- Research
- Open access
- Published:
On the ω-multiple Charlier polynomials
Advances in Difference Equations volume 2021, Article number: 119 (2021)
Abstract
The main aim of this paper is to define and investigate more general multiple Charlier polynomials on the linear lattice \(\omega \mathbb{N} = \{ 0,\omega ,2\omega ,\ldots \} \), \(\omega \in \mathbb{R}\). We call these polynomials ω-multiple Charlier polynomials. Some of their properties, such as the raising operator, the Rodrigues formula, an explicit representation and a generating function are obtained. Also an \(( r+1 )\)th order difference equation is given. As an example we consider the case \(\omega =\frac{3}{2}\) and define \(\frac{3}{2}\)-multiple Charlier polynomials. It is also mentioned that, in the case \(\omega =1\), the obtained results coincide with the existing results of multiple Charlier polynomials.
1 Introduction
In [3] the authors introduced the \(\delta _{\omega }\)-Appell polynomial sets which are defined by
where
They proved an equivalent definition in terms of the generating function:
where
It has also been shown in [3] that among all the \(\delta _{\omega }\)- Appell polynomials, d-orthogonal polynomial sets should have the generating function of the form
where \(H_{d}\) is a polynomial of degree d. In the special case
we have the polynomials generated as follows:
These polynomials can be called ω-Charlier polynomials, since the case \(\omega =1\) gives the usual Charlier polynomials.
On the other hand, in a recent paper, the multiple \(\Delta _{\omega }\)-Appell polynomials were defined [8] by the generating function
Inspired by these observations, in this paper we aim to introduce the ω-multiple Charlier polynomials starting from the multiple orthogonality relations with respect to the weight function of the form
and investigate certain of their properties such as raising operator, Rodrigues formula, explicit representation and generating function. We also obtain an \((r+1)\)th order difference equation and give some special examples for certain choices of ω. So it can be easily observed from the generating function of ω-multiple Charlier polynomials that these polynomials are examples of \(\Delta _{\omega }\)-multiple Appell polynomials.
We will start by recalling some basic knowledge about the discrete orthogonal and discrete multiple orthogonal polynomials.
The nth degree monic orthogonal polynomial \(p_{n}\) is defined by
where μ is a positive measure on the real line. In general, in the case of discrete orthogonal polynomials, the term \(x^{k}\) is replaced by \(( -x ) _{k}\), since \(\Delta ( -x ) _{k}=-k ( -x ) _{k-1}\), where
is the Pochhammer symbol and
is the forward difference operator.
The classical orthogonal polynomials (on a linear lattice) of a discrete variable are the Hahn, Meixner, Kravchuk and Charlier polynomials. The main concern of this paper is the Charlier polynomials.
The orthogonality measure (Poisson distribution) for Charlier polynomials is
with \(k\in \mathbb{N} \) (\(\mathbb{N} := \{ 0,1,2,\ldots \} \)) and \(a>0\).
The type II multiple orthogonal polynomials \(p_{\overrightarrow{n}}\) of degree \(\leq \vert \overrightarrow{n} \vert :=n_{1}+\cdots +n_{r}\) (\(r\geq 2 \)) with respect to r non-negative measures \(\mu _{1},\ldots ,\mu _{r}\) on \(\mathbb{R}\), are defined by
Here
and \(I_{i}\) (\(i=1,2,\ldots ,r \)) is the smallest interval containing \(\operatorname{supp} ( \mu _{i} ) \). Conditions (2) give \(\vert \overrightarrow{n} \vert \) linear equations for the \(\vert \overrightarrow{n} \vert +1\) unknown coefficients of \(p_{ \overrightarrow{n}}\). If \(p_{\overrightarrow{n}}\) is unique (up to a multiplicative factor) and has degree \(\vert \overrightarrow{n} \vert \), then \(\overrightarrow{n}\) is said to be normal. In general, the monic polynomials are considered.
