Summation Formulas Involving Binomial Coefficients, Harmonic Numbers, and Generalized Harmonic Numbers

Abstract

A variety of identities involving harmonic numbers and generalized harmonic numbers have been investigated since the distant past and involved in a wide range of diverse fields such as analysis of algorithms in computer science, various branches of number theory, elementary particle physics, and theoretical physics. Here we show how one can obtain further interesting and (almost) serendipitous identities about certain finite or infinite series involving binomial coefficients, harmonic numbers, and generalized harmonic numbers by simply applying the usual differential operator to well-known Gauss’s summation formula for ₂F₁(1).

1. Introduction and Preliminaries

The generalized harmonic numbers 𝐻_𝑛^(𝑠) of order 𝑠 which are defined by (cf. [1]; see also [2, 3], [4, page 156], and [5, Section 3.5]) 𝐻_𝑛^(𝑠)∶=^𝑛_𝑗=11𝑗^𝑠(𝑛∈ℕ;𝑠∈ℂ),𝐻_𝑛∶=𝐻_𝑛⁽¹⁾=^𝑛_𝑗=11𝑗(𝑛∈ℕ) (1)(2) are the harmonic numbers. Here ℕ and ℂ denote the set of positive integers and the set of complex numbers, respectively, and we assume that 𝐻₀∶=0,𝐻₀^(𝑠)({0,𝐻∶=0𝑠∈ℂ⧵})₀₍₀₎∶=1. (3) The generalized harmonic functions 𝐻_𝑛^(𝑠)(𝑧) are defined by (see [2, 6]; see also [7, 8]) 𝐻_𝑛^(𝑠)(𝑧)∶=^𝑛_𝑗=11𝑗+𝑧^𝑠(𝑛∈ℕ;𝑠∈ℂ⧵ℤ⁻;ℤ⁻∶={−1,−2,−3,…}), (4) so that, obviously, 𝐻_𝑛^(𝑠)(0)=𝐻_𝑛^(𝑠). (5)

Equation (1) can be written in the following form: 𝐻_𝑛^(𝑠)=𝜁(𝑠)−𝜁(𝑠,𝑛+1)(ℜ(𝑠)>1;𝑛∈ℕ), (6) by recalling the well-known (easily derivable) relationship between the Riemann zeta function 𝜁(𝑠) and the Hurwitz (or generalized) zeta function 𝜁(𝑠,𝑎) (see [4, equation 2.3(9)]): 𝜁(𝑠)=𝜁(𝑠,𝑛+1)+^𝑛_𝑘=1𝑘^−𝑠𝑛∈ℕ₀∶=ℕ∪{0}. (7)

The polygamma functions 𝜓^(𝑛)(𝑠)(𝑛∈ℕ) are defined by 𝜓^(𝑛)(𝑠)𝑑∶=^𝑛+1𝑑𝑧^𝑛+1(𝑠)=𝑑logΓ^𝑛𝑑𝑠^𝑛𝜓(𝑠)𝑛∈ℕ₀;𝑠∈ℂ⧵ℤ⁻₀∶=ℤ⁻∪{0}, (8) where Γ(𝑠) is the familiar gamma function and the psi-function 𝜓 is defined by 𝜓(𝑠)𝑑∶=(𝑠)𝑑𝑠logΓ,𝜓⁽⁰⁾(𝑠)=𝜓(𝑠)𝑠∈ℂ⧵ℤ⁻₀. (9)

A well-known (and potentially useful) relationship between the polygamma functions 𝜓^(𝑛)(𝑠) and the generalized zeta function 𝜁(𝑠,𝑎) is given by 𝜓^(𝑛)(𝑠)=(−1)^𝑛+1𝑛!^∞_𝑘=01(𝑘+𝑠)^𝑛+1=(−1)^𝑛+1𝑛!𝜁(𝑛+1,𝑠)𝑛∈ℕ;𝑠∈ℂ⧵ℤ⁻₀. (10) It is also easy to have the following expression (cf. [4, equation 1.2(54)]): 𝜓^(𝑚)(𝑠+𝑛)−𝜓^(𝑚)(𝑠)=(−1)^𝑚𝑚!𝐻_𝑛^(𝑚+1)(𝑠−1)𝑚,𝑛∈ℕ₀, (11) which immediately gives 𝐻_𝑛^(𝑠) another expression for 𝐻_𝑛^(𝑠) as follows (cf. [9, equation (20)]): 𝐻_𝑛^(𝑚)=(−1)^𝑚−1()!𝜓𝑚−1^(𝑚−1)(𝑛+1)−𝜓^(𝑚−1)(1)𝑚∈ℕ;𝑛∈ℕ₀. (12)

The following identity was discovered by Euler in 1775 and has a long history (see, e.g., [10, page 252 et seq.]): ^∞_𝑛=1𝐻_𝑛(𝑛+1)²=12^∞_𝑛=1𝐻_𝑛𝑛²=𝜁(3). (13) Identity (13) is a special case of the following more general sum due to Euler: 2^∞_𝑘=1𝐻_𝑘𝑘^𝑛=()𝜁()−𝑛+2𝑛+1_{𝑛−2𝑘=1}𝜁()𝜁()({1𝑛−𝑘𝑘+1𝑛∈ℕ⧵}) (14) or, equivalently, 2^∞_𝑘=1𝐻_𝑘(𝑘+1)^𝑛()−=𝑛𝜁𝑛+1_{𝑛−2𝑘=1}𝜁()𝜁()({1,𝑛−𝑘𝑘+1𝑛∈ℕ⧵}) (15) where (and in what follows) an empty sum is understood to be nil.

