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A Finite Sequences of Hahn-Type Discrete Orthogonal Polynomials Recurrence equations and their classical orthogonal polynomial solutions on a quadratic or q-quadratic lattice Daniel Duviol Tcheutia Institute of Mathematics, University of Kassel, Heinrich-Plett Str. 40, 34132 Kassel, Germany Abstract Every classical orthogonal polynomial system pn(x) satisfies a three-term recurrence relation of the type pn+1(x) = (An x + Bn)pn(x) Cn pn 1(x) (n = 0, 1, 2,..., p 1 0), − − − ≡ with CnAnAn 1 > 0. Moreover, Favard’s theorem states that the converse is true. A general − method to derive the coefficients An, Bn, Cn in terms of the polynomial coefficients of the divided- difference equations satisfied by orthogonal polynomials on a quadratic or q-quadratic lattice is recalled. The Maple implementations rec2ortho of Koorwinder and Swarttouw or retode of Koepf and Schmersau were developed to identify classical orthogonal polynomials given by their three-term recurrence relation as special functions. The two implementations rec2ortho and retode do not handle classical orthogonal polynomials on a quadratic or q-quadratic lattice. In this manuscript, the Maple implementation retode of Koepf and Schmersau is extended to cover classical orthogonal polynomials on quadratic or q-quadratic lattices and to answer as application an open problem submitted by Alhaidari during the 14th International Symposium on Orthogonal Polynomials, Special Functions and Applications. Keywords: Computer algebra, Maple, Three-term recurrence equation, Divided-difference equation. 2010 MSC: 33C45, 33D45, 33F99, 42C05 1. Introduction arXiv:1901.03672v1 [math.CA] 11 Jan 2019 Foupouagnigni showed in [4] that classical orthogonal polynomials on a quadratic or q-quadratic lattice satisfy a second-order divided difference equation of the form 2 φ(x(s))Dx pn(x(s)) + ψ(x(s))SxDx pn(x(s)) + λn pn(x(s)) = 0, (1) where φ(x) = ax2 + bx + c, ψ(x) = dx + e (d , 0), are polynomials of degree at most 2 and of degree one, respectively, the operators Dx and Sx are given by 1 1 1 1 f (x(s + 2 )) f (x(s 2 )) f (x(s + 2 )) + f (x(s 2 )) Dx f (x(s)) = − − , Sx f (x(s)) = − , x(s + 1 ) x(s 1 ) 2 2 − − 2 Email address: [email protected] (Daniel Duviol Tcheutia) Preprint submitted to January 14, 2019 and x(s) is a quadratic or q-quadratic lattice defined by [11] s s c1q + c2q− + c3 if 0 < q < 1, x(s) = 2 c1,..., c6 C. ( c4 s + c5 s + c6 if q = 1, ∈ Note that (1) is equivalent to a difference or q-difference equation of the form (see [9, chaps. 9, 14]) λ y(x(s)) = B(s)y(x(s + 1)) (B(s) + D(s))y(x(s)) + D(s)y(x(s 1)), (2) n − − with 1 1 1 φ(x(s)) = x(s + ) x(s ) (x(s + 1) x(s))B(s) + (x(s) x(s 1))D(s) , 2 2 2 − − − − − − ψ(x(s)) = (x(s) x(s + 1))B(s) + (x(s) x(s 1))D(s). − − − Following the work by Foupouagnigni [4], Njionou Sadjang et al. [13] proved that the Wilson and the continuous dual Hahn polynomials are solutions of a divided-difference equation of the form 2 φ(x)Dx pn(x) + ψ(x)SxDx pn(x) + λn pn(x) = 0, (3) where the operators Sx and the Wilson operator (see [3], [6]) Dx are defined by i i i i f x + 2 f x 2 f x + 2 + f x 2 D f (x) = − − , S f (x) = − . x 2ix x 2 Using the same approach, Tcheutia et al. [17] derived a divided–difference equation of type 2 φ(x)δxy(x) + ψ(x)Sxδxy(x) + λny(x) = 0, (4) satisfied by the continuous Hahn and the Meixner–Pollaczek polynomials, where the difference operator δx (see [15, p. 436], compare [9, p. 201 and 214], [12, 14], [18, Equation (1.15)]) is defined as follows: + i i f x 2 f x 2 δ f (x) = − − . x i (3) and (4) are equivalent to the difference equation (see [9, chap. 9]) λ y(x) = B(x)y(x + i) (B(x) + D(x))y(x) + D(x)y(x i), (5) n − − with φ(x) = x((2x + i)B(x) + (2x i)D(x)), ψ(x) = i((2x + i)B(x) (2x i)D(x)), − − − − and 1 φ(x) = (B(x) + D(x)), ψ(x) = i(B(x) D(x)), 2 − − respectively. The coefficients of the divided-difference equations given in the forms (1), (3) or (4) can be used for instance to compute the three–term recurrence relation or some structure formulae, the 2 inversion coefficients of classical orthogonal polynomials on a quadratic and q-quadratic lattice (see e. g. [5], [13], [16], [17] and references therein). Every