#### Abstract

We introduce a -deformation of the Yang and Youn matrix approach for Appell polynomials. This will lead to a powerful machinery for producing new and old formulas for -Appell polynomials, and in particular for -Bernoulli and -Euler polynomials. Furthermore, the --polynomial, anticipated by Ward, can be expressed as a sum of products of -Bernoulli and -Euler polynomials. The pseudo -Appell polynomials, which are first presented in this paper, enable multiple -analogues of the Yang and Youn formulas. The generalized -Pascal functional matrix, the -Wronskian vector of a function, and the vector of -Appell polynomials together with the -deformed matrix multiplication from the authors recent article are the main ingredients in the process. Beyond these results, we give a characterization of -Appell numbers, improving on Al-Salam 1967. Finally, we find a -difference equation for the -Appell polynomial of degree .

#### 1. Introduction

In this paper we will introduce the -Pascal and -Wronskian matrices in a general setting, with the aim of fruitful applications for -Appell polynomials. These two matrices contain a certain -difference operator, just like the -deformed Leibniz functional matrix from [1]. By the -Leibniz formula, products of such matrices contain a certain operator , just as in the previous article, but with a slightly different definition. However, since in almost all equations we compute the function value at , this operator will convert to ordinary matrix multiplication by formula (6). There is a connection to the -Pascal matrix [2] for the special case -exponential function. The Appell polynomials are seldom encountered in the literature; one exception was Carlitz’ article [3], where a formula for the product of two generating functions was given. Carlitz cleverly observed that the generating function for Appell polynomials can be inverted when . The NWA -addition seldom occurs in the literature, except for the author’s papers; it was first seen in an Italian paper by Nalli [4]. Corresponding to the two dual -additions, NWA and JHC, there are two dual -Bernoulli polynomials, with the same names; these are special cases of -Appell polynomials, the content of Section 2. It turns out that for certain purposes we will need pseudo -Appell polynomials, which correspond to the other basic -addition, JHC; these polynomials were first seen in [5]. The -Appell numbers form a certain algebraic structure, with two operations, which was first described by Al-Salam [6]. This structure will in a certain way be generalized to -Appell polynomials. The -Bernoulli and -Euler polynomials are presented along with the -Lucas polynomials; these definitions coincide with those given in [7]. In Section 3 we come to the core of the paper, which starts with the -Pascal matrix and the -Wronskian vector. In the beginning, we briefly pass to the shorter paper [8], to find a few -analogues of truncated -exponential generating function formulas. We mostly follow the ordering in [9]. Sometimes multiple -analogues follow, and these will usually come with the pseudo -Appell polynomials. A couple of general formulas for -Appell polynomials are given; sometimes these are specialized to -Bernoulli and -Euler polynomials. In the last part, we find a -difference equation for general -Appell polynomials. In an appendix we compare our formulas for Bernoulli and Euler polynomials with Hansen’s table [10], where another notation is used.

We now start with the definition of the umbral method from the recent book [11]. Some of the notations are well known and will be skipped.

*Definition 1. *Let the Gauss -binomial coefficient be defined by
Let and be any elements with commutative multiplication. Then the NWA -addition is given by
The JHC -addition is the function:
The -exponential function is defined by

The -derivative is defined by
Assume that , , and are the variables. If we want to indicate the variable on which the -difference operator is applied, we denote the operator in (5) by . To save space, we sometimes write instead of .

Theorem 2. *The following equation relates the th -difference operator at zero to the th derivative at zero for an analytic function [12],
*

For convenience, we will use the following two abbreviations for functions : represents the th -derivative of and represents the th power of . We will only consider functions in . Sometimes functions can be given in other forms, we then mean the Taylor expansion of the function around zero. In the sense of Milne-Thomson [13] and Rainville [14] we write to indicate the symbolic nature.

If , the ring of polynomials with complex coefficients, then the function is defined by We make the convention that all matrices are denoted by the first index . The rows and columns are denoted by and . The -Pascal matrix [2] is defined by The Polya-Vein matrix [2] is given by

#### 2. -Appell Polynomials

The Appell polynomials have been around for more than a century, but this general notion is still not so well known. The usual definition is The original article by Appell [15] started with the definition . Then Appell showed that multiplication of two Appell polynomials is commutative. He then considered the inverse of as a natural generalization of ordinary division. The -Appell polynomials form a natural generalization of the Appell polynomials.

