The Cauchy Operator and the Homogeneous Hahn Polynomials

The Cauchy operator plays important roles in the theory of basic hypergeometric series. As some applications, our purpose is mainly to show new proofs of the Mehler’s formula, the Rogers formula and the generating function for the homogeneous Hahn polynomials Φ n (x, y|q)) by making use of the Cauchy operator and its properties. In addition, some interesting results are also obtained, which include a formal extension of the generating function for Φ n (x, y|q)).

The q-exponential function is given by The q-binomial coefficients are defined by n k = (q;q)n (q;q) k (q;q) n−k , 0 ≤ k ≤ n, 0, k ≥ n.
The q-Hahn polynomials Φ (α) n (x|q) were first studied by Hahn, and then by Al-Salam and Carlitz [1,10,11]. Now, we restate the definition of the q-Hahn polynomials as follows.
Definition 1.1. The q-Hahn polynomials are defined by According to the q-Hahn polynomials, we can easily obtain the homogeneous Hahn polynomials Φ (α) n (x, y|q). In this paper, we need to give the following definition of the homogeneous Hahn polynomials, again.
Definition 1.2. The homogeneous Hahn polynomials are defined by n (x|q). The usual q-differential operator is defined by and we further define D 0 q,x {f (x)} = f (x), and for n ≥ 1, D n q,x {f (x)} = D q,x D n−1 q,x {f (x)} . By q-differential operator D q,x , VY. B. Chen and NS. S. Gu gave the argumentation operator T (bD q,x ) = ∞ n=0 b n D n q,x (q) n [4,5].
Based on the definitions for D q,x and T (a, b; D q,x ) and the Leibniz rule for D q,x , we can easily obtain the following formulas.
Three Cauchy operator identities established by VY. B. Chen and NS. S. Gu are restated as the following lemmas [3].
In [3], the authors derived Heine's 2 φ 1 transformation formula and Sears' 3 φ 2 transformation formula by the symmetric property of some parameters in the above operator identities (8) (9). And further they obtain extensions of the Askey-Wilson integral, the Askey-Roy integral, Sears' twoterm summation formula.
In this paper, our main purpose is to make use of the above Cauchy operator identities (4)-(9) to give new proofs of the Mehler's formula, the Rogers formula and the generating function for the homogeneous Hahn polynomials. In addition, some interesting results are also derived, which include a formal extension of the generating function for Φ (α) n (x, y|q)).

Some Applications
In Liu's recent paper [13], the following first two known results were proved by using the theory of analytic functions of several complex variables. In this section, we first make use of the Cauchy operator identities (4) and (9) to prove them. n (x, y|q)). If max {|xt|, |yt|} < 1, then, we have that Proof. The identity (1) can be rewritten as follows: ∞ n=0 (ty) n (q) n = 1 (ty) ∞ .
Applying T (α, x; D q,y ) to both sides of the above identity with respect to y, we get ∞ n=0 T (α, x; D q,y ) {y n } t n (q) n = T (α, x; D q,y ) 1 (ty) ∞ .
Then, taking y = 1 in the above identity, we derive our desired result. This completes the proof. n (x, y|q)). For max {|xs|, |xt|} < 1, then, we have that This completes the proof. Next, we make use of Cauchy operator properties to derive some interesting results relevant to Φ  (10)). For max {|ty|, |tx|} < 1, then, we have that Proof. We begin with the q-binomial theorem ∞ n=0 (a) n (ty) n (q) n = (aty) ∞ (ty) ∞ .
Applying T (α, x; D q,y ) to both sides of the above identity with respect to the variable y, we get ∞ n=0 (a) n t n (q) n T (α, x; D q,y ) {y n } = T (α, x; D q,y ) (aty) ∞ (ty) ∞ .
By means of (4) and (9), we obtain our desired result. This completes the proof. This completes the proof.