classical orthogonal polynomial system pn(x) satisfies a three-term recurrence relation of the type pn+1(x) = (An x + Bn)pn(x) Cn pn 1(x) (n = 0, 1, 2,..., p 1 0), (6) − − − ≡ with CnAnAn 1 > 0. Moreover, Favard’s theorem states that the converse is true. In Section 2, a general method− to derive such three–term recurrence relations for classical orthogonal polynomials on a quadratic or q-quadratic lattice in terms of the given polynomials φ(x) and ψ(x) will be recalled. Alhaidari [1] submitted (as open problem during the 14th International Symposium on Orthogonal Polynomials, Special Functions and Applications) two polynomials defined by their three-term recurrence relations and initial values. He was interested by the derivation of their weight functions, generating functions, orthogonality relations, etc.. In order to solve this problem as suggested in the comments by W. Van Assche in [2], we use the computer algebra system Maple to identify the polynomials from their recurrence relations, such as in the Maple implementation rec2ortho of Koorwinder and Swarttouw [10] or retode of Koepf and Schmersau [8]. The two implementations rec2ortho and retode do not handle classical orthogonal polynomials on a quadratic or q-quadratic lattice. In Section 3, we extend the Maple implementation retode of Koepf and Schmersau [8] to cover classical orthogonal polynomials on a quadratic or q-quadratic lattice and to answer the problem by Alhaidari [1] as application. 2. Three-term recurrence equation satisfied by classical orthogonal polynomials on a quadratic or q-quadratic lattices We recall the definition of the Pochhammer symbol (or shifted factorial) given by Γ(a + m) (a) = = a(a + 1)(a + 2) (a + m 1), m = 0, 1, 2,..., m Γ(a) ··· − and the q-Pochhammer symbol m 1 − (1 aq j) if m = 1, 2, 3,... (a; q)m = j=0 − Q 1 if m = 0. We set the following polynomial basis: n 1 s s − k 2 2k B (a, x) = (aq ; q) (aq− ; q) = (1 2axq + a q ), n 1, B (a, x) 1, (7) n n n − ≥ 0 ≡ Yk=0 qs + q s where x = x(s) = cos θ = − , qs = eiθ; 2 ϑ (a, x) = (a + ix) (a ix) ; (8) n n − n 3 n 1 x x+1 − 2k+1 k ξn(γ, δ, µ(x)) = (q− ; q)n(γδq ; q)n = (1 + γδq µ(x)q ), n 1, k=0 − ≥ (9) Q ξ0(γ, δ, µ(x)) 1, ≡ x x+1 with µ(x) =q− + γδq ; n 1 χ (γ, δ, λ(x)) = ( x) (x + γ + δ + 1) = − k(γ + δ + k + 1) λ(x) , n 1, n n n (10) − k=0 − ≥ χ (γ, δ, λ(x)) 1, Q 0 ≡ for λ(x) = x(x + γ + δ + 1). From the hypergeometric and the basic hypergeometric representations (see [9, chaps. 9, 14]) of classical orthogonal polynomials on a quadratic or q-quadratic lattice, their natural bases are B (a, x) , (a + ix) , ξ (γ, δ, µ(x)) or χ (γ,δ,λ(x)) whose elements are { n } { n} { n } { n } polynomials of degree n in the variables x, x, µ(x) or λ(x), respectively, and the basis ϑn(a, x) 2 { } whose elements are polynomials of degree n in the variable x . The operator Dx is appropriate for B (a, x), ξ (γ, δ, µ(x)) and χ (γ,δ,λ(x)), δ is appropriate for (a+ix) , whereas the corresponding n n n x { n} operator for the basis ϑn(a, x) is Dx. Starting from a di{fference} equation of type (2) or (5) given in [9], we deduce the divided- difference equation of type (1), (3) or(4) satisfied by each classical orthogonal polynomial on a quadratic or q-quadratic lattice. Some of them can be found in [4], [13], [17] and we recall all of them here to make the manuscript self-contained. They will also be recovered using the algorithms implemented in this manuscript. 2.1. Polynomials expanded in the basis ϑ (α, x) { n } In this basis are expanded: 1. the Wilson polynomials n, n + a + b + c + d 1, a + ix, a ix W (x2; a, b, c, d) = (a + b) (a + c) (a + d) F − − − 1 , n n n n4 3 a + b, a + c, a + d ! with φ(x) = (x2 (cd + ab + ac + bc + bd + ad) x + abcd), − ψ(x) = (a + b + c + d)x abc abd acd bcd, λ = n(n 1 + a + b + c + d); − − − − n − − 2. the continuous dual Hahn polynomials 2 n, a + ix, a ix S n(x ; a, b, c) = (a + b)n(a + c)n3F2 − − 1 , a + b, a + c ! with φ(x) = (a + b + c) x + abc, ψ(x) = x (bc + ab + ac), λ = n. − − n − The procedure to find the coefficients of the recurrence equation (6) (with x substituted by x2) in terms of the coefficients a, b, c, d, e of φ(x) and ψ(x) is as follows (cf.
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