In the case where we have r non-negative discrete measures on \(\mathbb{R} \):
where all \(x_{i,m}\) are different for each \(m=0,1,\ldots ,N_{i}\) (\(i=1,2,\ldots ,r \)), we have the discrete multiple orthogonal polynomials (on the linear lattice), and the above orthogonality conditions can be written as
where \(p_{\overrightarrow{n}}\) is a polynomial of degree \(\leq \vert \overrightarrow{n} \vert \).
In this paper, we pay attention to the AT system of r non-negative discrete measures; we recall its definition.
Definition 1.1
([1])
An AT system of r non-negative discrete measures is a system of measures
where \(\operatorname{supp} ( \mu _{i} ) \) (\(i=1,\ldots ,r \)) is the closure of \(x_{m}\) and the orthogonality intervals (2) are the same, namely I. It is also assumed that there exist r continuous functions \(w_{1},\ldots ,w_{r}\) on I with \(w_{i} ( x_{m} ) =\rho _{i,m}\) (\(m=1,\ldots ,N\), \(i=1,\ldots ,r \)) such that the \(\vert \overrightarrow{n} \vert \) functions
form a Chebyshev system on I for each multi-index \(\vert \overrightarrow{n} \vert < N+1\). This means that all the linear combinations of the form
where \(Q_{n_{i}-1}\) is a polynomial of degree \(\leq n_{i}-1\), has at most \(\vert \overrightarrow{n} \vert -1\) zeros on I.
Remark 1.1
If we have r continuous functions \(w_{1},\ldots ,w_{r}\) on I with \(w_{i} ( x_{m} ) =\rho _{i,m}\), then the orthogonality conditions (3) can be written as
As is pointed out in [1], in an AT system every discrete multiple orthogonal polynomials of type II corresponding to the multi-index \(\overrightarrow{n}\) has exact degree \(\vert \overrightarrow{n} \vert \), and every multi-index \(\overrightarrow{n}\) with \(\vert \overrightarrow{n} \vert < N+1\) is normal.
Recently, some discrete multiple orthogonal polynomials and their structural properties have been studied in [1]. Difference equations for discrete classical multiple orthogonal polynomials have been studied in [5]. In [7], the ratio asymptotics and the zeros of multiple Charlier polynomials have been investigated. Nearest neighbor recurrence relations for multiple orthogonal polynomials were investigated in [10]. The \((r+1)\)th order difference equations for the multiple Charlier and Meixner polynomials have been studied in [9]. Furthermore, in [2], the q-Charlier multiple orthogonal polynomials and some of their structural properties were studied.
The main aim of this paper is to extend the idea of discrete multiple orthogonality to more general linear lattice \(\omega \mathbb{N} = \{ 0,\omega ,2\omega ,\ldots \} \) for the ω-multiple Charlier polynomials. We note that in a recent paper this type of discrete orthogonality is used to define ω-multiple Meixner polynomials [11].
We organize the paper as follows: In Sect. 2, we define ω-multiple Charlier polynomials and obtain a raising operator and the Rodrigues formula for them. In Sect. 3, the explicit representation and generating function are given for the ω-multiple Charlier polynomials. In Sect. 4, recurrence relations are given. In Sect. 5, we obtain \(( r+1 )\)th order difference equations satisfied by ω-multiple Charlier polynomials. In Sect. 6, as an illustrative example, we consider the case \(\omega = \frac{3}{2}\) and exhibit our main results for this particular case. In the last section, it is shown that the special cases of the results obtained in Sects. 2, 3, 4 and 5 coincide with the corresponding results for multiple Charlier polynomials obtained in the earlier papers. Some concluding remarks are also stated.
2 Discrete ω-multiple Charlier orthogonal polynomials
In this section, we define ω-multiple Charlier polynomials. We present a raising operator and the Rodrigues formula for them. We start by defining the discrete multiple orthogonality on the linear lattice \(\omega \mathbb{N} = \{ 0,\omega ,2\omega ,\ldots \} \) (\(\omega >0 \)) and call them ω-multiple orthogonal polynomials.
Definition 2.1
The ω-multiple orthogonal polynomials are defined as
where ω is a fixed positive real number, \(\overrightarrow{n}= ( n_{1},\ldots ,n_{r} ) \) and \(p_{\overrightarrow{n}}\) is a polynomial of degree \(\vert \overrightarrow{n} \vert \) and
Now we choose the orthogonality measures as
where \(a_{1},\ldots ,a_{r}\) are different parameters and
is the k-gamma function [6].
