Power series expansion of a Hankel determinant Diego Dominici DK-ReportNo.2018-08 062018 A–4040 LINZ, ALTENBERGERSTRASSE 69, AUSTRIA Supported by Austrian Science Fund (FWF) Upper Austria Editorial Board: Bruno Buchberger Evelyn Buckwar Bert J¨uttler Ulrich Langer Manuel Kauers Peter Paule Veronika Pillwein Silviu Radu Ronny Ramlau Josef Schicho Managing Editor: Silviu Radu Communicated by: Peter Paule Manuel Kauers DK sponsors: • Johannes Kepler University Linz (JKU) • Austrian Science Fund (FWF) • Upper Austria Power series expansion of a Hankel determinant Diego Dominici ∗ Johannes Kepler University Linz Doktoratskolleg \Computational Mathematics" Altenberger Straße 69 4040 Linz Austria Permanent address: Department of Mathematics State University of New York at New Paltz 1 Hawk Dr. New Paltz, NY 12561-2443 USA June 11, 2018 Abstract In this article, we find a power series expansion for the Hankel de- terminant whose entries are the moments of a linear functional related to discrete semiclassical orthogonal polynomials. We provide explicit formulas that allow the computation of coefficients to arbitrary order, and give examples for the first few terms in the series. We reinterpret these results in the context of the theory of Young tableaux and Schur functions, and find closed-form expressions for the cases corresponding to the Charlier and Meixner polynomials. We also discuss further research directions. ∗e-mail: [email protected] 1 Keywords: Hankel determinants, orthogonal polynomials, integer parti- tions, Young tableaux, Schur polynomials. Subject Classification Codes: 15A15 (primary), 33C45, 05A17, 05E05 (secondary). 1 Introduction Let N denote the set of natural numbers and N0 the set of nonnegative integers, N = f1; 2; 3;:::g ; N0 = N [ f0g : If fµng is a sequence of complex numbers and L : C [x] ! C is the linear functional defined by n L [x ] = µn; n 2 N0; then L is called the moment functional [4] determined by the formal moment sequence fµng. The number µn is called the moment of order n. Suppose that fPng is a family of monic polynomials, with deg (Pn) = n: If the polynomials Pn (x) satisfy L [PnPm] = hnδn;m; n; m 2 N0; (1) where h0 = µ0; hn 6= 0 and δn;m is Kronecker's delta, then fPng is called an orthogonal polynomial sequence with respect to L. If we write n n X x = an;kPk (x) ; n 2 N0; k=0 we can define a lower triangular matrix An by (An)i;j = ai;j; 0 ≤ i; j ≤ n − 1: If we define the diagonal matrix Dn by (Dn)i;j = hiδi;j; 0 ≤ i; j ≤ n − 1; and the Hankel matrix Hn by (Hn)i;j = µi+j; 0 ≤ i; j ≤ n − 1; (2) 2 then we have the LDL decomposition [12, 4.1.2] T Hn = AnDnAn : (3) We define the Hankel determinants ∆n by ∆0 = 1 and ∆n = det (Hn) ; n 2 N: (4) Determinants have a long history and an extensive literature, see [1], [3], [21], [25], [29], [30], [31], [32], and the impressive monographs [17] and [18]. The theories of Hankel determinants and orthogonal polynomials are deeply connected, see for example [4], [6], [9], [10], [14], [15], [16] [19], and [28]. In [8], we studied the discrete semiclassical orthogonal polynomials of class 1; and considered linear functionals L that have the form 1 X k L [p] = p(k)! (k) z ; p(x) 2 C [x] ; (5) k=0 for z 2 C and some function ! with ! (k) 6= 0; k 2 N0: (6) In this case, the moments µi (z) are given by 1 X i k µi (z) = k ! (k) z ; i 2 N0; (7) k=0 and the entries of the Hankel matrix (2) are 1 X i+j k (Hn)i;j = µi+j (z) = k ! (k) z ; 0 ≤ i; j ≤ n − 1: (8) k=0 We note from (7) that all the moments µi (z) can be obtained from the first one µ0 (z) ; since 1 1 X i k X i k i µi (z) = k ! (k) z = ! (k) # z = # µ0 (z) ; k=0 k=0 3 where the differential operator # is defined by [23, Chapter 6] d # = z : dz It follows that if the first moment µ0 (z) is analytic in some disk jz0j < r; the same will be true for all the other moments. In fact, we showed in [7] that for all families of discrete semiclassical orthogonal polynomials of class s; the moments have the form s −i X µi (z) = (λ + τz) pk (z) µk (z) ; k=0 where the constants λ, τ 2 {−1; 0; 1g and the polynomials pk (z) depend on the given family. Since the Hankel determinants ∆n (z) are analytic functions of z (in the same domain as µ0); it is natural to consider the Taylor series 1 X m ∆n (z) = dm (n) z ; jzj < r; (9) m=0 and try to determine the coefficients