Generalized hyperharmonic number sums with reciprocal binomial coefficients

In this paper, we mainly show that generalized hyperharmonic number sums with reciprocal binomial coefficients can be expressed in terms of classical (alternating) Euler sums, zeta values and generalized (alternating) harmonic numbers.


Introduction and preliminaries
Let Z, N, N 0 and C denote the set of integers, positive integers, nonnegative integers and complex numbers, respectively. In the present paper, we mainly study the so-called generalized hyperharmonic numbers [11,15,19]  n are the classical hyperharmonic numbers introduced by Conway and Guy [7]. To see combinatorial interpretations of these hyperharmonic numbers and their connections with Stirling numbers, please find Benjamin et al's interesting paper [3]. For convenience, we recall the generalized alternating harmonic numbers which are defined as The harmonic numbers and their generalizations has caused many mathematicians' interest (see [9,11,10,12,13,15,16,17,18,20,21,22] and references therein), since they play an essential role in number theory, combinatorics, analysis of algorithms and many other areas (see e.g. [14]). One of the most famous result that obtained by Euler [12] is the following identity It is interesting that the Riemann zeta functions ζ(s) := ∞ n=1 n −s appear in such expressions. According to the recording of Ramanujan's Notebooks [4, p.253], Euler considered this type of infinite series containing harmonic numbers H n in response to a letter from Goldbach in 1742.
For convenience, we recall the definition of the well-known Hurwitz zeta function: From Euler's time on, infinite series containing harmonic numbers or their generalizations have been called Euler sums. It is a difficult task to give explicit evaluation for general Euler sums. Facilitated by numerical computations using an algorithm, Bailey, Borwein and Girgensohn [2] determined, with high confidence, whether or not a particular numerical value involving the generalized harmonic numbers H (m) n could be expressed as a rational linear combination of several given constants.
Flajolet and Salvy [12] developed the contour integral representation approach (the most powerful method in the corresponding area as far as the author knows, although restricted to parity principle) to the evaluation of Euler sums involving the classical (alternating) harmonic numbers. Note that, the contour integral representation approach can not only evaluate Euler sums, but also evaluate some infinite series involving hyperbolic functions.
Euler sums of hyperharmonic numbers had also attracted many mathematicians' attention. For instance, Mező and Dil [18] considered the Euler sums of type and showed that it could be reduced to infinite series involving the Hurwitz zeta function values. Later Dil and Boyadzhiev [10] extended this result to infinite series involving multiple sums of the Hurwitz zeta function values.
As a natural generalization, Dil, Mező and Cenkci [11] considered Euler sums of generalized hyperharmonic numbers of the form They proved that for positive integers p, r and m with m > r, ζ H (p,r) (m) could be reduced to infinite series of multiple sums of the Hurwitz zeta function values. For r = 1, 2, 3, ζ H (p,r) (m) were also written explicitly in terms of (multiple) zeta values. Although these results were interesting, Dil et al didn't give general formula for explicit evaluations of Euler sums of generalized hyperharmonic numbers. Fortunately, the author [15] found a combinatorial approach and proved that ζ H (p,r) (m) could be expressed as linear combinations of classical Euler sums. From Flajolet and Salvy's paper [12], we knew that the linear Euler sums could be reduced to zeta values. Thus for small values of p, r and m, we can determine the exact values of ζ H (p,r) (m). Motivated by Flajolet-Salvy's paper [12] and Dil-Mező-Cenkci's paper [11], the author [16] also introduced the notion of the generalized alternating hyperharmonic numbers and proved that Euler sums of the generalized alternating hyperharmonic numbers H (p,r,1) n could be expressed in terms of linear combinations of classical (alternating) Euler sums.
If we regard ∞ n=1 h (r) n /n s as a complex function in variable s, there are some more progresses toward this direction. For instance, Matsuoka [17] proved that ∞ n=1 h (1) n /n s admits a meromorphic continuation to the entire complex plane. Kamano [13] expressed the complex variable function ∞ n=1 h (r) n /n s in terms of the Riemann zeta functions, and showed that it could be meromorphically continued to the entire complex plane. In addition, the residue at each pole was also given.
There are some more interesting combinatorial properties about the generalized hyperharmonic numbers. For instance,Ömür and Koparal [19] defined two n × n matrices A n and B n with a i,j = H On the contrary, Euler sums of generalized harmonic numbers with reciprocal binomial coefficients had been studied by Sofo. In 2011, Sofo [20] proved that generalized harmonic number sums with reciprocal binomial coefficients of types ∞ In the present paper, we mainly show that generalized hyperharmonic number sums with reciprocal binomial coefficients of types