For each measure the weights form an extended Poisson distribution on \(\omega \mathbb{N} \) (\(\omega \mathbb{N} = \{ 0,\omega ,2\omega ,\ldots \} \)). It is easily seen from Example 2.1 in [1] that these r measures form a \(Chebyshev\) system on \(\mathbb{R} ^{+}\) for every \(\overrightarrow{n}= ( n_{1},\ldots ,n_{r} ) \in \omega \mathbb{N} ^{r}\) since the weight functions,
are continuous and they have no zeros on \(\mathbb{R} ^{+}\). So every multi-index is normal and the monic solution is unique.
The corresponding multiple orthogonality conditions are given on \(\omega \mathbb{N} \) as
where \(\overrightarrow{n}=(n_{1},\ldots ,n_{r})\) and \(\overrightarrow{a} = ( a_{1},\ldots ,a_{r} ) \). We represent these polynomials by \(C_{ \overrightarrow{n}}^{\overrightarrow{a}}\) and call them ω-multiple Charlier orthogonal polynomials.
Theorem 2.2
The raising relation for the ω-multiple Charlier polynomials is given as
where \(\nabla _{\omega }f ( x ) =f ( x ) -f ( x- \omega ) \) and \(\overrightarrow{e}_{i}= ( 0,\ldots ,0,1,\ldots ,0 ) \).
Proof
Applying the product rule \(\nabla _{\omega } [ f ( x ) g ( x ) ] =f ( x ) \nabla _{\omega }g ( x ) +g ( x-\omega ) \nabla _{\omega }f ( x ) \), we have
Since \(\nabla _{\omega }w_{i} ( x ) =w_{i} ( x ) [ 1- \frac{x}{a_{i}^{\omega }} ] \), we get by using (6)
Hence
Applying the ω-summation by parts formula, which is
we get
Since \(\Delta _{\omega } ( -\omega x ) _{j,\omega }=-\omega j ( -\omega x ) _{j-1,\omega }\), we have
Then, for \(j=0,\ldots ,n\), the summation on the left will be zero from the ω-multiple orthogonality conditions. Hence
By the uniqueness of the ω-multiple orthogonal polynomials, we have
Considering the above equality in (7), the proof is completed. □
Theorem 2.3
The Rodrigues formula for the ω-multiple Charlier polynomials is given by
Proof
We will give the proof for the case \(r=2\). The proof of the general case is similar. Repeatedly using the raising operators, we find, since \(C_{0,0}^{a_{1},a_{2}} ( x ) =1\), that
Hence, we get (9) for \(r=2\). □
3 Explicit representation and generating function
In this section, we use the Rodrigues type formula (9) to give the explicit representation of the multiple ω-Charlier polynomials. Furthermore, we obtain the generating function for these polynomials.
Theorem 3.1
The explicit representation for the ω-multiple Charlier polynomials is given by
Proof
We will give the proof for \(r=2\). The general case (10) can be proved in a similar manner. Using (9) for \(r=2\), we write
Since \(\nabla _{\omega }^{n}f ( x ) =\sum_{i=1}^{n} ( -1 ) ^{i}\binom{n}{i}f ( x-i\omega ) \), we have
Whence the result. □
Corollary 3.2
Equation (11) can be written as
where
is the second Appell hypergeometric functions of two variables [4].
Theorem 3.3
The ω-multiple Charlier polynomials have the following generating function
Proof
Using the explicit form of the polynomials given in Theorem 3.1, we can write
Using the Cauchy product of the series, we get for \(\sum_{i=1}^{r} \vert t_{i} \vert <\omega ^{-r}\)
Whence the result. □
Remark 3.1
It can be easily seen from (1) and (12) that ω-multiple Charlier polynomials are an example of the \(\Delta _{\omega }\)-multiple Appell polynomials.
