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
Proof. LHS of Theorem 25 is equal to This completes the proof.
Remark 26. Theorem 25 can be reduced to the common -Mehler’s formula (cf. [1, Equation (1.6)]) by setting and . Employing this formula, we can show that the polynomials defined by Andrews [4] can be expressed by . Using Sear’s nonterminating transformation formula [15, page 241, Equation III.10] then the right-hand side of (49) reduces to Then setting , , and , we have (cf. [4, Theorem 4.1])
Comparison of the above identity with Theorem 4.1 that appeared in [4] gives the following.
Corollary 27 (cf. [4, Corollary 3.1]). If , , and , then Combined with Theorem 6, if we replace by , respectively, and then equate the coefficient of , we have the following.
Corollary 28. One has
Theorem 29 (cf. [1, Equation (1.17)]). If , then provided that , , , .
Proof. Applying Lemma 17 and then using -binomial theorem, we get that the LHS of (56) comes to
Twice applying Lemma 22, the above identity is equal to
For (58), by using -Leibnitz rule three times and using -binomial theorem, we can achieve the proof after simplification.
Taking , , we have the following.
Corollary 30 (cf. [9, Equation (4.4)]). One has
Remark 31. Corollary 30 is equivalent to [9, Equation (4.4)] after applying -Heine transformation formula to simplify. Note that the process of proof is only using -Leibnitz rule, so we can remove the restrictive condition in [7, Theorem 3]. Employing the same method, we recover Theorem 7 as follows.
Theorem 32. If , , , then
Proof . Using Lemma 17, LHS of (60) is equal to
Exchanging the order of summation and using -binomial, theorem (61) is equal to
Applying -Leibnitz and exchanging the order, (62) is equal to
By induction, similar proof can be performed to get the desired result.
Taking , in (60), we have the following.
Corollary 33. If , , , then
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The author is supported by Jiangsu Overseas Research and Training Program for University Prominent Young and Middle-Aged Teachers and Presidents and University National Natural Science Foundation of Jiangsu (no. 14KJB110002). The author is also supported by the National Natural Science Foundation of China (nos. 11371163 and 11001098).