On the $q$-Charlier Multiple Orthogonal Polynomials

We introduce a new family of special functions, namely $q$-Charlier multiple orthogonal polynomials. These polynomials are orthogonal with respect to $q$-analogues of Poisson distributions. We focus our attention on their structural properties. Raising and lowering operators as well as Rodrigues-type formulas are obtained. An explicit representation in terms of a $q$-analogue of the second of Appell's hypergeometric functions is given. A high-order linear $q$-difference equation with polynomial coefficients is deduced. Moreover, we show how to obtain the nearest neighbor recurrence relation from some difference operators involved in the Rodrigues-type formula.


Introduction
Let µ = (µ 1 , . . . , µ r ) be a vector of r positive Borel measures on R, and let n = (n 1 , . . . , n r ) ∈ N r be a multi-index. By N we denote the set of all nonnegative integers. A type II multiple orthogonal polynomial P n , corresponding to the multi-index n, is a polynomial of degree ≤ | n| = n 1 + · · · + n r which satisfies the orthogonality conditions [16] Ωi P n (x)x k dµ i (x) = 0, k = 0, . . . , n i − 1, i = 1, . . . , r, (1.1) where Ω i is the smallest interval that contains supp (µ i ). In this paper we will consider the situation when P n is a monic multiple orthogonal polynomial and has exactly degree | n|. If the measures in (1.1) are discrete where δ x i,k denotes the Dirac delta function and x i1,k = x i2,k , k = 0, . . . , N i , whenever i 1 = i 2 , the corresponding polynomial solution is called discrete multiple orthogonal polynomial. In [5] was studied some type II discrete multiple orthogonal polynomials on the linear lattice x(s) = s. In particular, multiple Charlier polynomials were considered (when the component measures of µ are different Poisson distributions). In this paper we will introduce a q-analogue of such multiple orthogonal polynomials (when the component measures of µ are different q-Poisson distributions) and study their algebraic properties (structural properties). We are motivated by the recent applications that have been found for their predecessors: Multiple Charlier polynomials. In particular, these polynomials appear in remainder Padé approximation for the exponential function [18] and as common eigenstate of a set of r non-Hermitian oscillator Hamiltonians [13]. Furthermore, in [14] the authors pointed out a possible relationship of these polynomials to the orthogonal functions appearing in two speed totally asymmetric simple exclusion process (TASEP) [6]. Our new q-family of multiple orthogonal polynomials is likely to be of relevance in TASEP-like models as well as in q-extensions of the mathematical problems addressed in [13,18].
Recently, two families of q-multiple orthogonal polynomials and some of their structural properties have been studied [3,17]. Moreover, in [4,12,19] an (r + 1)-order difference equation for some discrete multiple orthogonal polynomials was obtained. An interesting fact is that these polynomials are common eigenfunctions of two distinct linear difference operators of order (r + 1) since multiple orthogonal polynomials also satisfy (r + 2)-term recurrence relations. The explicit expressions for the coefficients of this relation are the main ingredient for the study of some type of asymptotic behaviors for these polynomials. For instance, in [2] the weak asymptotics was studied for multiple Meixner polynomials of the first and second kind, respectively. The zero distribution of multiple Meixner polynomials was also studied. In [14] the recurrence relation is also used for studying the ratio asymptotic behavior of multiple Charlier polynomials (introduced in [5]) and from it the authors obtain the asymptotic distribution of the zeros after a suitable rescaling. Furthermore, this recurrence relation is a key-ingredient for attaining a Christoffel-Darboux kernel [7] among other applications, which plays important role in correlation kernel (for instance in the unitary random matrix model with external source).
In this paper, we will mainly focus on two structural properties satisfied by the multiple orthogonal polynomials introduced here, namely recurrence relations and the difference equation (with respect to the independent variable). The asymptotic analysis is out of the scope of this paper. Although we plan to address this question in a future publication.
The structure of the paper is as follows. Section 2 introduces the context and the background materials. In section 3 we will define the q-Charlier multiple orthogonal polynomials (new in the literature) and obtain raising operators and then the Rodrigues-type formula for these special functions when r orthogonality conditions are considered. The explicit series representation in terms of the second of q-Appell's hypergeometric functions is given for n = (n 1 , n 2 ). In section 4, an (r + 1)-order q-difference equation on a non-uniform lattice x(s) is obtained. Section 5 deals with the (r + 2)-term recurrence relations. In particular, the nearest neighbor recurrence relation is obtained. Explicit expressions for the recurrence coefficients are given. Finally, in section 6 some of our findings are summarized. A special limiting case when the parameter q involved in the components of the vector measure tends to one is pointed out. Under this limit the corresponding structural properties for the multiple Charlier polynomials are recovered.