4 Recurrence relations
The main aim of this section is to obtain some recurrence relations for ω-multiple Charlier polynomials. Throughout this section, we concentrate on the case \(r=2\), since the proof techniques for the general r will be similar.
Proposition 4.1
Let \(G ( x,t_{1},t_{2} ) = ( 1+\omega t_{1}+\omega t_{2} ) ^{\frac{x}{\omega }}e^{- ( a_{1}^{\omega }t_{1}+a_{2}^{ \omega }t_{2} ) }\). We have the properties
and
Proof
The proofs can be given by elementary calculations. □
Theorem 4.2
The recurrence relations
and
hold for the ω-multiple Charlier polynomials.
Proof
Using (13), we get
Comparing the coefficients of \(\frac{t_{1}^{n_{1}}}{n_{1}!}\frac{ t_{2}^{n_{2}}}{n_{2}!}\), (15) follows.
The left hand side of (14) can be written as
The right hand side of (14) will be
Combining (18) and (19), we get
From (15), replacing \(n_{2}\) by \(n_{2}-1\) and \(n_{1}\) by \(n_{1}-1\), we have
and
respectively.
Using (21) and (22), we get (16).
Using (22), we have
5 Difference equations for ω-multiple Charlier polynomials
In this section, we obtain the \(( r+1 )\)th difference equation for ω-multiple Charlier polynomials. As a corollary, we give the third order difference equation for the case \(r=2\). We start with the following theorem which will be needed for the main result.
Theorem 5.1
The raising operator can be rewritten as
where \(\overrightarrow{e_{i}}= ( 0,\ldots ,0,1,\ldots ,0 ) \) and \(L_{a_{i}} [ \cdot ] \) is defined by
Proof
From the raising relation (5), we have
Applying the ω-product rule, we can write
Since \(\nabla _{\omega }w_{i} ( x ) =w_{i} ( x ) [ 1- \frac{x}{a_{i^{\omega }}} ] \), we get by using (25)
Hence
and therefore
where
This completes the proof. □
Theorem 5.2
The lowering operator of the polynomials is determined from the following relation:
where \(\overrightarrow{e_{i}}= ( 0,\ldots ,1,\ldots ,0 ) \).
Proof
Applying \(\Delta _{\omega }\) on both sides of (12), we get
Comparing the coefficients of \(\frac{t_{1}^{n_{1}}t_{2}^{n_{2}}\ldots t_{r}^{n_{r}}}{n_{1}!n_{2}!\ldots n_{r}!}\), we get the result. □
Corollary 5.3
In particular, if \(r=2\),
Theorem 5.4
The ω-multiple Charlier polynomial \(\{ C_{\overrightarrow{n}}^{ \overrightarrow{a}} ( x ) \} _{ \vert n \vert =0}^{\infty }\) satisfies the following \(( r+1 ) \) order difference equation:
where \(L_{a_{i}} [ \cdot ] \) is the raising operator (\(i=1,\ldots,r \)) given in Theorem 5.1..
Proof
Applying \(L_{a_{1}}\ldots L_{a_{r}}\) to both sides of (26), we get
Since \(L_{a_{j}}L_{a_{k}} ( y ) =L_{a_{k}}L_{a_{j}} ( y ) \) for \(a_{j},a_{k}\in \mathbb{R} \), we obtain for \(i=1,2,\ldots ,r\)
Hence
Using (24) with \(\overrightarrow{n}\) replaced by \(\overrightarrow{n} -\overrightarrow{e_{i}}\), we get the result. □
Corollary 5.5
The ω-multiple Charlier polynomial \(\{ C_{n_{1},n_{2}}^{a_{1},a_{2}} ( x ) \} _{n_{1}+n_{2}=0}^{ \infty }\) satisfies the following difference equation:
6 Special cases of the ω-multiple Charlier polynomials
In this section, as an illustrative example of our new definition and its main results, we consider the case \(\omega =\frac{3}{2}\) and define \(\frac{3 }{2}\)-multiple Charlier polynomials. The corresponding consequences of our main results for \(\frac{3}{2}\)-multiple Charlier polynomials are also given.
