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Shifted Jacobi Polynomials and Delannoy Numbers

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Shifted Jacobi Polynomials and Delannoy Numbers SHIFTED JACOBI POLYNOMIALS AND DELANNOY NUMBERS GABOR´ HETYEI A` la m´emoire de Pierre Leroux Abstract. We express a weigthed generalization of the Delannoy numbers in terms of shifted Jacobi polynomials. A specialization of our formulas extends a relation between the central Delannoy numbers and Legendre polynomials, observed over 50 years ago [8], [13], [14], to all Delannoy numbers and certain Jacobi polynomials. Another specializa- tion provides a weighted lattice path enumeration model for shifted Jacobi polynomials and allows the presentation of a combinatorial, non-inductive proof of the orthogonality of Jacobi polynomials with natural number parameters. The proof relates the orthogo- nality of these polynomials to the orthogonality of (generalized) Laguerre polynomials, as they arise in the theory of rook polynomials. We also find finite orthogonal polynomial sequences of Jacobi polynomials with negative integer parameters and expressions for a weighted generalization of the Schr¨odernumbers in terms of the Jacobi polynomials. Introduction It has been noted more than fifty years ago [8], [13], [14] that the diagonal entries of the Delannoy array (dm,n), introduced by Henri Delannoy [5], and the Legendre polynomials Pn(x) satisfy the equality (1) dn,n = Pn(3), but this relation was mostly considered a “coincidence”. An important observation of our present work is that (1) can be extended to (α,0) (2) dn+α,n = Pn (3) for all α ∈ Z such that α ≥ −n, (α,0) where Pn (x) is the Jacobi polynomial with parameters (α, 0). This observation in itself is a strong indication that the interaction between Jacobi polynomials (generalizing Le- gendre polynomials) and the Delannoy numbers is more than a mere coincidence. In this paper we develop formulas for weighted Delannoy numbers, obtain lattice path enumera- tion models for Jacobi polynomials with integer parameters, and provide a combinatorial proof for the orthogonality of several sequences of such Jacobi polynomials. In Section 2 we introduce weighted Delannoy paths associating the same weight u to each east step, the same weight v to each north step and the same weight w to each 1991 Mathematics Subject Classification. Primary 05E35; Secondary 33C45, 05A15. Key words and phrases. Legendre polynomials, Delannoy numbers, Jacobi polynomials, rook polyno- mials, Laguerre polynomials. This work was supported by the NSA grant # H98230-07-1-0073. 1 2 GABOR´ HETYEI northeast step. We show that the total weight of all Delannoy paths in a rectangle may be easily expressed by substitution into a shifted Jacobi polynomial with the appropriate parameters. The shifting is achieved by replacing x with 2x − 1 in the definition of the Jacobi polynomial. In particular, a specific substitution into the parameters u, v, w yields the shifted Jacobi polynomials themselves. Using this observation, in Section 3 we provide a lattice (α,β) path enumeration proof for the orthogonality of the Jacobi polynomials Pn (x) when α, β ∈ N. The proof reduces the weight enumeration to a formula that is known in the classical theory of rook polynomials, and is used there to show the orthogonality of the (generalized) Laguerre polynomials. This connection is explored a little further in Section 4, but a lot of work remains to be done in this area. Using our weighted lattice path enumeration model, we can show a very simple formula (0,−β) expressing each Jacobi polynomial Pn (x) with negative integer second parameter and satisfying n ≥ β, in terms of a Jacobi polynomial with positive integer second parameter. (0,−β) The remaining sequence {Pn (x): n ≤ β − 1} is symmetric, and its first half forms a finite orthogonal polynomial sequence. A combinatorial proof of this result may be found in Section 5. Finally, in Section 6 we generalize Schr¨odernumbers in a way that is completely anal- ogous to our definition of weighted Delannoy numbers. We show that ordinary Schr¨oder numbers too may be obtained by substituting 3 into an appropriate Jacobi polynomial, and prove a formula for iterated antiderivatives of the shifted Legendre polynomials. Acknowledgments I wish to thank Mireille Bousquet-M´eloufor acquainting me with Viennot’s work [21]. This work was supported by the NSA grant # H98230-07-1-0073. 