Multiple Charlier polynomials
For the discrete measures (1.2) we have that supp (µ i ) is the closure of {x i,k } Ni k=0 and that Ω i is the smallest closed interval on R which contains {x i,k } Ni k=0 . Moreover, the above orthogonality conditions (1.1) give a linear system of | n| homogeneous equations for the | n|+1 unknown coefficients of P n (x). This polynomial solution P n always exists. We focus our attention on a unique solution (up to a multiplicative factor) with deg P n (x) = | n|. If this happen for every multi-index n, we say that n is normal [16]. If the above system of measures forms an AT system [16] then every multi-index is normal. Indeed, we will deal with such system of discrete measures, where Ω i = Ω for each i = 1, 2, . . . , r.
In [5] the authors consider a normal multi-index n ∈ N r , whenever α i > 0, i = 1, 2, . . . , r, and with all the α i different. Moreover, it was found the following raising operators and ▽f (x) = f (x) − f (x − 1) denotes the backward difference operator. As a consequence of (2.1) there holds the Rodrigues-type formula Two important structural properties are known for multiple Charlier polynomials [5], namely the (r + 1)-order linear difference equation [12] and the recurrence relation [5,10] where the multi-index e i is the standard r dimensional unit vector with the i-th entry equals 1 and 0 otherwise. Interestingly, the multiple Charlier polynomials C α n (x) are common eigenfunctions of the above two linear difference operators of order (r + 1), namely (2.3) and (2.4).

q-Charlier multiple orthogonal polynomials
Let us begin by recalling the definition of q-multiple orthogonal polynomials [3].
is said to be a q-multiple orthogonal polynomial of a multi-index n ∈ N r with respect to positive discrete measures µ 1 , µ 2 , . . . , µ r such that supp (µ i ) ⊂ Ω i ⊂ R, i = 1, 2, . . . , r, if there hold conditions (a) deg P n (x(s)) ≤ | n| = n 1 + n 2 + · · · + n r , Let us consider the following r positive discrete measures on R + , Here where α i > 0, i = 1, 2, . . . , r, with all the α i different. Recall that the q-Gamma function is given by See also [9,15] for the above definition of the q-Gamma function.
Lemma 3.2. The system of functions , form a Chebyshev system on R + for every n = (n 1 , . . . , n r ) ∈ N r .
Then, we conclude that the functions (3.4) form a Chebyshev system on R + .
As a consequence of Lemma 3.2 the system of measures µ 1 , µ 2 , . . . , µ r given in (3.2) forms an AT system on R + . Using this fact, we will rewrite Definition 3.1 for this system of measures. However, we first define the q-analogue of the Stirling polynomials denoted by [s] Definition 3.3. A polynomial C α q, n (s), with multi-index n ∈ N r and degree | n| that verifies the orthogonality conditions  .2)) is said to be the q-Charlier multiple orthogonal polynomial.
Let us point out some observations derived from the above definition. When r = 1 we recover the scalar q-Charlier polynomials [1]. The orthogonality conditions (3.1) have been written more conveniently as (3.6) since [s] Here, the symbol O(·) stands for big-O notation. Indeed, when q goes to 1, the symbol [s] (k) q converges to (−1) k (−s) k , where (s) k denotes the Pochhammer symbol. Hence, one can recover the multiple Charlier polynomials given in [5] as a limiting case, provided that the lattice x(s) = (q s − 1)/(q − 1). Moreover, in the sequel we will only use this lattice. Finally, we have an AT-system of positive discrete measures, then the q-Charlier multiple orthogonal polynomial C α q, n (s) has exactly | n| different zeros on R + (see [5,Theorem 2.1,). For monic q-Charlier multiple orthogonal polynomials we have r raising operators where , α/q = (α 1 /q, . . . , α r /q) . Furthermore, for any function f (s) defined on the discrete variable s. Notice that we call D αi q a raising operator since the i-th component of the multi-index n in (3.7) is increased by 1. For finding this operator we have replace [s] (k) q in (3.6) by the following finite-difference expression and then used summation by parts along with conditions υ αi q (−1) = υ αi q (∞) = 0. In the sequel we will only consider monic q-Charlier multiple orthogonal polynomials. In addition, the following difference operators will appear regularly and ▽x 1 (s) = ▽x(s + 1/2) = △x(s − 1/2) = q s−1/2 . Proposition 3.4. There holds the following q-analogue of Rodrigues-type formula C α q, n (s) = K n, α q q s| n| Γ q (s + 1)C α q, n 1 Γ q (s + 1) , Hence, using summation by parts and the raising operators (3.7) in a recursive way one obtains the Rodrigues-type formula (3.11).

High-order q-difference equation
In this section we will find a lowering operator for the q-Charlier multiple orthogonal polynomials. Then we will combine it with the raising operators (3.7) to get an (r + 1)-order q-difference equation (in the same fashion that [4] and [12]). is a basis for V.