Many different techniques have been used, in the vast mathematical literature, in order to evaluate harmonic sums of the types (13) and (15). For example, D. Borwein and J. M. Borwein [11] established the following interesting sums by applying Parseval's identity to a Fourier series and contour integrals to a generating function: ^∞_𝑛=1𝐻_𝑛𝑛+1²=114𝜁(4),_∞𝑛=1𝐻_𝑛𝑛²=174𝜁(4),_∞𝑛=1𝐻_𝑛𝑛³=54𝜁(4), (16)(17)(18) where, in light of Euler sum (18), nonlinear harmonic sums (16) and (17) are substantially the same, since it is easily verified that ^∞_𝑘=1𝐻_𝑘𝑘²=^∞_𝑘=1𝐻_𝑘𝑘+1²+32𝜁(4). (19)

Euler started this line of investigation in the course of his correspondence with Goldbach beginning in 1742 and he was the first to consider the linear harmonic sums: 𝒮_𝑝,𝑞∶=^∞_𝑛=1𝐻_𝑛^(𝑝)𝑛^𝑞. (20)

Euler, whose investigations were completed by Nielsen in 1906 (see Nielsen [12]), showed that the linear harmonic sums in (20) can be evaluated in terms of zeta values in the following special cases: 𝑝=1, 𝑝=𝑞, 𝑝+𝑞 odd, and 𝑝+𝑞 even, but with the pair (𝑝,𝑞) being restricted to a finite set of the so-called exceptional configurations {(2,4),(4,2)} (see Flajolet and Salvy [13]). Of these special cases, in the ones with 𝑝≠𝑞, if 𝒮_𝑝,𝑞 is known, then 𝒮_𝑞,𝑝 can be found by means of the symmetry relation: 𝒮_𝑝,𝑞+𝒮_𝑞,𝑝=𝜁𝑝𝜁𝑞+𝜁𝑝+𝑞, (21) and vice versa (see also [1, page 140, Proposition 6]. Some typical examples are ^∞_𝑛=1𝐻_𝑛⁽²⁾𝑛²=74𝜁(4),_∞𝑛=1𝐻_𝑛⁽²⁾𝑛⁵=5𝜁(2)𝜁(5)+2𝜁(3)𝜁(4)−10𝜁(7). (22)(23)

Rather extensive numerical search for linear relations between linear Euler sums and polynomials in zeta values (see Bailey et al. [14]; see also Flajolet and Salvy [13]) strongly suggests that Euler found all the possible evaluations of linear harmonic sums.

The nonlinear harmonic sums involve products of at least two harmonic numbers. Let 𝑃=(𝑝₁,…,𝑝_𝑘) be a partition of an integer 𝑝 into 𝑘 summands, so that 𝑝=𝑝₁+⋯+𝑝_𝑘 and 𝑝₁≦𝑝₂≦⋯≦𝑝_𝑘. The Euler sum of index 𝑃, 𝑞 is defined by 𝒮_𝑃,𝑞∶=^∞_𝑛=1𝐻^(𝑝1)_𝑛𝐻^(𝑝2)_𝑛⋯𝐻^(𝑝𝑘)_𝑛𝑛^𝑞, (24) where the quantity 𝑞+𝑝₁+⋯+𝑝_𝑘 is called the weight and the quantity 𝑘 is the degree. A few basic nonlinear sums were recently evaluated by de Doelder [15] by invoking their relations with the Eulerian beta integrals or with polylogarithm functions. A detailed numerical search was conducted by Bailey et al. [14] who showed the existence of many surprising evaluations like ^∞_𝑛=1𝐻_𝑛³𝑛⁴=231𝜁(7)−16514𝜁(3)𝜁(4)+2𝜁(2)𝜁(5). (25) Flajolet and Salvy [13] clearly and extensively analyzed most of the hitherto known evaluations for Euler sums and multiple zeta functions (see also Hoffman [16] and Zagier [17]). For a remarkably clear and insightful exposition of several important results and conjectures concerning multiple polylogarithms and the multiple zeta functions (including especially a broad survey of recent works on multiple zeta series and Euler sums of arbitrary degree), the interested reader should refer also to a survey-cum-expository paper by Bowman and Bradley [18], which contains a fairly comprehensive bibliography of as many as 83 further references on the subject.

Shen [19] investigated the connections between the Stirling numbers 𝑠(𝑛,𝑘) of the first kind and the Riemann zeta function 𝜁(𝑛) by means of the Gauss summation formula (28) for the hypergeometric series. In the course of his analysis, Shen [19] proved some known identities like (13), (18), and (22). In fact, by employing the univariate series expansion of classical hypergeometric formulas, Shen [19] and Choi and Srivastava [20, 21] investigated the evaluation of infinite series related to generalized harmonic numbers. On the other hand, more summation formulas have been systematically derived by Chu [22] and Chu and de Donno [23] who developed fully this approach to the multivariate case. We chose to recall two identities: ^∞_𝑛=1𝐻³_𝑛−1−3𝐻_𝑛−1𝐻⁽²⁾_𝑛−1+2𝐻⁽³⁾_𝑛−1𝑛²=6𝜁_∞𝑛=1(5),𝐻_𝑛⁽²⁾𝑛³+3𝐻_𝑛𝑛⁴=92𝜁(5). (26)

Many formulas of finite series involving binomial coefficients, the Stirling numbers of the first and second kinds, harmonic numbers, and generalized harmonic numbers have also been investigated in diverse ways (see, e.g., [2, 23–32]).

Here we show how one can obtain further interesting and (almost) serendipitous identities about certain finite or infinite series involving binomial coefficients, harmonic numbers, and generalized harmonic numbers by simply applying the usual differential operator to well-known Gauss’s summation formula for ₂𝐹₁(1). For example, see the identities in Corollary 5: ^∞_𝑛=1(𝐻_𝑛)³−𝐻_𝑛𝐻_𝑛⁽²⁾()((2)(3;𝑛+1)(𝑛+2=41+𝜁+𝜁))_∞𝑛=1(𝐻_𝑛)⁴−3(𝐻_𝑛)²𝐻_𝑛⁽²⁾+2𝐻_𝑛𝐻_𝑛⁽³⁾()((2)(3)(4.𝑛+1)(𝑛+2=121+𝜁+𝜁+𝜁)) (27) Relevant connections between some of the identities presented here with those in earlier works are also pointed out.