1. Preliminaries 1.1. Delannoy numbers. The Delannoy array (di,j : i, j ∈ Z) was introduced by Henri Delannoy [5]. It may be defined by the recursion formula (3) di,j = di−1,j + di,j−1 + di−1,j−1 with the conditions d0,0 = 1 and di,j = 0 if i < 0 or j < 0. The historic significance of these numbers is explained in the paper “Why Delannoy numbers?”[2] by Banderier and Schwer. The diagonal elements (dn,n : n ≥ 0) in this array are the central Delannoy numbers (A001850 of Sloane [16]). These numbers are known through the books of Comtet [4] and Stanley [18], but it is Sulanke’s paper [20] that gives the most complete enumeration of all known uses of the central Delannoy numbers. SHIFTED JACOBI POLYNOMIALS AND DELANNOY NUMBERS 3 (α,β) 1.2. Jacobi and Legendre polynomials. The n-th Jacobi polynomial Pn (x) of type (α, β) is defined as dn P (α,β)(x) = (−2)−n(n!)−1(1 − x)−α(1 + x)−β (1 − x)n+α(1 + x)n+β . n dxn (α,β) If α and β are real numbers satisfying α, β > −1, then the polynomials {Pn (x)}n≥0 form an orthogonal basis with respect to the inner product Z 1 hf, gi := f(x) · g(x) · (1 − x)α(1 + x)β dx. −1 The following formula, stated only slightly differently in [19, (4.21.2)], may be used to (α,β) extend the definition of Pn (x) to arbitrary complex values of α and β: n j X n + α + β + jn + α x − 1 (4) P (α,β)(x) = . n j n − j 2 j=0 The following important relation provides a way to “swap” the parameters α and β (see [3, Chapter V, (2.8)]). n (α,β) (β,α) (5) (−1) Pn (−x) = Pn (x). (0,0) The Legendre polynomials {Pn(x)}n≥0 are the Jacobi polynomials {Pn (x)}n≥0. They form an orthogonal basis with respect to the inner product Z 1 hf, gi := f(x) · g(x) dx. −1 Substituting α = β = 0 into (4) yields n j X n + j n x − 1 (6) P (x) = . n j n − j 2 j=0 As a specialization of (5) we obtain n (7) (−1) Pn(−x) = Pn(x). The shifted Legendre polynomials Pen(x) are defined by the linear substitution Pen(x) := Pn(2x − 1). They form an orthogonal basis with respect to the inner product Z 1 hf, gi := f(x) · g(x) dx. 0 They may be calculated using the following formula: n n X n−k n n + k k X n−k n + k 2k k (8) Pen(x) = (−1) x = (−1) x . k k n − k k k=0 k=0 A generalization of (8) is stated and shown in Proposition 2.4. Shifted Legendre polynomi- als are widely used, they even have a table entry in the venerable opus of Abramowitz and 4 GABOR´ HETYEI (α,β) Stegun [1, 22.2.11]. Their obvious generalization, the shifted Jacobi polynomials Pen (x), defined by the formula (α,β) (α,β) (9) Pen (x) := Pn (2x − 1), seem to occur much less frequently in the literature, a sample reference is the work of Gatteschi [6]. 1.3. Favard’s theorem. A good reference for the fundamental facts on orthogonal poly- nomials is Chihara’s book [3]. A moment functional L is a linear map C[x] → C.A ∞ sequence of polynomials {pn(x)}n=0 is an orthogonal polynomial sequence with respect 2 to L if pn(x) has degree n, L(pm(x)pn(x)) = 0 for m 6= n, and L(pn(x)) 6= 0 for all n. Such a sequence exists if and only if L is quasi-definite (see [3, Ch. I, Theorem 3.1], the term quasi-definite is introduced in [3, Ch. I, Definition 3.2]). Whenever an orthogonal polynomial sequence exists, each of its elements is determined up to a non-zero constant factor (see [3, Ch. I, Corollary of Theorem 2.2]). A way to verify whether a sequence of polynomials is orthogonal is Favard’s theorem [3, Chapter I, Theorem 4.4]. This states that a sequence of monic polynomials {pn(x)}n≥0 is an orthogonal polynomial sequence, if and only if it satisfies the recurrence formula (10) pn(x) = (x − cn)pn−1(x) − λnpn−2(x) n = 1, 2, 3,... where p−1(x) = 0, p0(x) = 1, the numbers cn and λn are constants, λn 6= 0 for n ≥ 2, and λ1 is arbitrary (see [3, Ch. I, Theorem 4.1]). Conversely, for every sequence of monic polynomials defined in the above way there is a unique quasi-definite moment functional ∞ L such that L(1) = λ1 and {Pn(x)}n=0 is the monic orthogonal polynomial sequence with respect to L. The original proof provides only a recursive description of L for a given {pn(x)}n≥0 satisfying (10). Viennot [21] gave a combinatorial proof of Favard’s theorem, upon which he has built a general combinatorial theory of orthogonal polynomials. In his theory, the values L(xn) are explicitly given as sums of weighted Motzkin paths. 1.4. Central Delannoy numbers and Legendre polynomials. Equation (1) linking the central Delannoy numbers to the Legendre polynomials has been known for over 50 years [8], [13], [14]. Until recently there was a consensus that this link is not very relevant.
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