Proof. From orthogonality relations
we have that polynomials C α/q −1 q, n− ei (s), i = 1, . . . , r, belong to V. Now, aimed to get a contradiction, let us assume that there exists constants λ i , i = 1, . . . , r, such that

Then, multiplying the previous equation by [s]
(n k −1) q υ α k /q −1 q (s) ▽ x 1 (s) and then taking summation on s from 0 to ∞, one gets

Thus, taking into account relations
we deduces that λ k = 0 for k = 1, . . . , r. Here δ i,k represents the Kronecker delta symbol. Therefore, is linearly independent in V. Furthermore, we know that any polynomial of V can be determined with | n| coefficients while (| n| − r) linear conditions are imposed on V, consequently the dimension of V is at most r. Hence, the system C spans V, which completes the proof. Now we will prove that operator (3.9) is indeed a lowering operator for the sequence of q-Charlier multiple orthogonal polynomials C α q, n (s). Lemma 4.2. There holds the following relation Proof. Using summation by parts we have is a polynomial of degree ≤ k + 1 in the variable x(s). Consequently, from the orthogonality conditions Hence, from Lemma 4.1, ∆C α q, n (s) ∈ V. Moreover, ∆C α q, n (s) can univocally be expressed as a linear combination of polynomials C

Multiplying both sides of the equation (4.4) by [s]
(n k −1) q υ α k /q −1 q (s) ▽ x 1 (s) and using relations (4.1) one has ∞ s=0 ∆C α q, n (s)[s] (n k −1) (4.6) Here we have used that x(s)[s − 1] (n k ) q + lower terms. On the other hand, from (3.7) one has that Then, by conveniently substituting (4.7) in the right-hand side of equation (4.6) and using once more summation by parts, we get ∞ s=0 ∆C α q, n (s)[s] (n k −1) Therefore, comparing this equation with (4.5) we obtain the coefficients in the expansion (4.4), i.e.
Theorem 4.3. The q-Charlier multiple orthogonal polynomial C α q, n (s) satisfies the following (r +1)-order q-difference equation Proof. Since operators (3.7) are commuting, we write Then, by acting on equation (4.2) with the product of operators (4.9), we obtain the following relation which proves (4.8).

Recurrence relation
In this section we will study two types of recurrence relations, namely the nearest neighbor recurrence relation for any multi-index n and a step-line recurrence relation for n = (n 1 , n 2 ). For the main result of this section we will make use of the following Lemma.
Lemma 5.1. Let n i be a positive integer and let f (s) be a function defined on the discrete variable s. The following relation is valid where difference operator C αi q,ni is given in (3.12). Proof. Let us act n i -times with backward difference operators (3.10) on the product of functions x(s)f (s). Assume momentarily that n i ≥ N > 1, Repeating this process -but on the second term of the right-hand side of equation (5.2)-yields or equivalently, Now, aimed to involve difference operator C αi q,ni in the above equation, we multiply equation (5.3) from the left side by (α i q ni ) −s and replace f (s) by (α i q ni ) s f (s). Thus, equation (5.3) transforms into (5.1), which completes the proof.
Proof. We will use Lemma 5.1. First, let us consider equation which can be rewritten in terms of difference operators (3.12) as follows .
Since operators (3.12) are commuting the multiplication of equation (5.5) from the left side by the product .
Finally, multiplying both sides of the previous expression by K n, α q q | n|s Γ q (s + 1) and using Rodriguestype formula (3.11) we obtain (5.4), which completes the proof.
Observe that other recurrence relations different from the above nearest neighbor recurrence relation (5.4) can be obtained. Indeed, from (3.15) a 4-term recurrence relation for n = (n 1 , n 2 ) can be obtained. In [5] it was given an approach for the recurrence relations of some discrete multiple orthogonal polynomials. This approach make use of the Rodrigues-type formulas along with some series representations (Kampé de Fériet series) for multiple orthogonal polynomials. Here we will proceed in the same fashion.
Here, when q → 1 we recover the recurrence coefficients given in [5].

Conclusions
In summary, let us recall some of our results. We have introduced a new family of special functions, i.e. q-Charlier multiple orthogonal polynomials. For these polynomials we have obtained a Rodrigues-type formula (3.11) as well as their explicit representation in terms of the second of q-Appell's hypergeometric functions of two variables (3.15). Moreover, these polynomials are common eigenfunctions of two different (r + 1)-order difference operators, namely (4.8) and (5.4). In the limiting situation q → 1 we recover the corresponding structural relations for multiple Charlier polynomials [5]. Indeed, our relations (3.11), Our algebraic approach for the nearest neighbor recurrence relation (5.4) does not require to introduce type I multiple orthogonality [20]. Indeed, the q-difference operators involved in the Rodrigues-type formula constitute the key-ingredient in our approach.
A description of the main term of the logarithm asymptotics of q-Charlier multiple orthogonal polynomials deserves our future attention. We expect to give it in terms of an algebraic function (see [2]). These results yield the Cauchy transform of the weak-star limit for scaling zero counting measure of the polynomials. In addition, the zero distribution of these type II multiple orthogonal polynomials will be studied.