##### 2.1. Definitions

We consider a certain value of the order () for the -Appell polynomials in [11]. The degree is only a name; it is not the degree of polynomials.

*Definition 3. *The -Appell polynomials of degree are defined by
We say that is the -Appell polynomial sequence for .

We put and obtain
where is said to have degree .

*Definition 4. *The pseudo -Appell polynomials of degree [5] are defined by
We say that is the pseudo -Appell polynomial sequence for .

By putting , we have
where is called a number of degree .

We have similarly

##### 2.2. -Analogue of Carlitz’ Product Formula (1963)

To motivate the generalized -Pascal functional matrix, we show that the Carlitz [3, page 247] formula for the product of generating functions for Appell polynomials has a -analogue involving the NWA -addition. Define the numbers by the inverse of (12) (this is allowable, since ): Then we have the following.

Theorem 5. *An inverse relation formula for the -Appell polynomials. *

*Proof. *By the umbral definition of -Appell polynomials we have
This completes the proof.

We introduce two other -Appell polynomials and by Let us consider Now put Because of (17) we get Introduce the notation where

Theorem 6. *With the previously mentioned notation, one has the following -analogue of Carlitz [3, page 247]:
*

*Proof. *Thus we have

##### 2.3. Characterization of -Appell Numbers

We will now present a characterization of -Appell numbers compare with Al-Salam [6].

*Definition 7. *We denote by the set of all -Appell-numbers.

Let and be two elements in . Then the operations + and are defined as follows:

Theorem 8. *The identity element in with respect to is . This corresponds to .*

*Proof. *Put and in (28).

Theorem 9 (Compare with [6, page 34]). *Let , and be three elements in corresponding to the generating functions , , and , respectively. Further assume that . Then
**
The associative law for addition reads:
*

*Proof. *This follows from the definitions.

Theorem 10 (Compare with [6, page 34]). *Let , , and be three elements in corresponding to the generating functions , , and , respectively. Then
**
The associative law for reads:
*

*Proof. *We prove (31) and (33) at the same time. We have in the umbral sense

Formulas (32) and (34) follow by the commutativity and the associativity of NWA.

##### 2.4. Specializations

*Definition 11. *The NWA -Bernoulli polynomials of degree are defined by

The NWA -Bernoulli polynomials are also given by
The pseudo--Bernoulli polynomials are defined by

The JHC -Bernoulli polynomials of degree are defined by

#### 3. -Pascal and -Wronskian Matrices

We now come to the most important part of the paper, which is of interest even for the case , since the original paper by Yang and Youn is not very widespread.

*Definition 12. *The generalized -Pascal functional matrix is given by

In the special case constant,

*Definition 13. *The -deformed Wronskian vector is given by

*Definition 14. *The -deformed Leibniz functional matrix is given by

In order to find a -analogue of the formula [16]
we infer that by the -Leibniz formula [1]
where in the matrix multiplication for every term which includes , we operate with on . We denote this by . This operator can also be iterated, compare with [1].

Theorem 15 (a -analogue of [9, page 67]). *
Consider**
where the -Pascal matrix is defined in (8).**The mappings and are linear, that is,
**One has the following two matrix multiplication formulas: the first one is a -analogue of [8, page 232],
**
The scaling properties can be stated as follows:
*

*Proof. *The formula (49) is proved by the -Leibniz theorem. For (50) use the chain rule.

*Definition 16. *Assume that . If the inverse exists for , we can define the -inverse of the generalized -Pascal functional matrix as

*Example 17 (a -analogue of [8, pages 231–234]). *Let
Put
a truncated -exponential generating function of -binomial type. We find that
By a routine computation we obtain
The function argument is induced from the corresponding -addition.

In the same way we obtain
By the last formula we can define the inverse of as

We now return permanently to [9].

*Definition 18 (compare with [9, page 71]). *Let be the -Appell polynomial corresponding to in (11). The vector of -Appell polynomials for is given by

Theorem 19 (a -analogue of [9, page 69]). *Let be analytic in the neighbourhood of , with . The -difference equation in ,
**
with initial value , has the following solution:
*

*Proof. *Since is equal to the unit matrix, the solution of (60) is
By (47) and (48)
where the power is to be interpreted as products with .

