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Discrete Multiple Orthogonal Polynomials on Shifted Lattices

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Discrete Multiple Orthogonal Polynomials on Shifted Lattices Discrete multiple orthogonal polynomials on shifted lattices ALEXANDER DYACHENKO AND VLADIMIR LYSOV Abstract. We introduce a new class of polynomials of multiple orthogonality with respect to the product of r classical discrete weights on integer lattices with noninteger shifts. We give explicit representations in the form of the Rodrigues formulas.The case of two weights is described in detail. 1. Introduction Orthogonal polynomials of discrete variable have numerous applications. One the most well-known applications is related to the theory of representations of the 3D rotation group [10]: it is based on a deep connection between polynomials orthogonal with respect to classical discrete and continuous measures. The similarity between these polynomials is nicely exposed in the monograph [15]. The current paper focuses on multiple orthogonal polynomials induced by a system of discrete measures that generalize the classical ones. Our study makes use of the analogy with the continuous case. For that reason, we first remind some properties of classical polynomials orthogonal with respect to continuous measures. 1.1. Classical orthogonal polynomials. Classical orthogonal polynomials are the polyno- mials by Hermite,Laguerre, and Jacobi (the last also include polynomials by Chebyshev and Gegenbauer as special cases). They constitute a simple yet quite important class of special functions.These polynomials arise in a wide range of problems in quantum mechanics, mathem- atical physics, numerical analysis and approximation theory. Their classification and derivation of their main properties base upon the Pearson equation for the weight of orthogonality. Ac- cording to the definition in [16], classical orthogonal polynomials are the polynomials Pn of degree n that satisfy the orthogonality relations with respect to the continuous weight ρ on the interval Sρ ⊂ R: Z k (1) Pn(x)x ρ(x) dx = 0; k = 0; 1; : : : ; n − 1; Sρ where ρ satisfies the Pearson differential equation: d (2) σ(x)ρ(x) = τ(x)ρ(x) dx arXiv:1908.11467v1 [math.CA] 29 Aug 2019 with polynomials σ; τ of degrees deg σ 6 2 and deg τ = 1. Up to shifts and rescaling, there are precisely three1 types of classical orthogonal polynomials, see Table 1. The polynomials Pn may be written explicitly through the Rodrigues formula: dn (3) P (x)ρ(x) = C σn(x)ρ(x); n n dxn Date: 2nd September 2019. 2010 Mathematics Subject Classification. 33C45 (42C05). The first author is supported by the research fellowship DY 133/1-1 of the German Research Founda- tion (DFG). The second author is supported by Russian Science Foundation, grant No. 19-71-30004. 1Some authors additionally include in this list the Bessel polynomials which are orthogonal on the cardioid − 1 in C with respect to e x . For our goal, it is enough only to consider orthogonality on subsets of the real line R. 1 2 ALEXANDER DYACHENKO AND VLADIMIR LYSOV Table 1. Classical orthogonal polynomials α α,β Pn Hn, Hermite Ln, Laguerre Pn , Jacobi ρ(x) e−x2 xαe−x (1 + x)α(1 − x)β Sρ (−∞; 1) (0; 1)(−1; 1) σ(x) 1 x x2 − 1 τ(x) −2x −x + α + 1 (α + β + 2)x − α + β C1 — α > −1 α > −1, β > −1 where Cn 6= 0 stands for a normalization constant. Condition C1 in Table 1 yields positivity and integrability of the weight ρ. All zeros of Pn are simple and belong to Sρ. 1.2. Multiple orthogonal polynomials. Some applications related to rational approxima- tions, random matrices, Diophantine approximations give rise to multiple orthogonal polyno- mials, see e.g. [4, 13, 14, 19, 23]. These polynomials are determined by orthogonality relations with respect to more than one weight. Let ρ1; : : : ; ρr be the weight functions determined on the intervals Sρ1 ;:::;Sρr of the real line. A polynomial Pn is called a (type II) multiple orthogonal polynomial if Pn 6≡ 0, deg Pn 6 rn and the orthogonality relations: Z k (4) Pn(x)x ρj(x) dx = 0; k = 0; : : : ; n − 1; j = 1; : : : ; r Sρj are satisfied.The case r = 1 corresponds to the standard orthogonality (1).Two ways to determine classical multiple orthogonal polynomials are currently in use.The first one [3] is when the intervals Sρj coincide, i.e. Sρj = ··· = Sρj , and the weights ρj satisfy the Pearson equation (2) with distinct τ (j).In this case, the restrictions