On Polynomials Orthogonal with Respect to an Inner Product Involving Higher-Order Differences: The Meixner Case

In this contribution we consider sequences of monic polynomials orthogonal with respect to Sobolev-type inner product \[ \left\langle f,g\right\rangle= \langle {\bf u}^{\tt M},fg\rangle+\lambda \mathscr T^j f (\alpha)\mathscr T^{j}g(\alpha), \] where ${\bf u}^{\tt M}$ is the Meixner linear operator, $\lambda\in\mathbb{R}_{+}$, $j\in\mathbb{N}$, $\alpha \leq 0$, and $\mathscr T$ is the forward difference operator $\Delta$, or the backward difference operator $\nabla$. We derive an explicit representation for these polynomials. The ladder operators associated with these polynomials are obtained, and the linear difference equation of second order is also given. In addition, for these polynomials we derive a $(2j+3)$-term recurrence relation. Finally, we find the Mehler-Heine type formula for the $\alpha\le 0$ case.


Introduction
The Meixner orthogonal polynomials, usually denoted in the literature as M n (x; β, c), constitute a family of classical orthogonal polynomials, introduced by J. Meixner in 1934, in his seminal paper [15]. When β > 0 and 0 < c < 1 they are orthogonal with respect to the well-known negative binomial distribution of the probability theory, i.e. in such a case character, their finite differences constitute as well an orthogonal polynomial family, their corresponding orthogonality weight satisfy a Pearson-type difference equation, and even they satisfy two different kinds structure relations. A distinctive and interesting characteristic of this family is that they have certain dual character, that is, every formula one can derives for M n (x; β, c) has a dual formula with x and n interchanged c m−n n!(1 + β) m−1 M n (m; β, c) = c n−m m!(1 + β) n−1 M m (n; β, c), (see, for example [7, Ch. VI, Sec. 3], [11,Sec. 6.1], [18,Sec. 2.4], and the references therein).
On the other hand, since the first paper [1] on Sobolev orthogonal polynomials published by Althammer, until the present time, the results connected to these polynomials have attracted the attention of several mathematicians. The name of Sobolev orthogonal polynomials was given to those families of polynomials orthogonal with respect to inner products involving positive Borel measures supported on infinite subsets of the real line, and also involving regular derivatives. Moreover, in the c ase that the derivatives appear only on function evaluations on a finite discrete set, the corresponding families are called Sobolev-type or discrete Sobolev orthogonal polynomial sequences. For a recent and comprehensive survey on the subject, see [14] and the references therein. In the last decade of the past century, H. Bavinck introduced the study of orthogonal polynomials with respect to the inner product involving differences instead of derivatives where λ ∈ R + , c ∈ R and ψ is a distribution function with infinite spectrum, see [3,4]. Moreover, in these works Bavinck obtained algebraic properties and some results connected to the location of the zeros of the orthogonal polynomials with respect to the inner product (1). On the other hand, in [4] he proved that the orthogonal polynomials with respect to inner product defined in equation(1) satisfy a five term recurrence relation. Furthermore, in [5] the author considered the inner product where β > 0, 0 < c < 1, λ > 0 and P denote the linear space of all polynomials with real coefficients. Here, he obtained a second order difference equation with polynomial coefficients, which the orthogonal polynomials with respect to (2) satisfy. Then, in [6] the author showed that the Sobolev type Meixner polynomials orthogonal with respect to the inner product where β > 0, 0 < c < 1 and M, N ≥ 0, are eigenfunctions of a difference operator. Other results a little most recent, connected with the Sobolev Meixner polynomials, can be found in [12,16]. The structure of the paper is the following: In Section 2, we introduce some preliminary results about Meixner polynomials which will be very useful in the analysis presented. In Section 3, we obtain the connection formula between the Meixner polynomials and the polynomials orthogonal with respect to the Sobolev-type inner product where u M is the Meixner linear operator, λ ∈ R + , j ∈ N, α ≤ 0, and T is the forward or the backward difference operator, as well as we deduce the hypergeometric representation of such polynomials. In Section 4, we find the ladder (creation and annihilation) operators for the sequence of orthogonal polynomials of Sobolev type. As a consequence, the second order linear difference equation associated with them is deduced. And on the other hand, in Section 5, we determine the (2j + 3)-term recurrence relation that these polynomials satisfy. Finally, in Section 6, we determine the Mehler-Heine type formula for the especial case α = 0. Indeed, the techniques used in Sections 3, 4 and 5 are based on those used in [2,10,11], respectively.

Preliminaries
We adopt the following set notations: N 0 := {0} ∪ N = {0, 1, 2, 3, . . .}, and we use the sets Z, R, C which represent the integers, real numbers and complex numbers respectively. P denotes the vector space of univariate, complex-valued, polynomials, and let P denote its algebraic dual space.
We also adopt the following notation and conventions. We denote by u, p the duality bracket for u ∈ P and p ∈ P.
Definition 2.1. For u ∈ P, π ∈ P, and c ∈ C, let πu, (x − c) −1 u, ∇u, and ∆u be the linear functional defined by πu, p := u, πp , p ∈ P, and thus ∆u, p := − u, ∇p , where ∇ and ∆ are the backward and forward difference operator defined as: The Dirac delta functional, δ c , is the functional defined by δ c , p := p(c), p ∈ P.
In order to obtain our derived identities, we rely on properties of the Pochhammer symbol (shifted factorial). For any n ∈ N 0 , a ∈ C, the Pochhammer symbol is defined as (a) n := (a)(a + 1) · · · (a + n − 1), n ∈ N 0 , Furthermore, define for all a, b ∈ C, where a + b ∈ −N 0 , and we will also use the common notational product convention The hypergeometric series, which we will often use, is defined for z ∈ C such that |z| < 1, where, of course, the parameters must be such that the denominator factors in the terms of the series are never zero.