2. Infinite Series Involving Binomial Coefficients, Harmonic Numbers, and Generalized Harmonic Numbers

We begin by recalling well-known Gauss’s summation formula for ₂𝐹₁(1): ₂𝐹₁(𝑎,𝑏;𝑐;1)∶=^∞_𝑛=0(𝑎)_𝑛(𝑏)_𝑛𝑛!(𝑐)_𝑛=Γ(𝑐)Γ(𝑐−𝑎−𝑏)Γ()Γ()ℜ()𝑐−𝑎𝑐−𝑏𝑐−𝑎−𝑏>0;𝑐∉ℤ⁻₀, (28) where (𝛼)_𝑛 denotes the Pochhammer symbol defined (for 𝛼∈ℂ) by (𝛼)_𝑛(),𝛼()⋯(),().∶=1,𝑛=0𝛼+1𝛼+𝑛−1𝑛∈ℕ (29)

For convenient reference, without proof, we collect a set of easily derivable formulas necessary to provide further interesting identities about certain finite or infinite series involving binomial coefficients, harmonic numbers, and generalized harmonic numbers asserted as in the following lemma.

Lemma 1. Each of the following identities holds true: 𝑑𝑑𝛼(𝛼)_𝑛=(𝛼)_𝑛𝐻_𝑛⁽¹⁾(𝛼−1)𝑛∈ℕ₀;𝛼∈ℂ⧵ℤ⁻₀;𝑑1𝑑𝛼(𝛼)_𝑛𝐻=−^𝑛(1)(𝛼−1)(𝛼)_𝑛𝑛∈ℕ₀;𝛼∈ℂ⧵ℤ⁻₀;𝑑²𝑑𝛼²(𝛼)_𝑛=(𝛼)_𝑛𝐻_𝑛⁽¹⁾(𝛼−1)²−𝐻_𝑛⁽²⁾(𝛼−1)𝑛∈ℕ₀;𝛼∈ℂ⧵ℤ⁻₀;𝑑²𝑑𝛼²1(𝛼)_𝑛=1(𝛼)_𝑛𝐻_𝑛⁽¹⁾(𝛼−1)²+𝐻_𝑛⁽²⁾(𝛼−1)𝑛∈ℕ₀;𝛼∈ℂ⧵ℤ⁻₀;𝑑𝑑𝑧𝐻_𝑛^(𝑠)(𝑧)=(−1)(𝑠)𝐻_𝑛^(𝑠+)(𝑧)𝑛∈ℕ;∈ℕ₀;𝑠∈ℂ⧵ℤ⁻;𝑑Γ(𝑐)Γ()𝑑𝑎𝑐−𝑎−𝑏Γ()Γ()=Γ(𝑐)Γ()𝑐−𝑎𝑐−𝑏𝑐−𝑎−𝑏Γ()Γ()𝜓()(;𝑑𝑐−𝑎𝑐−𝑏𝑐−𝑎−𝜓𝑐−𝑎−𝑏)Γ(𝑐)Γ()𝑑𝑐𝑐−𝑎−𝑏Γ()Γ()=Γ(𝑐)Γ()𝑐−𝑎𝑐−𝑏𝑐−𝑎−𝑏Γ()Γ()⋅𝜓(𝑐)()()(.𝑐−𝑎𝑐−𝑏+𝜓𝑐−𝑎−𝑏−𝜓𝑐−𝑎−𝜓𝑐−𝑏) (30)

Now we are ready to present certain general identities of infinite series involving binomial coefficients, harmonic numbers, and generalized harmonic numbers as in the following theorem.