for the degrees of σ and τ (j) remain (j) the same: deg σ 6 2 and deg τ = 1 There is also another way [2, 5, 20], which we consider in the particular case r = 2. Here, the weight functions ρ1; ρ2 are proportional to restrictions of a certain analytic function ρ to disjoint intervals S ;S , that is: ρ := c ρj for a constant c .The function ρ is additionally ρ1 ρ2 j j Sρj j assumed to satisfy the Pearson equation (2) with the polynomials σ and τ of degrees deg σ 6 3 and τ = 2.This construction yields the Angelesco-Jacobi [1, 11], Jacobi-Laguerre [21] and Laguerre-Hermite [22] polynomials.The basic characteristics of these polynomials are stated in Table 2. Condition MC1 (Multiple Continuous) provides positivity and integrability of the weights, as well as that Sρ1 \ Sρ2 = ?.The orthogonality (4) immediately implies that the polynomial Pn has n simple zeros in Sρj . One of the most remarkable properties of this construction is that the resulting polynomials also satisfy the Rodrigues formula (3). 1.3. Classical orthogonal polynomials of discrete variable. Up until this point we dis- cussed the continuous case. Now, let us consider orthogonality with respect to discrete measures. Let Sρ be a discrete subset of R, e.g. Sρ = Z+ := f0; 1;::: g or Sρ = f0; 1;:::;Ng for some integer N > 0.The function ρ is supposed to be nonnegative on Sρ. Put the charge ρ(x) in each point x 2 Sρ and consider the discrete measure: X (5) µ(y) = ρ(x)δ(y − x); x2Sρ where δ is the Dirac delta. Then the function ρ is called the discrete weight of the measure µ. Discrete multiple orthogonal polynomials on shifted lattices 3 Consider a sequence of polynomials Pn of degree n, which are orthogonal with respect to the measure µ: Z k X k (6) Pn(x)x dµ(x) = Pn(x)x ρ(x) = 0; k = 0; 1; : : : ; n − 1: x2Sρ Denote by ∆f and rf the left and the right finite differences of f, respectively: ∆f(x) = f(x + 1) − f(x) and rf(x) = f(x) − f(x − 1): As above, given two polynomials σ and τ such that deg σ 6 2 and deg τ = 1 one may also consider the (difference) Pearson equation: (7) ∆σ(x)ρ(x) = τ(x)ρ(x): Up to trivial transformations, its regular solutions determine precisely four (see [15]) classes of classical orthogonal polynomials of discrete variable: the Charlier and Mexner polynomials are orthogonal on Z+, while the Kravchuk (Krawtchouk) and Hahn polynomials are orthogonal on f0; 1;:::;Ng. Polynomials of these classes satisfy the Rodrigues formula: n (8) Pn(x)ρ(x) = Cnr ρn(x); where Cn is a normalizing constant and ρn is a shifted weight, see Table 3.In this table, the Hahn weight is expressed via the so-called Pochhammer symbol: Γ(x + α) (α) := x Γ(α) and, in particular, (α)x = (α)(α + 1) ··· (α + x − 1) for x 2 Z+.The Kravchuk and Hahn polynomials additionally require that n 6 N. Condition D1 in Table 3 yields positivity2 and finiteness of all moments for the measure µ from (5).The orthogonality relations immediately imply that all zeros of Pn are simple and lie in the convex hull of Sρ. Between two consecutive lattice points there is at most one zero of Pn. 2For the Hahn case, the inequality α; β < N − 1 in Condition D1 induces that the weight keeps the same N sign (−1) on the whole support Sρ. In general, our Table 3 and Condition D1 may be compared to [17]. Table 2. Classical multiple orthogonal polynomials Pn Laguerre-Hermite Jacobi-Laguerre Angelesco-Jacobi ρ(x) xβe−x2 (a + x)αxβe−x (a + x)αxβ(1 − x)γ β −x2 α β −x α β γ ρj(x) jxj e (a + x) jxj e (a + x) jxj (1 − x) Sρ1 ;Sρ2 (−∞; 0), (0; 1)(−a; 0), (0; 1)(−a; 0), (0; 1) σ(x) x (x + a)x (x + a)x(x − 1) −x2 (α + β + γ + 3)x2 2 τ(x) −2x + β + 1 +(α + β + 2 − a)x + (β + γ + 2)a +(β + 1)a − (α + β + 2) x − (β + 1)a MC1 β > −1 α > −1, β > −1, α > −1, β > −1, a > 0 γ > −1, a > 0 4 ALEXANDER DYACHENKO AND VLADIMIR LYSOV Table 3. Classical orthogonal polynomials of a discrete variable Pn Charlier Meixner Kravchuk Hahn bx bxΓ(x + α) x (α) (β) ρ(x) b x N−x Γ(x + 1) Γ(x + 1) Γ(x+1)Γ(N−x+1) Γ(x + 1)Γ(N − x + 1) Sρ Z+ Z+ f0; 1;:::;Ng f0; 1;:::;Ng bx bxΓ(x + α + m) x (α + m) (β + m) ρ (x) b x N−m−x m Γ(x + 1) Γ(x + 1) Γ(x+1)Γ(N−m−x+1) Γ(x + 1)Γ(N − m − x + 1) ρ (x) (x + α + m)(N − m − x) m+1 1 x + α + m N − m − x ρm(x) (α + m)(β + m) ρm+1(x − 1) x x x x(N − x + β) ρm(x) b b b (α + m)(β + m) D1 b > 0 α>0; 0<b<1 b > 0 α; β > 0 or α; β < 1 − N 0 < jbj < 1, α; β2 = f0; −1;:::; 1 − Ng, D2 b 6= 0 b 6= 0, b 6= −1 α2 = −Sρ −α − β2 = f0; 1;:::; 2N − 2g 1.4.
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