Taking the weight function as
we can define the \(\frac{3}{2}\)-multiple Charlier polynomial by the following orthogonality conditions:
Their explicit representation can be written from Theorem 3.1 as
The generating function of the \(\frac{3}{2}\)-multiple Charlier polynomials is written from Theorem 3.3 as
Their recurrence relations can be written from Theorem 4.2 as
and
The difference equation of the \(\frac{3}{2}\)-multiple Charlier polynomials for the case \(r=2\) is
7 Concluding remarks and observations
The multiple Charlier polynomials \(C_{\overrightarrow{n}}^{\overrightarrow{a} }\) were introduced in [1]. The raising operators and Rodrigues formula were obtained. The explicit representation, recurrence relation and generating function were investigated in [1] and [7]. Also an \(( r+1 )\)th order difference equation was investigated in [5].
In this paper, we define the ω-multiple Charlier polynomials by the orthogonality condition (4). We obtain the raising relation, the Rodrigues formula, an explicit representation, a recurrence relation and a generating function. Also an \(( r+1 )\)th order difference equation was obtained. All our results coincide in the case \(\omega =1\) with the corresponding versions of the multiple Charlier polynomials. For instance, this is so in the case \(\omega =1\).
The raising relation (5) coincides with the raising operators given in ([1], pp. 30). That is
where
The Rodrigues formula (9) coincides with the Rodrigues formula given in ([1], pp. 31). That is,
The explicit representation (10) coincides with the explicit representation given in ([7], pp. 824). That is,
The recurrence relation (\(r=2 \)) (16) coincides with the recurrence relation (\(r=2 \)) given in ([1], pp. 32). That is,
The generating function (12) coincides with the generating function given in ([7], pp. 825). That is,
The third order difference equation (27) coincides with the third order difference equation in ([5], pp. 137). That is,
Availability of data and materials
Data sharing not applicable to this paper as no datasets were generated or analyzed during the current study.
References
Arvesu, J., Coussement, J., Van Assche, W.: Some discrete multiple orthogonal polynomials. J. Comput. Appl. Math. 153, 19–45 (2003)
Arvesu, J., Ramirez-Aberasturis, A.M.: On the q-Charlier multiple orthogonal polynomials. SIGMA 11, 026 (2015)
Cheikh, B.Y., Zaghouani, A.: Some discrete d-orthogonal polynomial sets. J. Comput. Appl. Math. 156, 253–263 (2003)
Erdelyi, A.: Higher Transcendental Functions, Vol. I. McGraw-Hill, New York (1953)
Lee, D.W.: Difference equations for discrete classical multiple orthogonal polynomials. J. Approx. Theory 150, 132–152 (2008)
Mubeen, S., Rehman, A.: A note on k-gamma function and Pochhammer k-symbol. J. Inform. Math. Sci. 6, 93–107 (2014)
Ndayiragije, F., Van Assche, W.: Asymptotics for the ratio and the zeros of multiple Charlier polynomials. J. Approx. Theory 164, 823–840 (2012)
Sadjang, N.P., Mboutngam, S.: On multiple \(\Delta _{\omega }\)- Appell polynomials. arXiv:1806.00032v1 [math.CA]
Van Assche, W.: Difference equations for multiple Charlier and Meixner polynomials. In: Proceedings of the Sixth International Conference on Difference Equations, Augsburg, Germany, pp. 549–557 (2001)
Van Assche, W.: Nearest neighbor recurrence relations for multiple orthogonal polynomials. J. Approx. Theory 163, 1427–1448 (2011)
Zorlu, O.S., Elidemir, İ.: On the ω-multiple Meixner polynomials of the first kind. J. Inequal. Appl. 2020, 167 (2020)
Funding
Not available.
Author information
Authors and Affiliations
Contributions
The authors declare that the study was realized in collaboration with equal responsibility. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Ethics approval and consent to participate
Not applicable.
Competing interests
The authors declare that they have no competing interests.
Consent for publication
Not applicable.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Özarslan, M.A., Baran, G. On the ω-multiple Charlier polynomials. Adv Differ Equ 2021, 119 (2021). https://doi.org/10.1186/s13662-021-03278-z
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13662-021-03278-z