Remarks on Homogeneous Al-Salam and Carlitz Polynomials

Several multilinear generating functions of the homogeneous Al-Salam and Carlitz polynomials are derived from -operator. In addition, two interesting relationships of product of this kind of polynomials are obtained. 1. Introduction The Al-Salam and Carlitz polynomials have been studied by many researchers for a long time. The history of these polynomials may go back to Al-Salam and Carlitz in 1965. Since then, these polynomials have been studied by many mathematicians [1–14]. Recently, Cao [7] used Carlitz’s -operators to study the following homogeneous Al-Salam and Carlitz polynomials: and he gave some linear generating functions of them. In this paper, we will research these polynomials by some construction of -operator. With this method, some new multilinear generating functions can be easily derived. Firstly, we give the following three results which originated from the results about which appeared in [1, 2, 5–7, 9, 10]. Theorem 1. If , , then Theorem 2. If , , then Theorem 3 (cf. [7, Equation (1.9)]). If , , then Now further using our method, we can deduce more results of multilinear generating functions. Theorem 4. If , , , then provided that , where . Theorem 5. If , , then provided that , , , , . Theorem 6. If , , , then Theorem 7. If , , , , then Theorem 8. If , , , , then Theorem 9. If , , , then where , , and . Theorem 10. If , , , and , then provided that , where . Polynomials (2) evidently reduce to the Rogers-Szeg？ polynomials (cf. [9]) when and . And when they reduce to the common Al-Salam and Carlitz polynomials (1) (cf. [9]). So now we take some special cases for checking. Let , in (3); we have the following. Corollary 11 (cf. [6, Theorem 1.1] or [9, Equation (4.1)]). If , then provided that . Remark 12 (from [6, Equation (3.1)]). We know that Corollary 11 is equivalent to Theorem 1.1 given in [6]. And if we take and , (3) turns to [9, Equation (1.2)]. Taking in (4), we have the following. Corollary 13 (cf. [6, Theorem 1.2] or [9, Equation (1.3)]). If , then provided that . Remark 14. Using Hall’s transformation [15, Equation ( )] we find that Corollary 13 is equivalent to Theorem 1.2 given in [6]. And if we take , then, with simplifying, (4) turns to [9, Equation (3.5)]. Taking and in (5), we have the following. Corollary 15 (cf. [6, Theorem 1.3] or [9, Equation (1.4)]). If , then provided that . The rest of this paper is organized as follows. In Section 2, we will give some notations and lemmas. In Section 3, we give the proofs of theorems. Section 4 describes the relationship between (cf. [4, Equation ]) and . In addition, an