a, b)with their differences being determined by·∆Bk = −hk(a, b) + hk+1(a, b)ak + b,·∆B′k = −1ak + b− 1ak + a+ b.In order to be able to apply Lemma 1, we will consider couples of se-quences {Ak, A′k} satisfying the property Ak ·∆Bk = B′k∇A′k which isstronger than the condition (4).First, it is almost trivial to verify that the two sequences {Ak, A′k}given below satisfy the condition of Lemma 1Ak :=−aak + a+ band A′k :=−1ak + a+ b;Infinite Series Involving Harmonic Numbers 289in view of their differences∇Ak = a2(ak + b)(ak + a+ b),∇A′k =a(ak + b)(ak + a+ b);as well as the limiting relations[AB]+ = 0, A0B1 =−ab2(a+ b);[A′B′]+ = 0, A′0B′1 =−2a− 3bb(a+ b)2.Writing explicitly the equality corresponding to (5) and then simplifyingthe result, we get the first formula (6).The second formula (7) can be shown analogously by choosing anothercouple of sequences {Ak, A′k} byAk :=−2a(ak + a+ b)(ak + 2a+ b),A′k :=−1(ak + a+ b)(ak + 2a+ b);and then computing their differences∇Ak = 4a2(ak + b)(ak + a+ b)(ak + 2a+ b),∇A′k =2a(ak + b)(ak + a+ b)(ak + 2a+ b);as well the limiting relations[AB]+ = 0, A0B1 =−2ab2(a+ b)(2a+ b);[A′B′]+ = 0, A′0B′1 =−2a− 3bb(a+ b)2(2a+ b).The following interesting infinite series identities containing quadraticharmonic numbers can be easily deduced from Theorem 2.290 X. Wang, W. ChuCorollary 3 (a = b = 1 in Theorem 2).∑k≥1H2k(k + 1)(k + 2)= 1 +pi26,(8)∑k≥1H2k(k + 1)(k + 2)(k + 3)=pi224− 316.(9)Corollary 4 (a = 2 and b = 1 in Theorem 2).∑k≥1O2k(2k + 1)(2k + 3)=18+pi232,(10)∑k≥1O2k(2k + 1)(2k + 3)(2k + 5)=pi2256− 196.(11)Theorem 5 (Five infinite series identities).∑k≥12a2h¯2k(a, b)(ak+a+b)(ak+2a+b)=∑k≥01(ak + b)2− 3a+ 2b2ab(a+ b),(12)∑k≥12a2h¯2k(a, b)(ak−2a+b)(ak−3a+b) =5a− 2b2a(a−b)(2a−b) +∑k≥01(ak+b)2,(13)∑k≥1a2(3ak+2a+3b)h¯2k(a, b)(ak+b)(ak+a+b)(ak+2a+b)=2∑k≥0(−1)kak+b2− 12(a+b)b,(14)∑k≥1(−1)k4a2h¯2k(a, b)(ak + b)(ak + 2a+ b)=12b(a+ b)−∑k≥0(−1)k(ak + b)2,(15)∑k≥1(−1)ka(2ak+a+2b)h¯2k(a, b)(ak + b)(ak + a+ b)=1ab−∑k≥0(−1)k(ak+b)2− 2a∑k≥0(−1)kak+b.(16)Proof: Following the proof of Theorem 2, we first fix the sequences{Bk, B′k} byBk := h¯2k(a, b) and B′k := h¯k(a, b) + h¯k+1(a, b)Infinite Series Involving Harmonic Numbers 291and determine their differences by·∆Bk = (−1)kak + b{h¯k(a, b) + h¯k+1(a, b)},·∆B′k =(−1)kak + b− (−1)kak + a+ b.To prove Theorem 5, it suffices to figure out the difference couples{Ak, A′k} according to Lemma 1. They are sketched below so that thereader will not have difficulty to follow the approach, even though thefinal confirmations for these identities are made by combining partialfraction decompositions and the telescoping method.Formula (12): Sequence coupleAk :=2ak + 2a+ band A′k =(−1)k(ak + a+ b)(ak + 2a+ b);with their differences∇Ak = −2a(ak + a+ b)(ak + 2a+ b),∇A′k =2(−1)k(ak + b)(ak + 2a+ b);as well as limiting relations[AB]+ = 0, A0B1 =2b2(2a+ b);[A′B′]+ = 0, A′0B′1 =−1b(a+ b)2.Formula (13): Sequence coupleAk :=2ak − 2a+ b and A′k :=(−1)k(ak + b)(ak − a+ b) ;with their differences∇Ak = −2a(ak − 2a+ b)(ak − 3a+ b) ,∇A′k =2(−1)k(ak + b)(ak − 2a+ b) ;292 X. Wang, W. Chuas well as limiting relations[AB]+ = 0, A0B1 =2b2(b− 2a) ;[A′B′]+ = 0, A′0B′1 =2a+ bb2(a2 − b2) .Formula (14): Sequence coupleAk :=(ak + b)(2ak + 3a+ 2b)(ak + a+ b)(ak + 2a+ b)and A′k :=(−1)kak + 2a+ b;with their differences∇Ak = a2(3ak + 2a+ 3b)(ak + b)(ak + a+ b)(ak + 2a+ b),∇A′k =(−1)k(2ak + 3a+ 2b)(ak + a+ b)(ak + 2a+ b);as well as limiting relations[AB]+ = 2∑k≥0(−1)kak + b2, A0B1 =3a+ 2bb(a+ b)(2a+ b);[A′B′]+ = 0, A′0B′1 =−1b(a+ b).Formula (15): Sequence coupleAk :=(−1)k−12a(ak + a+ b)(ak + 2a+ b),A′k :=1(ak + a+ b)(ak + 2a+ b);with their differences∇Ak = (−1)k−14a(ak + b)(ak + 2a+ b),∇A′k =−2a(ak + b)(ak + a+ b)(ak + 2a+ b);Infinite Series Involving Harmonic Numbers 293as well as limiting relations[AB]+ = 0, A0B1 =−2ab2(a+ b)(2a+ b);[A′B′]+ = 0, A′0B′1 =−1b(a+ b)2.Formula (16): Sequence coupleAk :=(−1)k−1aak + a+ band A′k :=1ak + a+ b;with their differences∇Ak = (−1)k−1a(2ak + a+ 2b)(ak + b)(ak + a+ b),∇A′k =−a(ak + b)(ak + a+ b);as well as limiting relations[AB]+ = 0, A0B1 =−ab2(a+ b);[A′B′]+ = 0, A′0B′1 =−2a− bb(a+ b)2.As examples of Theorem 5, we record ten remarkable infinite seriesidentities.Corollary 6 (a = b = 1 in Theorem 5).∑k≥1H¯2k(k + 2)(k + 3)=pi212− 58,(17)∑k≥3H¯2k(k − 1)(k − 2) =pi212− 14,(18)∑k≥1(3k + 5)H¯2k(k + 1)(k + 2)(k + 3)= 2(ln 2)2 − 14,(19)∑k≥1(−1)kH¯2k(k + 1)(k + 3)=116− pi248,(20)∑k≥1(−1)k(2k + 3)H¯2k(k + 1)(k + 2)= 1− pi212− 2 ln 2.(21)294 X. Wang, W. ChuCorollary 7 (a = 2 and b = 1 in Theorem 5).∑k≥1O¯2k(2k + 3)(2k + 5)=pi264− 112,(22)∑k≥1O¯2k(2k − 5)(2k − 3) =pi264+112,(23)∑k≥1(6k + 7)O¯2k(2k + 1)(2k + 3)(2k + 5)=pi232− 124,(24)∑k≥1(−1)kO¯2k(2k + 1)(2k + 5)=196− G16,(25)∑k≥1(−1)k(k + 1)O¯2k(2k + 1)(2k + 3)=116− G8− pi32.(26)In the last two series, we have used G to denote the Catalan constant:G =∑k≥0(−1)k/(2k + 1)2 = 0.9159 . . .3. Infinite series involving cubic harmonic numbersUnder the same condition of Lemma 1, we may apply (2) further.Let {A′′k , B′′k} be the third difference pair subject to the conditionΩ′′ :=∑k≥1A′k ·∆B′k = ρ+∑k≥1B′′k∇A′′k ,where ρ is a constant. Then we can manipulate the Ω′′-sum again by (2)in the following manner:Ω′′ =∑k≥1A′k ·∆B′k = ρ+ [A′′B′′]+ −A′′0B′′1 +∑k≥1A′′k ·∆B′′k .Substituting this Ω′′ into (5), we find, after