Theorem 20 (a -analogue of [9, page 71]). *Let be a vector of length . Then is the -Appell polynomial vector for if and only if
*

*Proof. *We know from [2, 17] that
We thus obtain

*Definition 21 (A -analogue of [9, page 74]). *Let be the -Appell polynomial vector for . Then the -Pascal matrix for the -Appell polynomial vector for is defined by

Can be compared with (53).

Theorem 22 (a -analogue of [9, pages 74-75]). *Let , , and be the -Appell polynomials corresponding to , , and , respectively.**Further assume that .**Then
*

*Proof. *By formula (49) we get

By (11) the left-hand side of (69) is equal toBy (11) the right-hand side of (69) is equal to
The theorem follows by equating the last rows of formulas (71) and (70).

Formula (68) is almost a generalization of (28).

There is a variation of this formula with JHC, which enables inverses.

Theorem 23 (another -analogue of [9, pages 74-75]). *Let and be two -Appell polynomial sequences for and , respectively. Let be the pseudo -Appell polynomial for . Furthermore assume that . Then
*

*Proof. *By formula (49) we get

By (11) and (13) the left-hand side of (73) is equal toBy (11) the right-hand side of (73) is equal to
The theorem follows by equating the last rows of formulas (75) and (74).

The following formula from [18, equation (50), page 133] for follows immediately from (68).

Corollary 24 ([11, 4.242]). *When ,
**
where we assume that operates on .*

Yang and Youn have found a new identity between Bernoulli and Euler polynomials: The -analogue looks as follows.

Corollary 25 (a -analogue of [9, page 75 (18)]). *Consider
*

*Proof. *We put in and in (68). Then
This is obviously the same as the left hand side of the generating function for -numbers.

Corollary 26 (a -analogue of [9, pages 85-86]). *Let be a -Appell polynomial and a pseudo -Appell polynomial. Then
**
Let and be -Appell polynomials. Then
*

*Proof. * Formula (80) is proved as follows. We put in (72). The RHS of (72) is then equal to the LHS of (80). We finish the proof by using formula (49). Formula (81) is proved as follows. Use (68) with and . Or use an equivalent umbral reasoning.

Corollary 27. *One has the following special cases:
*

Corollary 28 (a -analogue of [9, page 87]). *Consider the -Appell polynomial and the pseudo--Appell polynomial . Then*

*Proof. * Let be the -Appell polynomial for and the pseudo--Appell polynomial for . By formula (50),On the other hand, by formula (49), we obtainThe theorem follows by multiplying both sides of (85) by and equating the last rows of formulas (86) and (87).

We can easily write down specializations of (85) to (pseudo) -Bernoulli and -Euler polynomials. One example is

Theorem 29 (the -connection constant theorem, a -analogue of [9, page 77]). *Let and be the -Appell vectors corresponding to and , respectively. Then
*

*Proof. *By formula (49) we get

Corollary 30 (a -analogue of [9, page 77]). *A relationship between Bernoulli and Euler polynomials
*

*Proof. *We put in and in (89). Then
We find that
The result follows by (89).

Theorem 31 (a -analogue of [9, page 91]). *Let be the -Appell polynomial sequence for . Then satisfies the following -difference equation:
**
where .*

*Proof. * We put :By equating the first and last expression for the last rows of formula (95) we get ()
Operating with on this and putting ,
where we have used the following identities:

#### 4. Conclusion

By finding -analogues of Yang and Youn [9], we have also shown that Yang-Youn paper contains a lot of new and interesting formulas. It is our hope that this paper together with future articles will point out the interplay between special functions and linear algebra. The formulas in Corollary 28 are in the same style as Nørlund’s [18] formulas, which are proved using umbral calculus. However, our formulas (78), Corollary 30, and (91) are of a more intricate nature. This shows that the matrix approach in this paper is not just a restatement of Nørlund’s [18] umbral calculus in -deformed form but a generalization. We also see that in order to obtain -analogues of the Yang and Youn [9, pages 85-87] formulas, we had to use (pseudo) -Bernoulli and -Euler polynomials, which were first mentioned in [5].

#### Appendix

#### Bernoulli Number Formulas in Hansen

In Hansens impressive table [10, pages 331–347], several formulas for Bernoulli and Euler numbers are listed. We show that some of these are connected to our formulas. For , formula (83) can be written as follows: Hansen [10, page 341, 50.11.2] writes this as follows: Hansen [10, page 346, 51.6.2] also has a corresponding result for the Euler polynomials.