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.
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  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 .
So now we take some special cases for checking.
Let , in (3); we have the following.
Taking in (4), we have the following.
Remark 14. Using Hall’s transformation [15, Equation ()] we find that Corollary 13 is equivalent to Theorem 1.2 given in . And if we take , then, with simplifying, (4) turns to [9, Equation (3.5)].
Taking and in (5), we have the following.
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 interesting relationship between the polynomial multiplications is given.
2. Notations and Some Lemmas
In this paper, we apply the standard notations that follow from  and assume that ; the -shifted factorial and its compact factorials are defined, respectively, by and , where . And we use to denote the set of nonnegative integers and to denote the set of positive integers.
For any complex number , we have
The -binomial coefficient and the -binomial theorem are given by respectively.
The basic hypergeometric series is defined by
From definition, we can easily obtain ; by mathematical induction, we have the following proposition.
Proposition 16 (cf. [20, Equation (6)]). If is any analytical function, then
Lemma 17. One has
Proof. For we have which completes the proof.
It is the reason why we employ this operator to study the properties of homogeneous Al-Salam and Carlitz polynomials.
From Proposition 16, we may easily obtain the following identity.
Lemma 18. If , , and is any analytical function, then
Remark 19. To calculate the inner sum of the above, under the condition , -binomial theorem is usable.
Lemma 20 (cf. [20, Lemma 1]). If , , then
Proof. LHS of Lemma 20 equates to We complete the proof.
The special cases of Lemma 20 as taking , , and , respectively, will be frequently used.
Lemma 21. For ,
Lemma 22. If , then
Lemma 23. For , ,
Throughout this paper, we also often use the following property.
Lemma 24. If , , then
Proof. We find that LHS of Lemma 24 equates to This completes the proof.
3. Proof of Theorems
4. Some Other Cases and a Homogeneous -Mehler’s Formula
In this section we obtain Mehler’s formula involving a series and then give an interesting relationship between the polynomial multiplications.