dm (n) : Surprisingly enough, we haven't been able to find many references on this topic. In [24], Rusk considered at the nth−derivative of a general determinant, whose elements are functions of a variable t; and found some connection with a symbolic version of the multinomial theorem. In [5], Christiano and Hall used the general Leibniz rule and obtained a formula for the nth−derivative in terms of determinants. In [13], Hochstadt considered the derivative of a Wronskian determinant W (t) = det X (t) ; where X (t) satisfies the matrix equation d X (t) = A (t) X (t) ; dt and obtained d W (t) = tr (A) W (t) ; dt 4 where tr (A) denotes the trace of A: In [11], Golberg generalized this result and proved Jacobi's formula d d det A(t) = tr adjA A ; dt dt where adj (A) is the adjugate of A defined by adj (A) A = det (A) I: Lastly, in [33] Withers and Nadarajah studied the Taylor series of a determi- nant with general entries, and obtained a formula involving traces of powers of the matrix and the complete exponential Bell polynomials. In this article, we obtain explicit expressions for the coefficients dm (n) in the Taylor series (9) when the entries of the Hankel matrix are given by (8). The paper is organized as follows: in Section 2, we derive a formula for dm (n) when ! (k) is a general function. We give some examples for special cases of ! (k) : In Section 3 we relate the results from Section 2 to the theory of Schur polynomials. We obtain exact evaluations of dm (n) for the Charlier and Meixner polynomials. Finally, in Section 4 we give a summary of results and point out some future directions. 2 Main result 2.1 Vandermonde polynomials Definition 1 The Vandermonde determinant V (x1; x2; : : : ; xn) is defined by n j−1 X Y j−1 V (x1; x2; : : : ; xn) = det xi = sgn (σ) x ; 1≤i;j≤n σ(j) σ2Sn j=1 where sgn (σ) denotes the sign of the permutation σ: Remark 2 It is well known that the Vandermonde determinant V (x1; x2; : : : ; xn) is an alternating multivariate polynomial given by [31, 4.1.2] Y V (x1; x2; : : : ; xn) = (xj − xi) ; n > 1: (10) 1≤i<j≤n 5 Definition 3 If a; m 2 Z and m ≤ 0; we define V (xa; xa+1; : : : ; xa+m) = 1: (11) Next, we derive some basic results about Vandermonde polynomials. Lemma 4 If a 2 Z and m 2 N; then m j−1 YY V (xa; xa+1; : : : ; xa+m) = (xj+a − xi+a) ; (12) j=0 i=0 Proof. Using (10), we have a+mj−1 Y Y Y V (xa; xa+1; : : : ; xa+m) = (xj − xi) = (xj − xi) a≤i<j≤a+m j=a i=a m j+a−1 m j−1 Y Y YY = (xj+a − xi) = (xj+a − xi+a) : j=0 i=a j=0 i=0 Corollary 5 If m 2 N0; then V (0; 1; 2; : : : ; m) = G (m + 1) ; (13) where the function G (m) is defined by G (0) = 1 and m−1 Y G (m) = (i!) ; m 2 N: (14) i=0 Note that G (m) satisfies the recurrence G (m + 1) = m!G (m) : 6 Proof. If we set a = 0 and xi = i; 0 ≤ i ≤ m in (12), we get m j−1 m j YY YY V (0; 1; 2; : : : ; m) = (j − i) = k (k = j − i) j=0 i=0 j=0k=1 m Y = j! = G (m + 1) : j=0 Remark 6 The function G (m) can be written as G (m) = G(m + 1); where G(m) is Barnes' G-Function [22, 5.17], defined by G(1) = 1 and G (z + 1) = Γ (z) G(z); z 2 C: Proposition 7 For all n 2 N and r 2 N0; we have V (x1; x2; : : : ; xn) = V (x1; x2; : : : ; xn−r) V (xn−r+1; : : : ; xn) (15) n n−r Y Y × (xj − xi) ; j=n−r+1 i=1 where empty products are assumed to be equal to 1: Proof. Setting a = 1 and m = n − r − 1 in (12), we get n−r−1j−1 Y Y V (x1; x2; : : : ; xn−r) = (xj+1 − xi+1) : j=0 i=0 Similarly, if we set a = n − r + 1 and m = r − 1 in (12), we have r−1j−1 YY V (xn−r+1; : : : ; xn) = (xj+n−r+1 − xi+n−r+1) : j=0 i=0 7 Thus, V (x1; x2; : : : ; xn) V (x1; x2; : : : ; xn−r) V (xn−r+1; : : : ; xn) n−1j−1 YY (xj+1 − xi+1) j=0 i=0 = "n−r−1j−1 #"r−1j−1 # Y Y YY (xj+1 − xi+1) (xj+n−r+1 − xi+n−r+1) j=0 i=0 j=0 i=0 n−1 j−1 Y Y (xj+1 − xi+1) j=n−ri=0 = : r−1j−1 YY (xj+n−r+1 − xi+n−r+1) j=0 i=0 But r−1j−1 n−1 j−n−r−1 YY Y Y (xj+n−r+1 − xi+n−r+1) = (xj+1 − xi+n−r+1) j=0 i=0 j=n−r i=0 n−1 j−1 Y Y = (xj+1 − xi+1) : j=n−ri=n−r Hence, V (x1; x2; : : : ; xn) V (x1; x2; : : : ; xn−r) V (xn−r+1; : : : ; xn) j−1 Y n−1 (xj+1 − xi+1) n−1 n−r−1 Y i=0 Y Y = = (x − x ) : j−1 j+1 i+1 j=n−r Y j=n−r i=0 (xj+1 − xi+1) i=n−r We conclude that n n−r V (x1; x2; : : : ; xn) Y Y = (x − x ) : V (x ; x ; : : : ; x ) V (x ; : : : ; x ) j i 1 2 n−r n−r+1 n j=n−r+1 i=1 8 We now obtain a representation for ∆n (z) in terms of Vandermonde poly- nomials.