Theorem 2. Each of the following summation formulas holds true: ^∞_𝑛=1(𝑎)_𝑛(𝑏)_𝑛𝑛!(𝑐)_𝑛𝐻_𝑛⁽¹⁾()=Γ(𝑐)Γ()𝑎−1𝑐−𝑎−𝑏Γ()Γ()𝜓()();𝑐−𝑎𝑐−𝑏𝑐−𝑎−𝜓𝑐−𝑎−𝑏_∞𝑛=1(𝑎)_𝑛(𝑏)_𝑛𝑛!(𝑐)_𝑛𝐻_𝑛⁽¹⁾(𝑎−1)²−𝐻_𝑛⁽²⁾(=Γ(𝑐)Γ()𝑎−1)𝑐−𝑎−𝑏Γ()Γ()⋅𝑐−𝑎𝑐−𝑏(𝜓(𝑐−𝑎)−𝜓(𝑐−𝑎−𝑏))²−𝜓^′(𝑐−𝑎)−𝜓^′(𝑐−𝑎−𝑏_∞𝑛=1);(𝑎)_𝑛(𝑏)_𝑛𝑛!(𝑐)_𝑛𝐻_𝑛⁽¹⁾(𝑎−1)³−3𝐻_𝑛⁽¹⁾(𝑎−1)𝐻_𝑛⁽²⁾(𝑎−1)+2𝐻_𝑛⁽³⁾(=Γ(𝑐)Γ()𝑎−1)𝑐−𝑎−𝑏Γ()Γ()𝜓()()𝑐−𝑎𝑐−𝑏𝑐−𝑎−𝜓𝑐−𝑎−𝑏³𝜓()()×𝜓−3𝑐−𝑎−𝜓𝑐−𝑎−𝑏_′(𝑐−𝑎)−𝜓^′(+𝜓𝑐−𝑎−𝑏)₍₂₎(𝑐−𝑎)−𝜓⁽²⁾(𝑐−𝑎−𝑏_∞𝑛=1);(𝑎)_𝑛(𝑏)_𝑛𝑛!(𝑐)_𝑛𝐻_𝑛⁽¹⁾(𝑎−1)⁴−6𝐻_𝑛⁽¹⁾(𝑎−1)²𝐻_𝑛⁽²⁾(𝑎−1)+8𝐻_𝑛⁽¹⁾(𝑎−1)𝐻_𝑛⁽³⁾()𝐻𝑎−1+3_𝑛⁽²⁾(𝑎−1)²−6𝐻_𝑛⁽⁴⁾(=Γ(𝑐)Γ()𝑎−1)𝑐−𝑎−𝑏Γ()Γ()𝜓()()𝑐−𝑎𝑐−𝑏𝑐−𝑎−𝜓𝑐−𝑎−𝑏⁴−6𝜓(𝑐−𝑎)−𝜓(𝑐−𝑎−𝑏)²×𝜓^′(𝑐−𝑎)−𝜓^′()𝜓()(×𝜓𝑐−𝑎−𝑏+4𝑐−𝑎−𝜓𝑐−𝑎−𝑏)₍₂₎(𝑐−𝑎)−𝜓⁽²⁾()𝜓𝑐−𝑎−𝑏+3^′(𝑐−𝑎)−𝜓^′(𝑐−𝑎−𝑏)²−𝜓⁽³⁾(𝑐−𝑎)−𝜓⁽³⁾(𝑐−𝑎−𝑏_∞𝑛=1);(𝑎)_𝑛(𝑏)_𝑛𝑛!(𝑐)_𝑛𝐻_𝑛⁽¹⁾(𝑎−1)𝐻_𝑛⁽¹⁾()=Γ(𝑐)Γ()𝑏−1𝑐−𝑎−𝑏Γ()Γ()⋅𝜓()(×𝜓()()𝑐−𝑎𝑐−𝑏𝑐−𝑎−𝜓𝑐−𝑎−𝑏)𝑐−𝑏−𝜓𝑐−𝑎−𝑏+𝜓^′(𝑐−𝑎−𝑏)_∞𝑛=1;(𝑎)_𝑛(𝑏)_𝑛𝑛!(𝑐)_𝑛𝐻_𝑛⁽¹⁾(𝑎−1)𝐻_𝑛⁽¹⁾(𝑏−1)²−𝐻_𝑛⁽¹⁾(𝑎−1)𝐻_𝑛⁽²⁾(=Γ(𝑐)Γ()𝑏−1)𝑐−𝑎−𝑏Γ()Γ()𝜓()()×𝜓()()𝑐−𝑎𝑐−𝑏𝑐−𝑎−𝜓𝑐−𝑎−𝑏𝑐−𝑏−𝜓𝑐−𝑎−𝑏²+2𝜓(𝑐−𝑎)−𝜓(𝑐−𝑎−𝑏)×𝜓^′(𝑐−𝑎−𝑏)−𝜓⁽²⁾()−𝜓()()×𝜓𝑐−𝑎−𝑏𝑐−𝑎−𝜓𝑐−𝑎−𝑏_′(𝑐−𝑏)−𝜓^′(𝑐−𝑎−𝑏_∞𝑛=1);(𝑎)_𝑛(𝑏)_𝑛𝑛!(𝑐)_𝑛𝐻_𝑛⁽¹⁾(𝑎−1)𝐻_𝑛⁽¹⁾(𝑏−1)³−3𝐻_𝑛⁽¹⁾(𝑎−1)𝐻_𝑛⁽¹⁾(𝑏−1)𝐻_𝑛⁽²⁾(𝑏−1)+2𝐻_𝑛⁽¹⁾(𝑎−1)𝐻_𝑛⁽³⁾(=Γ(𝑐)Γ()𝑏−1)𝑐−𝑎−𝑏Γ()Γ()𝜓()()×𝜓()()𝑐−𝑎𝑐−𝑏𝑐−𝑎−𝜓𝑐−𝑎−𝑏𝑐−𝑏−𝜓𝑐−𝑎−𝑏³+3𝜓(𝑐−𝑏)−𝜓(𝑐−𝑎−𝑏)²×𝜓^′(𝑐−𝑎−𝑏)+𝜓⁽³⁾()𝜓()()×𝜓()()⋅𝜓𝑐−𝑎−𝑏−3𝑐−𝑎−𝜓𝑐−𝑎−𝑏𝑐−𝑏−𝜓𝑐−𝑎−𝑏_′(𝑐−𝑏)−𝜓^′()𝜓()()𝑐−𝑎−𝑏−3𝑐−𝑏−𝜓𝑐−𝑎−𝑏×𝜓⁽²⁾(𝑐−𝑎−𝑏)−3𝜓^′()×𝜓𝑐−𝑎−𝑏_′(𝑐−𝑏)−𝜓^′()+𝜓()()×𝜓𝑐−𝑎−𝑏𝑐−𝑎−𝜓𝑐−𝑎−𝑏₍₂₎(𝑐−𝑏)−𝜓⁽²⁾(𝑐−𝑎−𝑏_∞𝑛=1);(𝑎)_𝑛(𝑏)_𝑛𝑛!(𝑐)_𝑛𝐻_𝑛⁽¹⁾(𝑎−1)𝐻_𝑛⁽¹⁾(𝑏−1)⁴−6𝐻_𝑛⁽¹⁾(𝑎−1)𝐻_𝑛⁽¹⁾(𝑏−1)²𝐻_𝑛⁽²⁾(𝑏−1)+8𝐻_𝑛⁽¹⁾(𝑎−1)𝐻_𝑛⁽¹⁾(𝑏−1)𝐻_𝑛⁽³⁾(𝑏−1)+3𝐻_𝑛⁽¹⁾(𝑎−1)𝐻_𝑛⁽²⁾(𝑏−1)²−6𝐻_𝑛⁽¹⁾(𝑎−1)𝐻_𝑛⁽⁴⁾(=Γ(𝑐)Γ()𝑏−1)𝑐−𝑎−𝑏Γ()Γ()𝜓()(×𝜓()()𝑐−𝑎𝑐−𝑏𝑐−𝑎−𝜓𝑐−𝑎−𝑏)𝑐−𝑏−𝜓𝑐−𝑎−𝑏⁴+4𝜓(𝑐−𝑏)−𝜓(𝑐−𝑎−𝑏)³×𝜓^′(𝑐−𝑎−𝑏)−𝜓⁽⁴⁾()𝜓()(×𝜓()()𝑐−𝑎−𝑏−6𝑐−𝑎−𝜓𝑐−𝑎−𝑏)𝑐−𝑏−𝜓𝑐−𝑎−𝑏²⋅𝜓^′(𝑐−𝑏)−𝜓^′()𝜓()()𝑐−𝑎−𝑏−6𝑐−𝑏−𝜓𝑐−𝑎−𝑏²×𝜓⁽²⁾()𝜓()()×𝜓𝑐−𝑎−𝑏−12𝑐−𝑏−𝜓𝑐−𝑎−𝑏_′(𝑐−𝑏)−𝜓^′(𝑐−𝑎−𝑏)×𝜓^′()𝜓()()×𝜓()()⋅𝜓𝑐−𝑎−𝑏+4𝑐−𝑎−𝜓𝑐−𝑎−𝑏𝑐−𝑏−𝜓𝑐−𝑎−𝑏₍₂₎(𝑐−𝑏)−𝜓⁽²⁾()𝜓()()𝑐−𝑎−𝑏+4𝑐−𝑏−𝜓𝑐−𝑎−𝑏×𝜓⁽³⁾()𝜓()()×𝜓𝑐−𝑎−𝑏+3𝑐−𝑎−𝜓𝑐−𝑎−𝑏_′(𝑐−𝑏)−𝜓^′(𝑐−𝑎−𝑏)²+6𝜓^′(𝑐−𝑏)−𝜓^′(𝑐−𝑎−𝑏)×𝜓⁽²⁾(𝑐−𝑎−𝑏)+4𝜓^′()×𝜓𝑐−𝑎−𝑏₍₂₎(𝑐−𝑏)−𝜓⁽²⁾()−𝜓()()×𝜓𝑐−𝑎−𝑏𝑐−𝑎−𝜓𝑐−𝑎−𝑏₍₃₎(𝑐−𝑏)−𝜓⁽³⁾(𝑐−𝑎−𝑏). (31)(32)(33)(34)(35)(36)(37)(38)

