Journal of Mathematics

Journal of Mathematics / 2014 / Article

Research Article | Open Access

Volume 2014 |Article ID 523013 | 12 pages | https://doi.org/10.1155/2014/523013

Remarks on Homogeneous Al-Salam and Carlitz Polynomials

Academic Editor: Fawang Liu
Received02 Apr 2014
Accepted04 Jul 2014
Published17 Jul 2014

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 [114].

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, 57, 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, 1623]), acting on the variable , are defined by

The -Leibnitz rule for the product of two functions (cf. [1725]) 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