#### Abstract

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-Szegö 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 interesting relationship between the polynomial multiplications is given.

#### 2. Notations and Some Lemmas

In this paper, we apply the standard notations that follow from [15] 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

The -derivative operator and -shifted operator (cf. [1, 6, 7, 9, 10, 16–23]), acting on the variable , are defined by

The -Leibnitz rule for the product of two functions (cf. [17–25]) is given 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
*

In [19, 20], we had established the following -operator structure: For convenient, we use to denote the operator acting on variable . From this definition we can easily get the following Lemma.

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
*

*Proof. *Applying Proposition 16, LHS of Lemma 18 comes to

We complete the proof.

*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

*Proof of Theorem 1. *LHS of (3) is equal to
We complete the proof.

*Proof of Theorem 2. *LHS of (4) is equal to
then using Lemma 24 and simplifying, we can get the desired result.

*Proof of Theorem 3. *LHS of (5) is equal to
This completes the proof.

*Proof of Theorem 4. *LHS of (6) is equal to

Setting
then by applying Lemma 18 and simplifying, we get
substituting it into (41), we complete the theorem.

*Proof of Theorem 5. *LHS of (7) is equal to

Applying Lemma 18 and simplifying, we get the theorem. This completes the proof.

*Proof of Theorems 6 and 7. *Using Lemma 17, then Theorem 6 can be obtained by Lemma 20 directly.

Using Lemma 17 and then applying Lemma 18, we can get Theorem 7.

*Proof of Theorem 8. *LHS of (10) is equal to
we complete the proof.

*Proof of Theorem 9. *LHS of (11) equates to
If we set
using Lemma 24, we get ; then employing Lemma 18, we can get the desired result.

*Proof of Theorem 10. *LHS of (12) equates to
then using Lemma 18 four times we can get the result.

#### 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.

Theorem 25 (cf. [1, Equation (1.17)] or [7, Lemma 12]). *One has
*