Proof. Differentiating each side of (28) with respect to the variable 𝑎 and using some suitable identities in Section 1 and Lemma 1, we obtain (31). Differentiating each side of (31) with respect to the variable 𝑎, we get (32). Similarly we prove (33) and (34). Differentiating each side of (31) with respect to the variable 𝑏 and using some suitable identities in Section 1 and Lemma 1, we obtain (35). Similarly we prove (36), (37), and (38).

Setting 𝑐=2 and 𝑎=𝑏=1 in (31) to (34) and using some suitable identities in Section 1, we obtain some interesting identities involving harmonic numbers and generalized harmonic numbers given in the following corollary.

Corollary 3. Each of the following identities holds true:^∞_𝑛=1𝐻_𝑛(𝑛+1)(𝑛+2)=1;_∞𝑛=11(𝑛+1)(𝑛+2)(𝐻_𝑛)²−𝐻_𝑛⁽²⁾=2=1⋅2;_∞𝑛=11(𝑛+1)(𝑛+2)(𝐻_𝑛)³−3𝐻_𝑛𝐻_𝑛⁽²⁾+2𝐻_𝑛⁽³⁾=6=2⋅3;_∞𝑛=11()𝐻𝑛+1)(𝑛+2^𝑛4−6(𝐻_𝑛)²𝐻_𝑛⁽²⁾+8𝐻_𝑛𝐻_𝑛⁽³⁾+3(𝐻_𝑛⁽²⁾)²−6𝐻_𝑛⁽⁴⁾=24=6⋅4. (39)

Differentiating only the left-hand side of (34) with respect to the variable 𝑎 twice and setting 𝑐=3 and 𝑎=𝑏=1 in each expression, in view of the identities (39), we may guess two identities asserted by the following conjecture.

Conjecture 4. Each of the following identities may hold true: ^∞_𝑛=11(𝑛+1)(𝑛+2)(𝐻_𝑛)⁵−10(𝐻_𝑛)³𝐻_𝑛⁽²⁾+20(𝐻_𝑛)²𝐻_𝑛⁽³⁾+15𝐻_𝑛𝐻_𝑛⁽²⁾²−30𝐻_𝑛𝐻_𝑛⁽⁴⁾−20𝐻_𝑛⁽²⁾𝐻_𝑛⁽³⁾+24𝐻_𝑛⁽⁵⁾=120=24⋅5;_∞𝑛=11()𝐻𝑛+1)(𝑛+2^𝑛6−15𝐻_𝑛⁴𝐻_𝑛⁽²⁾+40𝐻_𝑛³𝐻_𝑛⁽³⁾+45𝐻_𝑛²𝐻_𝑛⁽²⁾²−90(𝐻_𝑛)²𝐻_𝑛⁽⁴⁾−120𝐻_𝑛𝐻_𝑛⁽²⁾𝐻_𝑛⁽³⁾+144𝐻_𝑛𝐻_𝑛⁽⁵⁾−15𝐻_𝑛⁽²⁾³+90𝐻_𝑛⁽²⁾𝐻_𝑛⁽⁴⁾+40(𝐻_𝑛⁽³⁾)²−120𝐻_𝑛⁽⁵⁾=720=120⋅6. (40)

Setting 𝑐=3 and 𝑎=𝑏=1 in (35) to (44) and using some suitable identities in Section 1 and Lemma 1, we obtain a set of very interesting identities in the following corollary.

