Orthogonal Polynomials Satisfying Higher-Order Difference Equations

Abstract

We introduce a large class of measures with orthogonal polynomials satisfying higher-order difference equations with coefficients independent of the degree of the polynomials. These measures are constructed by multiplying the discrete classical weights of Charlier, Meixner, Krawtchouk, and Hahn by certain variants of the annihilator polynomial of a finite set of numbers.

Appendix

The purpose of this paper is not to study the structural properties of the orthogonal polynomials introduced above. Nevertheless, we would like to finish with a couple of comments about that issue.

Firstly, we point out here that they can be defined by means of modified Rodrigues’ formulas. Since their orthogonalizing measures are defined by applying a Christoffel transform to the discrete classical measures, we can also get expressions for these polynomials in terms of the discrete classical polynomials. Grünbaum and Haine (et al.) proved that polynomials satisfying fourth- or higher-order differential equations can be generated by applying a Darboux transform to certain choices of the classical polynomials (see [12, 14, 15]). That is also the case of our polynomials: one can obtain them by one (or several) Darboux transforms applied to certain instances of the discrete classical polynomials. Here is an example to illustrate this last statement.

The formula

$$p_n(x)=\Delta^n \biggl(\bigl[\bigl(n(a-1)-c\bigr)(x+c)-(c+n)\bigr]\frac{a^x\varGamma (x+c)}{(x-n)!} \biggr)\big/\rho(x), \quad n\ge1,$$

defines a sequence of orthogonal polynomials with respect to the measure ρ=∑ _x∈ℕ ρ(x)δ _x defined by (1.11).

The above Rodrigues’ formula allows us to expand the orthogonal polynomials p _n , n≥0, in terms of the Meixner polynomials. Indeed, write $\omega_{a,c} (x)=\frac{a^{x}\varGamma(x+c)}{x!}$ so that ω _a,c =∑ _x∈ℕ ω _a,c (x)δ _x is the Meixner measure (1.6), and consider the following Rodrigues’ formula for Meixner polynomials:

$$m_{n,a,c}(x)=\Delta^n \biggl(\frac{a^x\varGamma(x+c)}{(x-n)!}\biggr)\big/\omega_{a,c} (x).$$

It is then easy to see that the polynomials p _n , n≥1, are equal to

$$p_n(x)=\frac{(n(a-1)-c)(x+c)m_{n,a,c+1}(x)-(c+n)m_{n,a,c}(x)}{x+c+1}.$$

Since the leading coefficient of m _n,a,c is (a−1)ⁿ, we get that the polynomials

$$\hat{p}_n=\frac{p_n}{(n(a-1)-c)(a-1)^n}$$

are monic.

Since ρ=(x+c+1)ω _a,c , we have that ρ is a Christoffel transform of the Meixner measure ω _a,c . Using (2.5), we then have

$$\hat{p}_n(x)=\frac{m_{n+1,a,c}(x)-A_nm_{n,a,c}(x)}{(a-1)^{n+1}(x+c+1)},$$

where

$$A_n=\frac{m_{n+1,a,c}(-c-1)}{m_{n,a,c}(-c-1)}=(c+n)\frac {c-(n+1)(a-1)}{c-n(a-1)}.$$

Assume that the Jacobi matrix J associated with the sequence of monic orthogonal polynomials with respect to a measure μ can be decomposed as

$$J=AB+\lambda I,$$

where

The three-diagonal matrix $\tilde{J}=BA+\lambda I$ is then called a Darboux transform of J. If $\tilde{J}_{i+1,i}\not=0$ for i=1,2,…, then $\tilde{J}$ is also the Jacobi matrix associated with a certain measure $\tilde{\mu}$. It was proved by G.J. Yoon that the measures μ and $\tilde{\mu}$ are related by the formula $(x-\lambda)\tilde{\mu}=\mu$ (see [35], Theorem 2.4). The measure $\tilde{\mu}$ is sometimes called a Geronimus transform of the measure μ (Geronimus transform is reciprocal to the Christoffel transform; see [10] or, for a modern systematic treatment, [5, 30]).

Since the measure ρ (1.11) can be written in the form ρ=ω _a,c+2/(x+c), we have that the orthogonal polynomials (p _n ) _n with respect to ρ can be obtained by applying a Darboux transform to the Meixner polynomials m _n,c+2. The explicit expressions for the matrices A and B in this case are

From this we can get the three-term recurrence coefficients for the sequence (p _n ) _n :

$$x\hat{p}_n(x)=\hat{p}_{n+1}(x)+b_n\hat{p}_n(x)+c_n\hat{p}_{n-1}(x),$$

where