Generalized hyperharmonic number sums
In this section, we develop closed form representations for generalized hyperharmonic number sums with reciprocal binomial coefficients of types Before going further, we introduce some notations and lemmata. Following Flajolet-Salvy's paper [12], we write four types of classical linear (alternating) Euler sums as We now recall Faulhaber's formula on sums of powers. It is well known that the sum of powers of consecutive intergers 1 k + 2 k + · · · + n k can be explicitly expressed in terms of Bernoulli numbers or Bernoulli polynomials. Faulhaber's formula can be written as where Bernoulli numbers B + n are determined by the recurrence formula or by the generating function and Bernoulli polynomials B n (x) are defined by the following generating function Definition 1. For p ∈ Z and m, r, t ∈ N, define the quantities S(p, m, t, r, 0) and S(p, m, 1, r, 1) as is understood to be the sum 1 p + 2 p + · · · + n p .
Lemma 2. Let p, m, r ∈ N, then we have Let m, r ∈ N, p ∈ N 0 and m ≥ p + 2, then we have Proof. When p, m, r ∈ N, we can obtain that With the help of Lemma 1, we get the desired result. When m, r ∈ N, p ∈ N 0 and m ≥ p + 2, we have .
With the help of partial fraction expansion we get the desired result.
Proof. By a change of counter, we have we get the desired result.
Let m, r ∈ N, p ∈ N 0 and m ≥ p + 1, then we have Proof. When p, m, r ∈ N, we can obtain that With the help of Lemma 3, we get the desired result. When m, r ∈ N, p ≥ 0 and m ≥ p + 1, we have .
With the help of partial fraction expansion we get the desired result. The coefficients a(r, m, j) satisfy the following recurrence relations: where D(r, m, j, y) = j ℓ=max{0,m−y−1} The initial value is given by a(1, 0, 0) = 1.
Now we are able to prove our main theorems of this section.

Quadratic generalized hyperharmonic number sums
In this section, we develop closed form representations for quadratic generalized hyperharmonic number sums with reciprocal binomial coefficients of Before going further, we introduce some notations and lemmata. Following Flajolet-Salvy's paper [12], we write classical (alternating) quadratic Euler sums as Lemma 6 (Abel's lemma on summation by parts [1,6]). Let {f k } and {g k } be two sequences, and define the forward difference and backward difference, respectively, as then, there holds the relation: Proof. Set n and g n := 1 n + 1 + · · · + 1 n + r , by using Lemma 6, we have When p ≥ 0, H (−p) n is understood to be the sum 1 p + 2 p + · · · + n p .
Let p 1 , m, r ∈ N, p 2 ∈ N 0 and m ≥ p 2 + 2, then we have Let m, r ∈ N, p 1 , p 2 ∈ N 0 and m ≥ p 1 + p 2 + 3, then we have Proof. When p 1 , p 2 , m, r ∈ N, we can obtain that n H (p 2 ) n n(n + r) .
With the help of Lemma 7, we get the desired result. When p 1 , m, r ∈ N, p 2 ∈ N 0 and m ≥ p 2 + 2, we have With the help of Lemma 2, we get the desired result. When m, r ∈ N, p 1 , p 2 ∈ N 0 and m ≥ p 1 + p 2 + 3, we have .
Proof. By a change of counter, we have we get the desired result.
Lemma 10. Let p 1 , p 2 , r ∈ N and m ∈ N 0 , then we have Let p 1 , m, r ∈ N, p 2 ∈ N 0 and m ≥ p 2 + 1, then we have T (p 1 , −p 2 , m, 1, r, 1) Let m, r ∈ N, p 1 , p 2 ∈ N 0 and m ≥ p 1 + p 2 + 2, then we have 1, r, 1) . Proof. When p 1 , p 2 , r ∈ N and m ∈ N 0 , we can obtain that With the help of Lemma 9, we get the desired result. When p 1 , m, r ∈ N, p 2 ∈ N 0 and m ≥ p 2 + 1, we have With the help of Lemma 4, we get the desired result. When m, r ∈ N, p 1 , p 2 ∈ N 0 and m ≥ p 1 + p 2 + 2, we have Comparing the real part and the imaginary part on both sides, we have .
It is known (see [9]) that De Doelder [9] also considered the function g(z) = log z/(z 2 − 1) along the same contour. Then the following results could be established: .
We now consider the function f (z) = − log z/(2 − z) along the same contour, since within this contour there are no singularities, by using the Cauchy residue theorem we can obtain that It follows that

Some formulas for harmonic numbers
In this section, we develop some formulas for harmonic numbers in terms of binomial coefficients. We begin by recalling a known result for harmonic numbers [21]. For n ∈ N 0 , the following result holds: − H n+1 n + 1 = 1 0 y n log ydy .
L(n, m, x) = x n+1 n + 1 log m x − m n + 1 With the help of Lemma11, we get the desired result.