Corollary 5. Each of the following identities holds true: ^∞_𝑛=1𝐻_𝑛²()((2;𝑛+1)(𝑛+2=21+𝜁))_∞𝑛=1(𝐻_𝑛)³−𝐻_𝑛𝐻_𝑛⁽²⁾()((2)(3;𝑛+1)(𝑛+2=41+𝜁+𝜁))_∞𝑛=1(𝐻_𝑛)⁴−3(𝐻_𝑛)²𝐻_𝑛⁽²⁾+2𝐻_𝑛𝐻_𝑛⁽³⁾()((2)(3)(4;𝑛+1)(𝑛+2=121+𝜁+𝜁+𝜁))_∞𝑛=1(𝐻_𝑛)⁵−6(𝐻_𝑛)³𝐻_𝑛⁽²⁾+8𝐻_𝑛²𝐻_𝑛⁽³⁾+3𝐻_𝑛(𝐻_𝑛⁽²⁾)²−6𝐻_𝑛𝐻_𝑛⁽⁴⁾×((𝑛+1)(𝑛+2))⁻¹=48(1+𝜁(2)+𝜁(3)+𝜁(4)+𝜁(5)). (41)(42)(43)(44)

Differentiating only the left-hand side of (38) with respect to the variable 𝑏 twice and setting 𝑐=3 and 𝑎=𝑏=1 in each expression, in view of the identities (41) to (44), we may guess two identities asserted by the following conjecture.

Conjecture 6. Each of the following identities may hold true: ^∞_𝑛=11(𝑛+1)(𝑛+2)(𝐻_𝑛)⁶−10(𝐻_𝑛)⁴𝐻_𝑛⁽²⁾+20(𝐻_𝑛)³𝐻_𝑛⁽³⁾+15(𝐻_𝑛)²(𝐻_𝑛⁽²⁾)²−30(𝐻_𝑛)²𝐻_𝑛⁽⁴⁾−20𝐻_𝑛𝐻_𝑛⁽²⁾𝐻_𝑛⁽³⁾+24𝐻_𝑛𝐻_𝑛⁽⁵⁾=240(1+𝜁(2)+𝜁(3)+𝜁(4)+𝜁(5)+𝜁(6))_∞𝑛=1;1(𝑛+1)(𝑛+2)(𝐻_𝑛)⁷−15(𝐻_𝑛)⁵𝐻_𝑛⁽²⁾+40(𝐻_𝑛)⁴𝐻_𝑛⁽³⁾+45(𝐻_𝑛)³(𝐻_𝑛)²−90(𝐻_𝑛)³𝐻_𝑛⁽⁴⁾−120𝐻_𝑛²𝐻_𝑛⁽²⁾𝐻_𝑛⁽³⁾+144𝐻_𝑛²𝐻_𝑛⁽⁵⁾−15𝐻_𝑛𝐻_𝑛⁽²⁾³+90𝐻_𝑛𝐻_𝑛⁽²⁾𝐻_𝑛⁽⁴⁾+40𝐻_𝑛(𝐻_𝑛⁽³⁾)²−120𝐻_𝑛𝐻_𝑛⁽⁶⁾=1440(1+𝜁(2)+𝜁(3)+𝜁(4)+𝜁(5)+𝜁(6)+𝜁(7)). (45)(46)

Remark 7. It is interesting to observe that the number of terms in the numerator of each of the left-hand sides of (41) to (46) is equal to the number of partitions of 𝑘−1  (𝑘=2,3,4,5,6,7), respectively. For example, for (46), all the partitions of 7−1=6 are as follows: 7=1+1+1+1+1+1+1=1+1+1+1+1+2=1+1+1+2+2=1+2+2+2=1+1+1+1+3=1+1+2+3=1+3+3=1+1+1+4=1+1+5=1+2+4, (47) where 𝑘 denotes 𝐻_𝑛^(𝑘)  (𝑘=1,2,3,4,5) and + is translated into a multiplication of its corresponding 𝐻_𝑛^(𝑘). The coefficient of each of the right-hand sides of (41) to (46) has the following rule: 2,4=2⋅2,12=4⋅3,48=12⋅4,240=48⋅5,1440=240⋅6,…. (48) The remaining thing is to find a rule that dominates the coefficients of each term of the numerator of the left-hand sides of (41) to (46).

Differentiating each side of (28) with respect to the variable 𝑐 successively and using some suitable identities in Section 1 and Lemma 1, we obtain a set of infinite series involving binomial coefficients, harmonic numbers, and generalized harmonic numbers different from those in Theorem 2 as in the following theorem.