having applied thrice themodified Abel lemma on summation by parts, another transformationformula.Lemma 8. For the three difference pairs {Ak, Bk}, {A′k, B′k}, and{A′′k , B′′k} satisfying (4) and(27)∑k≥1A′k ·∆B′k = ρ+∑k≥1B′′k∇A′′k ,Infinite Series Involving Harmonic Numbers 295the following infinite series formula holds(28) Ω :=∑k≥1Bk∇Ak = [AB]+ + [A′B′]+ + [A′′B′′]++ β + ρ−A0B1 −A′0B′1 −A′′0B′′1 +∑k≥1A′′k ·∆B′′kprovided that the three limits [AB]+, [A′B′]+, and [A′B′]+ exist, andone of the two series displayed in (28) is convergent.By employing this lemma, we can prove the following general theorem.Theorem 9 (Two infinite series identities).∑k≥12a3h3k(a, b)(2ak + a+ 2b)(2ak − 3a+ 2b) =4a(2b− a) +∑k≥01(ak + b)2,(29)∑k≥12a3(2ak − a+ 2b)h3k(a, b)(ak+b)(ak+a+b)(ak−a+b)(ak−2a+b) =1ab(b− a)(30)+∑k≥01(ak + b)3.Proof: Similar to the preceding proofs, we shall fix the sequences {Bk,B′k, B′′k} byBk := h3k(a, b),B′k := h2k(a, b) + hk(a, b)hk+1(a, b) + h2k+1(a, b),B′′k := hk+1(a, b);with their differences being given by·∆Bk =h2k(a, b) + hk(a, b)hk+1(a, b) + h2k+1(a, b)−(ak + b) ,·∆B′k =a(2ak + a+ 2b)(ak + b)2(ak + a+ b)2− 3(2ak + a+ 2b)hk+1(a, b)(ak + b)(ak + a+ b),·∆B′′k =−1ak + a+ b.296 X. Wang, W. ChuThen the first formula (29) can be confirmed by specifying in Lemma 8the triple sequences {Ak, A′k, A′′k} byAk :=−(ak + b)a(2ak + a+ 2b)(2ak − a+ 2b) ,A′k :=−12a2(2ak + a+ 2b),A′′k :=−32a3(ak + a+ b);and evaluating their differences∇Ak = 1(2ak + a+ 2b)(2ak − 3a+ 2b) ,∇A′k =1a(2ak + a+ 2b)(2ak − a+ 2b) ,∇A′′k =32a3(ak + b)(ak + a+ b);as well as limiting relations[AB]+ = 0, A0B1 =1ab2(a2 − 4b2) ;[A′B′]+ = 0, A′0B′1 =−(3a2 + 9ab+ 7b2)2a2b2(a+ b)2(a+ 2b);[A′′B′′]+ = 0, A′′0B′′1 =−3(a+ 2b)2a3b(a+ b)2.However, one should be careful about the constants β = 0 and ρ, whichcan be evaluated by combining partial fractions with the telescopingmethod:ρ =−12a∑k≥11(ak + b)2(ak + a+ b)2=∑k≥1{1a4(ak + b)− 1a4(ak + a+ b)}−∑k≥1{12a3(ak + b)2+12a3(ak + a+ b)2}=a+ 2b2a4(a+ b)2− 1a3∑k≥11(ak + a+ b)2.Infinite Series Involving Harmonic Numbers 297As for the second formula (30), it can be done by letting alternativelyin Lemma 8 the triple sequences {Ak, A′k, A′′k} byAk :=−1a(ak − a+ b)(ak + a+ b) ,A′k :=−12a2(ak + b)(ak + a+ b),A′′k :=−32a3(ak + a+ b)2;and then determining their differences∇Ak = 2ak − a+ 2b(ak + b)(ak + a+ b)(ak − a+ b)(ak − 2a+ b) ,∇A′k =1a(ak + b)(ak + a+ b)(ak − a+ b) ,∇A′′k =3(2ak + a+ 2b)2a2(ak + b)2(ak + a+ b)2;as well as the limiting relations[AB]+ = 0, A0B1 =1ab3(a2 − b2) ;[A′B′]+ = 0, A′0B′1 =−(3a2 + 9ab+ 7b2)2a2b3(a+ b)3;[A′′B′′]+ = 0, A′′0B′′1 =−3(a+ 2b)2a3b(a+ b)3.Also in this case, we have β = 0 and another non vanishing constantρ =−12a∑k≥12ak + a+ 2b(ak + b)3(ak + a+ b)3=∑k≥1{12a4(ak + b)2− 12a4(ak + a+ b)2}−∑k≥1{12a3(ak + b)3+12a3(ak + a+ b)3}=b2a4(a+ b)3− 1a3∑k≥11(ak + a+ b)3.298 X. Wang, W. ChuAs consequences of Theorem 9, we display below four infinite seriesidentities involving cubic harmonic numbers.Corollary 10 (a = b = 1 in Theorem 9).