Some combinatorial identities concerning harmonic numbers and binomial coefficients

School of Mathematics and Statistics, Zhoukou Normal University, Henan, China (Received: 6 September 2021. Received in revised form: 8 October 2021. Accepted: 21 October 2021. Published online: 25 October 2021.) c © 2021 the author. This is an open access article under the CC BY (International 4.0) license (www.creativecommons.org/licenses/by/4.0/). Abstract An elementary proof for a binomial identity of Batir [Integers 20 (2020) #A25] is presented and several combinatorial identities concerning harmonic numbers and binomial coefficients are derived.


Introduction
It is well-known that the harmonic numbers are defined by H 0 = 0 and H n = n k=1 1 k for n ∈ N.
For n, ∈ N and for an indeterminate x, they are generalized (cf. [3,6]) by which reduce, when = 1, to H n (x) (cf. [4,7,8]) and to the ordinary harmonic numbers H n , respectively. Here and forth, we use N, Z − and C to denote the sets of positive integers, negative integers and complex numbers, respectively. Given a differentiable function f (x), denote two derivative operators by Then it is not difficult to get the derivative of binomial coefficients (cf. [5,6]) which can be stated in terms of the generalized harmonic numbers as The binomial coefficients can be expressed as gamma function The beta function B(s, t), defined by has a closed relationship with gamma function There are many nice identities about harmonic numbers and binomial coefficients. For example, by making use of integral, it is not hard to derive (cf. [1, Recently, Batir [2] obtained a series of identities based on the binomial transformation formula n k=0 s + n k Throughout the paper, we use the rising and falling factorials, defined by (x) 0 = 1 and (x) n = x(x + 1) · · · (x + n − 1) for n ∈ N, The rest of the paper is organized as follows. In the next section, we firstly give a proof of Lemma 2.1, which is equivalent to Batir's binomial relation (5), and then derive some consequences by integration and derivation method. The paper ends with the third section, where several identities are established.

Main results
The identity in the following lemma is equivalent to Batir's identity (5). Here, we present an elementary proof, instead of induction.
Lemma 2.1. For n ∈ N, the following relation holds Proof. Applying the Vandermonde convolution to the right side of (6) gives which completes the proof.
Proof. The proof follows from differentiating with respect to x both sides of (7).
Proof. Separating the term of k = 0 from the right-hand sum of (6), and then replacing k by k + 1, we obtain Putting the first term of the sum on the left side to the right, and then dividing the resulting equation by z, we have Integrating from z = 0 to z = 1, we get n k=1 x + k k we complete the proof.
Proof. Integrating both sides of (10) from z = 0 to z = u, we get n k=1 x + k k Integrating again the above equation from u = 0 to u = 1, after dividing by u, we find that n k=1 x + k k By means of integration by parts, we obtain Applying (9), we complete the proof.
Proof. The proof follows from differentiating with respect to x both sides of (11).
Proof. The proof follows by differentiating with respect to z both sides of (6), and then letting z = 1.
Proof. The result can be obtained by deriving x for the identity in Corollary 2.6.

Identities
Based on the results obtained in Section 2, we can establish several interesting identities by taking special values of the parameter x, including certain well-known identities. For example, letting x = − 1 2 in (7) and x = 0 in (8), we recover the two well-known identities Further examples can analogously be derived as follows. Then the identity follows by Proof. Replacing x by x − n − 1 in (7), we have . (8), we get (20) with the help of (16).
If setting m = −1, 0 in (20), we get, respectively, the following two identities n k=0 2k k In view of we complete the proof.
Two special cases corresponding to m = 1, 2 in (23) are recorded below Proof. The identity follows by taking x = − 1 2 + m in (13) and using (29).
A special case of (38) corresponding to m = 0 is recorded below Besides the identities derived above, by choosing other different special values of parameter x, it is possible to derive more finite series identities on harmonic numbers and binomial coefficients. The interested reader is encouraged to make further attempts.