The Meixner polynomials
Let β and c be two complex numbers such that c = 0, 1 and β is not a negative integer. We write {M n (x; β, c)} n≥0 for the sequence of Meixner polynomials defined by [7] These polynomials are orthogonal with respect to the linear functional u M ∈ P which is a classical functional since it fulfills the Pearson difference equation which is equivalent to the Pearson difference equation Remark 2.2. Observe that when β > 0 and 0 < c < 1, then which is a positive definite linear functional. Observe this definition can be extended to |c| < 1 and β ∈ C nor a negative integer. Moreover, since then one can extend the Meixner functional for |c| > 1. In[8, Proposition 9] the authors obtained an integral representation for this operator for β, c ∈ C, with c ∈ [0, ∞] and −β ∈ N: where C is a complex contour from −∞i to ∞i separating the increasing poles {0, 1, 2, . . . } from the decreasing poles When c = 0, 1, they satisfy the following three term recurrence formula: where which can be explicitly in terms of hypergeometric series as[17, (9.10.1)] Next, we summarize some basic properties of Meixner orthogonal polynomials to be used in the sequel.
2. Structure relations. For every n ∈ N, 3. Squared norm. For every n ∈ N, 4. Value in the initial extreme of the orthogonality interval, 5. Forward and backward difference operators. For every n, k ∈ N, where 6. Mehler-Heine type formula [9, eq. 35] To complete this section we present some useful results we need along the paper.
Then, for all n ∈ N, Taking into account the inner product we have considered, then it is natural to consider the partial derivatives of K n (x, y) we will use the following notation: and Proposition 2.5. (cf. [13, p. 454]) The following identities hold: Proof . We are going to prove the first identity. After applying to (15) the difference operator ∇ j with respect to y we obtain Using a analogue of the Leibnitz's rule and since for any positive integer k we have the result follows after a straightforward calculation. The proof of the second identity is analogous and it will be omitted. Hence the result follows.
Proposition 2.6. The following identity holds for the kernel associated to the Meixner polynomials: Proof . Having (12)-(15) into account and by definition (17) we have By using some identities of the pochhammer symbols we obtain from this expression it is a direct calculation to get the desired expression.
Corollary 2.7. The following limit for the kernels associated to the Meixner polynomials holds:

The Sobolev-type Meixner polynomials
We start this section introducing the Sobolev-type inner product (3) f, g λ,j, = u M , f g + λT j f (α)T j g(α), where u M is the Meixner linear operator, j ∈ N, α ≤ 0, and T is the operator ∇ when = 1, and it is the operator ∆ when = 2. We denote by M j, n (x; β, c; λ) n∈N 0 the sequence of monic polynomials, orthogonal with respect to the inner product (22). These polynomials are said to be Sobolev-type Meixner polynomials.

Connection formula and hypergeometric representation
We first express the Sobolev-type Meixner polynomials in terms of the monic Meixner polynomials and the Kernel polynomials associated to the Meixner polynomials. Taking into account the Fourier expansion and using orthogonality conditions of (M n (x)) and (M j, n (x; β, c; λ)) we obtain (see cf. where From these identities we can express the Sobolev-type Meixner polynomials in terms of hypergeometric series. where f n (x) is given in(31) and Proof . Taking into account (−x) k = 0 if x < k as well as (7) and (24) we deduce By using the identity we get Thus, we have And after a straightforward calculation and by using (30) with a → −f n (x) − 1 the identity (29) follows. This completes the proof.

Second Order Linear Difference equation
In this section we obtain a second order linear difference equation that the sequence {M j, n (x; β, c; λ)} n≥0 satisfies. In order to do that, we will find the ladder (creation and annihilation) operators, using the connection formula (24), the three-term recurrence relation (6), and the structure relations (9), (10) satisfied by them.
Thus, by using the definitions of the operators a and a † , taking into account the identity and after tedious calculations we obtain(40). Hence the result follows.

The (2j + 3)-term recurrence relation
In this section we find the (2j + 3)-term recurrence relation that the sequence of monic Sobolev-type Meixner polynomials (29) fulfill. For this purpose, we use the fact, which is a straightforward consequence of (3), that the multiplication operator by x − α j+1 i is a symmetric operator with respect to such a discrete Sobolev inner product. Indeed, for any p, q ∈ P we have for = 1 and for = 2 Taking these identities into account and by using the three-term recurrence relation(6) we can state the following result.
By using the connection formula (43) and taking into account that A n (x) is a monic polynomial of degree j + 1 and B n (x) is a polynomial of degree j with the leading coefficient b j, n we deduce