A note on a generalization of Riordan’s combinatorial identity via a hypergeometric series approach

: In this note, an attempt has been made to generalize the well-known and useful Riordan’s combinatorial identity via a hypergeometric series approach.


Introduction
We begin by recalling the definition of Gauss's hypergeometric series or simply a hypergeometric series defined as follows [5]: Here (a) n is the well-known Pochhammer's symbol defined for any complex number a (̸ = 0) by (a) n =

 
The series defined in (1) is convergent for all values of z if |z| < 1 and divergent if |z| > 1. When z = 1, the series is convergent if ℜ(c − a − b) > 0 and divergent if ℜ(c − a − b) ≤ 0. Also, when z = −1, the series is absolutely convergent if ℜ(c − a − b) > 0 and is convergent but not It should be remarked here that whenever a hypergeometric series reduces to the gamma function, the results are very important from the application point of view. Thus the classical summation theorems such as those of Gauss, Gauss second, Kummer, Bailey, and a few terminating summations play an important role.
However, in our present investigation, we shall require the following hypergeometric summation theorem and identities [3,5] each valid for n ∈ N 0 .
Gauss second summation theorem [5] 2 F 1 a, b 1 2 (a + b + 1) Hypergeometric identities [3,5] On the other hand, one of the applications of the hypergeometric series is to solve binomial sums [7]. Using this method, the given sum is first converted into a hypergeometric series. This is accomplished by combining any terms that are polynomial in the summation index with binomials, expanding the binomials into factorials, and transferring the factorials into Pochhammer symbols. If successfully converted, the resulting hypergeometric series is compared with known summation theorems and a closed-form sum may be obtained when a suitable match is found.
In this context, we first mention here the following famous combinatorial sum known as Knuth's old sum viz.
The above combinatorial identity is, in the literature, also known as Reed Dawson identity because Reed Dawson presented this identity in a private communication to Riordan who recorded it in his well-known book [6, p. 71].
It is not out of place to mention here that from time to time, several different proofs of the above sums have been given in the literature. In this regard, we would like to refer to a survey paper by Prodinger [4]. In 1974, Andrews [1, p. 478] established the identity given in (7) by employing Gauss second summation theorem (2). Later, in 2004, Choi et al. [2] established by utilizing the hypergeometric identities (3) and (4).
Riordan [6], by the method of inverse relations, established the following identity closely related to (7) viz.
The aim of this note is to provide a natural generalization of the Riordan identity (8) via a hypergeometric series approach by employing the hypergeometric identities (5) and (6). The same is given in the next section.

Generalization of Riordan identity
In this section, we shall establish the natural generalization of the Riordan identity (8) asserted in the following theorem.
Theorem 2.1. For i ∈ N, the following identity holds true.
( 1 2 ) n (1)n , n is even, (2)n , n is odd. (9) Proof. In order to establish the result (9) asserted in the theorem, we proceed as follows. Denoting the left-hand side of (9) by S i , we have Upon converting binomial coefficients into Pochhammer symbols viz.
we have, after some simplification Finally, summing up the series with the help of the equation (1), we have Now, changing n to 2n and 2n + 1, respectively, and using the results (5) and (6), we easily arrive at the right-hand side of (9). This completes the proof of the generalization of the Riordan identity (9) asserted in the theorem.

Corollaries
In this section, we shall mention known as well as some new identities of our main findings.
Corollary 3.2. In (9), if we set i = 2, we get the following interesting identity.

Concluding remark
In this short note, we have provided a generalization of the well-known and useful identity due to Riordan via a hypergeometric series approach.