Theorem 8. Each of the following summation formulas holds true: ^∞_𝑛=1(𝑎)_𝑛(𝑏)_𝑛𝑛!(𝑐)_𝑛𝐻_𝑛⁽¹⁾()=Γ(𝑐)Γ()𝑐−1𝑐−𝑎−𝑏Γ()Γ()×𝜓()()(𝑐)(;𝑐−𝑎𝑐−𝑏𝑐−𝑎+𝜓𝑐−𝑏−𝜓−𝜓𝑐−𝑎−𝑏)_∞𝑛=1(𝑎)_𝑛(𝑏)_𝑛𝑛!(𝑐)_𝑛𝐻_𝑛⁽¹⁾(𝑐−1)²+𝐻_𝑛⁽²⁾(=Γ(𝑐)Γ()𝑐−1)𝑐−𝑎−𝑏Γ()Γ()×𝜓()()(𝑐)()𝑐−𝑎𝑐−𝑏𝑐−𝑎+𝜓𝑐−𝑏−𝜓−𝜓𝑐−𝑎−𝑏²−𝜓^′(𝑐−𝑎)+𝜓^′(𝑐−𝑏)−𝜓^′(𝑐)−𝜓^′(𝑐−𝑎−𝑏_∞𝑛=1);(𝑎)_𝑛(𝑏)_𝑛𝑛!(𝑐)_𝑛𝐻_𝑛⁽¹⁾(𝑐−1)³+3𝐻_𝑛⁽¹⁾(𝑐−1)×𝐻_𝑛⁽²⁾(𝑐−1)+2𝐻_𝑛⁽³⁾(=Γ(𝑐)Γ()𝑐−1)𝑐−𝑎−𝑏Γ()Γ()×𝜓()()(𝑐)()𝑐−𝑎𝑐−𝑏𝑐−𝑎+𝜓𝑐−𝑏−𝜓−𝜓𝑐−𝑎−𝑏³𝜓()()(𝑐)()⋅𝜓−3𝑐−𝑎+𝜓𝑐−𝑏−𝜓−𝜓𝑐−𝑎−𝑏_′(𝑐−𝑎)+𝜓^′(𝑐−𝑏)−𝜓^′(𝑐)−𝜓^′()+𝜓𝑐−𝑎−𝑏₍₂₎(𝑐−𝑎)+𝜓⁽²⁾(𝑐−𝑏)−𝜓⁽²⁾(𝑐)−𝜓⁽²⁾(𝑐−𝑎−𝑏_∞𝑛=1);(𝑎)_𝑛(𝑏)_𝑛𝑛!(𝑐)_𝑛𝐻_𝑛⁽¹⁾(𝑐−1)⁴+6𝐻_𝑛⁽¹⁾(𝑐−1)²×𝐻_𝑛⁽²⁾(𝑐−1)+8𝐻_𝑛⁽¹⁾(𝑐−1)𝐻_𝑛⁽³⁾()𝐻𝑐−1+3_𝑛⁽²⁾(𝑐−1)²+6𝐻_𝑛⁽⁴⁾(=Γ(𝑐)Γ()𝑐−1)𝑐−𝑎−𝑏Γ()Γ()×𝜓()()(𝑐)()𝑐−𝑎𝑐−𝑏𝑐−𝑎+𝜓𝑐−𝑏−𝜓−𝜓𝑐−𝑎−𝑏⁴−6𝜓(𝑐−𝑎)+𝜓(𝑐−𝑏)−𝜓(𝑐)−𝜓(𝑐−𝑎−𝑏)²⋅𝜓^′(𝑐−𝑎)+𝜓^′(𝑐−𝑏)−𝜓^′(𝑐)−𝜓^′()𝜓()()(𝑐)()⋅𝜓𝑐−𝑎−𝑏+4𝑐−𝑎+𝜓𝑐−𝑏−𝜓−𝜓𝑐−𝑎−𝑏₍₂₎(𝑐−𝑎)+𝜓⁽²⁾(𝑐−𝑏)−𝜓⁽²⁾(𝑐)−𝜓⁽²⁾()𝜓𝑐−𝑎−𝑏+3^′(𝑐−𝑎)+𝜓^′(𝑐−𝑏)−𝜓^′(𝑐)−𝜓^′(𝑐−𝑎−𝑏)²−𝜓⁽³⁾(𝑐−𝑎)+𝜓⁽³⁾(𝑐−𝑏)−𝜓⁽³⁾(𝑐)−𝜓⁽³⁾(𝑐−𝑎−𝑏). (49)

Setting 𝑐=1 and 𝑎=𝑏=−1/2 in (49) and using some suitable identities in Section 1 and special values of 𝜓-function (see, e.g., [4, Section 1.2] and [5, Section 1.3]), we obtain a set of interesting infinite series involving binomial coefficients and harmonic numbers given in the following corollary.

Corollary 9. Each of the following identities holds true: (2𝑛−1)^∞_𝑛=1^𝑛2𝑛222^4𝑛𝐻_𝑛=4𝜋3−4log2_{∞𝑛=1𝑛2𝑛2};(2𝑛−1)²2^4𝑛(𝐻_𝑛)²+𝐻_𝑛⁽²⁾=4𝜋(3−4log2)²+7−4𝜁(2)_{∞𝑛=1𝑛2𝑛2};(2𝑛−1)²2^4𝑛(𝐻_𝑛)³+3𝐻_𝑛𝐻_𝑛⁽²⁾+2𝐻_𝑛⁽³⁾=4𝜋3−4log2³((2))(3);−33−4log24𝜁−7−24𝜁+30_{∞𝑛=1𝑛2𝑛2}(2𝑛−1)²2^4𝑛(𝐻_𝑛)⁴+6(𝐻_𝑛)²𝐻_𝑛⁽²⁾+8𝐻_𝑛𝐻_𝑛⁽³⁾+3(𝐻_𝑛⁽²⁾)²+6𝐻_𝑛⁽⁴⁾=4𝜋3−4log2⁴−63−4log2²((2))((3((2))4𝜁−7+243−4log25−4𝜁))+34𝜁−7²+186−168𝜁(4). (50)

3. Finite Series Involving Binomial Coefficients, Harmonic Numbers, and Generalized Harmonic Numbers

Setting 𝑏=−𝑁∈ℕ in some chosen formulas in Theorems 2 and 8 and using some suitable identities in Section 1 and the following known and easily derivable formula: (−𝑁)_𝑛=(−1)^𝑛𝑁!()!,(),(),=𝑁−𝑛0≤𝑛≤𝑁;𝑁∈ℕ0,𝑛>𝑁(−1)^𝑛𝑁𝑛,(),(),𝑛!0≤𝑛≤𝑁;𝑁∈ℕ0,𝑛>𝑁 (51) we obtain a set of finite series involving binomial coefficients, harmonic numbers, and generalized harmonic numbers given in the following theorem.