∑k≥1H3k(2k − 1)(2k + 3) =24 + pi212,(31)∑k≥2(2k + 1)H3kk(k + 1)(k + 2)(k − 1) =6ζ(3) + 1912.(32)Corollary 11 (a = 2 and b = 1 in Theorem 9).∑k≥2O3k(k + 1)(k − 1) =26 + pi28,(33)∑k≥1kO3k(2k − 3)(2k − 1)(2k + 1)(2k + 3) =7ζ(3)− 4512.(34)Acknowledgement. The first author has been supported partially byLiaoning Educational Commission of China (No. JDL2016018) duringthis research.References[1] D. Borwein and J. M. Borwein, On an intriguing integral andsome series related to ζ(4), Proc. Amer. Math. Soc. 123(4) (1995),1191–1198. DOI: 10.2307/2160718.[2] K. N. Boyadzhiev, Harmonic number identities via Euler’s trans-form, J. Integer Seq. 12(6) (2009), Article 09.6.1, 8 pp.[3] X. Chen and W. Chu, The Gauss 2F1(1)-summation theoremand harmonic number identities, Integral Transforms Spec. Funct.20(12) (2009), 925–935. 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Ge, T.-T. Liu, and Q.-M. Luo, A new identity via par-tial fraction decomposition, Filomat 30(7) (2016), 1957–1967. DOI:10.2298/FIL1607957G.[17] H. Liu and W. Wang, Harmonic number identities via hypergeo-metric series and Bell polynomials, Integral Transforms Spec. Funct.23(1) (2012), 49–68. DOI: 10.1080/10652469.2011.553718.[18] H. Prodinger, Identities involving harmonic numbers that are ofinterest for physicists, Util. Math. 83 (2010), 291–299.[19] A. Sofo and M. Hassani, Quadratic harmonic number sums,Appl. Math. E-Notes 12 (2012), 110–117.[20] W. Wang, Riordan arrays and harmonic number identities, Com-put. Math. Appl. 60(5) (2010), 1494–1509. DOI: 10.1016/j.camwa.2010.06.031.300 X. Wang, W. Chu[21] D.-Y. Zheng, Further summation formulae related to generalizedharmonic numbers, J. Math. Anal. Appl. 335(1) (2007), 692–706.DOI: 10.1016/j.jmaa.2007.02.002.Xiaoyuan Wang:School of ScienceDalian Jiaotong UniversityDalian 116028P. R. ChinaE-mail address: xiaoyuanwang27@yahoo.comWenchang Chu:School of Mathematics and StatisticsZhoukou Normal UniversityZhoukou (Henan)P. R. ChinaE-mail address: chu.wenchang@unisalento.itPrimera versio´ rebuda el 21 de novembre de 2016,darrera versio´ rebuda el 20 d’abril de 2017.

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Infinite series identities involving quadratic and cubic harmonic numbers

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