Three Identities of the Catalan-Qi Numbers

In the paper, the authors find three new identities of the Catalan-Qi numbers and provide alternative proofs of two identities of the Catalan numbers. The three identities of the Catalan-Qi numbers generalize three identities of the Catalan numbers.


Introduction
It is stated in [1] that the Catalan numbers C n for n ≥ 0 form a sequence of natural numbers that occur in tree enumeration problems such as "In how many ways can a regular n-gon be divided into n − 2 triangles if different orientations are counted separately?" (for other examples, see [2,3]) the solution of which is the Catalan number C n−2 .The Catalan numbers C n can be generated by Three of explicit equations of C n for n ≥ 0 read that where is the generalized hypergeometric series defined for complex numbers a i ∈ C and b i ∈ C \ {0, −1, −2, . . .}, for positive integers p, q ∈ N, and in terms of the rising factorials (x) n defined by A generalization of the Catalan numbers C n was defined in [4][5][6] by for n ≥ 1.The usual Catalan numbers C n = 2 d n are a special case with p = 2.
In (Remark 1 [21]), an alternative and analytical generalization of the Catalan numbers C n and the Catalan function C x was introduced by It is clear that In (Theorem 1.1 [22]), among other things, it was deduced that where x denotes the floor function the value of which is the largest integer less than or equal to x.
In 2009, J. Koshy ([33] p. 322) provided another recursive equation We observe that Identity (4) can be rearranged as where x stands for the ceiling function which gives the smallest integer not less than x.
The aims of this paper are to generalize Identities ( 2)-( 4) for the Catalan numbers C n to ones for the Catalan-Qi numbers C(a, b; n).
Our main results can be summarized up as the following theorem.
Theorem 1.For a, b > 0, n ∈ N, and n ≥ 2m ≥ 0, the Catalan-Qi numbers C(a, b; n) satisfy and As by-products, alternative proofs for Identities (2) and ( 4) are also supplied in next section.

Proofs
We are now in a position to prove Theorem 1 and Identities ( 2) and ( 4).(5).By the definition (1), we have

Proof of Identity
Further using the relations The proof of Identity ( 5) is thus complete.(6).By the definition (1), we have

Using the relations
(−z) r = (−1) r (z − r + 1) r and (z) r+s = (z) r (z + r) s we have As a result, it follows that which can be reformulated as Identity (7).The proof of Identity ( 7) is complete.
Proof of Identity (2).Putting a = 1 2 and b = 2 in Equation ( 5) results in The proof of Identity (2) is complete.
Proof of Identity (4).Putting a = 1 2 and b = 2 in Equation (7) gives Further employing the duplication equation Remark 2. Please note, we recommend a newly-published paper [35] which is closely related to the Catalan numbers C n .
Remark 3. paper is a slightly revised version of the preprint [36] and has been reviewed by the survey article [37].

Conclusions
Three new identities for the Catalan-Qi numbers are discovered and alternative proofs of two identities for the Catalan numbers are provided.The three identities for the Catalan-Qi numbers generalize three identities for the Catalan numbers.
For the uniqueness and convenience of referring to the quantity C(a, b; x), we call the quantity C(a, b; x) the Catalan-Qi function and, when taking x = n ≥ 0, call C(a, b; n) the Catalan-Qi numbers.