Theorem 10. Each of the following finite summation formulas holds true: ^𝑁_𝑛=1(−1)^𝑛+1(𝑎)_𝑛(𝑐)_𝑛𝑁𝑛𝐻_𝑛⁽¹⁾()=Γ()Γ(𝑐)𝑎−1𝑐−𝑎+𝑁Γ()Γ()𝐻𝑐−𝑎𝑐+𝑁^𝑁(1)(𝑐−𝑎−1)𝑁∈ℕ₀, (52) where the empty sum is (as usual) understood to be nil throughout this paper, ^𝑁_𝑛=1(−1)^𝑛(𝑎)_𝑛(𝑐)_𝑛𝑁𝑛𝐻_𝑛⁽¹⁾(𝑎−1)²−𝐻_𝑛⁽²⁾(=Γ()Γ(𝑐)𝑎−1)𝑐−𝑎+𝑁Γ()Γ()𝐻𝑐−𝑎𝑐+𝑁^𝑁(1)(𝑐−𝑎−1)²−𝐻_𝑁⁽²⁾(𝑐−𝑎−1)𝑁∈ℕ₀;_𝑁𝑛=1(−1)^𝑛+1(𝑎)_𝑛(𝑐)_𝑛𝑁𝑛𝐻_𝑛⁽¹⁾(𝑎−1)³−3𝐻_𝑛⁽¹⁾(𝑎−1)×𝐻_𝑛⁽²⁾(𝑎−1)+2𝐻_𝑛⁽³⁾(=Γ()Γ(𝑐)𝑎−1)𝑐−𝑎+𝑁Γ()Γ()𝐻𝑐−𝑎𝑐+𝑁^𝑁(1)(𝑐−𝑎−1)³−3𝐻_𝑁⁽¹⁾(𝑐−𝑎−1)×𝐻_𝑁⁽²⁾(𝑐−𝑎−1)+2𝐻_𝑁⁽³⁾(𝑐−𝑎−1)𝑁∈ℕ₀;_𝑁𝑛=1(−1)^𝑛(𝑎)_𝑛(𝑐)_𝑛𝑁𝑛𝐻_𝑛⁽¹⁾(𝑎−1)⁴−6𝐻_𝑛⁽¹⁾(𝑎−1)²×𝐻_𝑛⁽²⁾(𝑎−1)+8𝐻_𝑛⁽¹⁾(𝑎−1)×𝐻_𝑛⁽³⁾(𝑎−1)+3𝐻_𝑛⁽²⁾(𝑎−1)²−6𝐻_𝑛⁽⁴⁾(=Γ()Γ(𝑐)𝑎−1)𝑐−𝑎+𝑁Γ()Γ()𝐻𝑐−𝑎𝑐+𝑁^𝑁(1)(𝑐−𝑎−1)⁴−6𝐻_𝑁⁽¹⁾(𝑐−𝑎−1)²×𝐻_𝑁⁽²⁾(𝑐−𝑎−1)+8𝐻_𝑁⁽¹⁾(𝑐−𝑎−1)𝐻_𝑁⁽³⁾()𝐻𝑐−𝑎−1+3_𝑁⁽²⁾(𝑐−𝑎−1)²−6𝐻_𝑁⁽⁴⁾(𝑐−𝑎−1)𝑁∈ℕ₀;_𝑁𝑛=1(−1)^𝑛(𝑎)_𝑛(𝑐)_𝑛𝑁𝑛𝐻_𝑛⁽¹⁾()=Γ()Γ(𝑐)𝑐−1𝑐−𝑎+𝑁Γ()Γ()𝐻𝑐−𝑎𝑐+𝑁^𝑁(1)(𝑐−1)−𝐻_𝑁⁽¹⁾(𝑐−𝑎−1)𝑁∈ℕ₀;_𝑁𝑛=1(−1)^𝑛𝑁𝑛(𝑎)_𝑛(𝑐)_𝑛𝐻_𝑛⁽¹⁾(𝑐−1)²+𝐻_𝑛⁽²⁾(=Γ()Γ(𝑐)𝑐−1)𝑐−𝑎+𝑁Γ()Γ()𝐻𝑐−𝑎𝑐+𝑁^𝑁(1)(𝑐−1)−𝐻_𝑁⁽¹⁾(𝑐−𝑎−1)²+𝐻_𝑁⁽²⁾(𝑐−1)−𝐻_𝑁⁽²⁾(𝑐−𝑎−1)𝑁∈ℕ₀. (53)

Setting 𝑎=1 and 𝑐=2 in (52) to (53) and using some suitable identities in Section 1, we obtain a set of interesting identities involving binomial coefficients, harmonic numbers, and generalized harmonic numbers given in the following corollary.

Corollary 11. Each of the following identities holds true: ^𝑁_𝑛=1(−1)^𝑛+1𝑁𝑛𝐻𝑛+1^𝑛=𝐻_𝑁𝑁+1𝑁∈ℕ₀;_𝑁𝑛=1(−1)^𝑛𝑁𝑛𝑛+1(𝐻_𝑛)²−𝐻_𝑛⁽²⁾=(𝐻_𝑁)²−𝐻_𝑁⁽²⁾𝑁+1𝑁∈ℕ₀;_𝑁𝑛=1(−1)^𝑛+1𝑁𝑛𝐻𝑛+1^𝑛3−3𝐻_𝑛𝐻_𝑛⁽²⁾+2𝐻_𝑛⁽³⁾=1𝑁+1(𝐻_𝑁)³−3𝐻_𝑁𝐻_𝑁⁽²⁾+2𝐻_𝑁⁽³⁾𝑁∈ℕ₀;_𝑁𝑛=1(−1)^𝑛𝑁𝑛𝐻𝑛+1^𝑛4−6𝐻_𝑛²𝐻_𝑛⁽²⁾+8𝐻_𝑛𝐻_𝑛⁽³⁾+3(𝐻_𝑛⁽²⁾)²−6𝐻_𝑛⁽⁴⁾=1𝑁+1(𝐻_𝑁)⁴−6(𝐻_𝑁)²𝐻_𝑁⁽²⁾+8𝐻_𝑁𝐻_𝑁⁽³⁾+3(𝐻_𝑁⁽²⁾)²−6𝐻_𝑁⁽⁴⁾𝑁∈ℕ₀;_𝑁𝑛=1(−1)^𝑛+1𝑁𝑛𝐻𝑛+1^𝑛(1)(1)=𝑁(𝑁+1)²𝑁∈ℕ₀;_𝑁𝑛=1(−1)^𝑛+1𝑁𝑛𝐻𝑛+1^𝑛(1)(1)²+𝐻_𝑛⁽²⁾(1=)2𝑁(𝑁+1)³𝑁∈ℕ₀. (54)

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by Dongguk University Research Fund of 2013. Also, it was supported, in part, by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (2010-0011005).