Power Series Expansion of a Hankel Determinant

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 determinant 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 partitions, 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 = {1, 2, 3,...} , N0 = N ∪ {0} .

If {µn} is a sequence of complex numbers and L : C [x] → C is the linear functional deﬁned by n L [x ] = µn, n ∈ N0, then L is called the moment functional [4] determined by the formal moment sequence {µn}. The number µn is called the moment of order n. Suppose that {Pn} is a family of monic polynomials, with deg (Pn) = n. If the polynomials Pn (x) satisfy

L [PnPm] = hnδn,m, n, m ∈ N0, (1) where h0 = µ0, hn 6= 0 and δn,m is Kronecker’s delta, then {Pn} is called an orthogonal polynomial sequence with respect to L. If we write n n X x = an,kPk (x) , n ∈ 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 ∈ 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

∞ X k L [p] = p(k)ω (k) z , p(x) ∈ C [x] , (5) k=0 for z ∈ C and some function ω with

ω (k) 6= 0, k ∈ N0. (6)

In this case, the moments µi (z) are given by

∞ X i k µi (z) = k ω (k) z , i ∈ N0, (7) k=0 and the entries of the Hankel matrix (2) are

∞ X i+j k (Hn)i,j = µi+j (z) = k ω (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

∞ ∞ X i k X i k i µi (z) = k ω (k) z = ω (k) ϑ z = ϑ µ0 (z) , k=0 k=0

3 where the diﬀerential operator ϑ is deﬁned by [23, Chapter 6]

d ϑ = z . dz

It follows that if the ﬁrst moment µ0 (z) is analytic in some disk |z0| < 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 ∈ {−1, 0, 1} 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

∞ X m ∆n (z) = dm (n) z , |z| < 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 determinant 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 ω (k) is a general function. We give some examples for special cases of ω (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) σ∈Sn 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

5 Deﬁnition 3 If a, m ∈ 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 ∈ Z and m ∈ 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

Corollary 5 If m ∈ 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 ∈ 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 ∈ C.

Proposition 7 For all n ∈ N and r ∈ 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 polynomials.

Theorem 8 Let the moments µi (z) be deﬁned by

∞ X i k µi (z) = k ω (k) z , k=0 and let ∆n (z) denote the Hankel determinant

∆n (z) = det (µi+j−2) . 1≤i,j≤n

Then,

" n # X Y ki 2 ∆n (z) = ω (ki) z V (k1, k2, . . . , kn) . (16)

0≤k1

Proof. This result is essentially contained in the proof of the Theorem in Section 6.10.4 of [31]. We reproduce the main steps for the purpose of completion. Let n ≤ N and

N ! X i+j−2 ki ∆n,N (z) = det ki ω (ki) z . 1≤i,j≤n ki=0 Using the linearity of the determinant, we have

N " n #" n # X Y ki Y i−1 j−1 ∆n,N (z) = ω (ki) z ki det ki 1≤i,j≤n k1,...,kn=0 i=1 i=1 N " n #" n # X Y ki Y i−1 = ω (ki) z ki V (k1, k2, . . . , kn) . k1,...,kn=0 i=1 i=1

Since n Y ki ω (ki) z i=1

9 is a symmetric function of (k1, k2, . . . , kn) , we get

N " n # n ! 1 X Y X Y ∆ (z) = ω (k ) zki ki−1 V k , . . . , k . n,N n! i σ(i) σ(1) σ(n) k1,...,kn=0 i=1 σ∈Sn i=1

Using the identity [31, 4.1.9]

n ! X Y i−1 2 kσ(i) V kσ(1), . . . , kσ(n) = V (k1, k2, . . . , kn) , σ∈Sn i=1 we obtain

N " n # 1 X Y ∆ (z) = ω (k ) zki V 2 (k , k , . . . , k ) . n,N n! i 1 2 n k1,...,kn=0 i=1

Finally, since 2 V (k1, k2, . . . , kn) is a symmetric function of (k1, k2, . . . , kn) that vanishes if ki = kj, we can rewrite ∆n,N (z) as

" n # X Y ki 2 ∆n,N (z) = ω (ki) z V (k1, k2, . . . , kn) .

0≤k1

2.2 Integer partitions In this section, we will collect powers of z in (16) to ﬁnd a power series for ∆n (z) . First, we deﬁne the concept of integer partitions.

Definition 9 A partition Λ is any (finite or infinite) weakly decreasing sequence Λ = (λ1, λ2, . . . , λr,...) (17) of non-negative integers

λi ≥ λi+1 ≥ 0, i ∈ N,

10 containing only ﬁnitely many non-zero terms. We do not distinguish between two partitions which diﬀer only by a string of zeros at the end. For example,

(2, 1) = (2, 1, 0) = (2, 1, 0, 0).

The non-zero elements λi in (17) are called the parts of Λ. The number of parts is the length of Λ, denoted by l (Λ)

l (Λ) = card {λi ∈ Λ | λi 6= 0} < ∞.

The sum of the parts is the weight of Λ, denoted by kΛk ,

∞ l(Λ) X X kΛk = λi = λi. i=1 i=1 If kΛk = m, we say that Λ is a partition of m. The set of all partitions of m is denoted by P (m). In particular, P (0) consists of a single element, the unique partition of zero, which we denote by Λ = (0) . We will denote by Pn (m) the set of all partitions of m into at most n parts, Pn (m) = {Λ | kΛk = m and l (Λ) ≤ n} .

We now establish a connection between the sum in (16) and those indexed by partitions.

Lemma 10 Let n ∈ N and f be an arbitrary function. Then, X X f (k1, k2, . . . , kn) = f (λn, λn−1 + 1, . . . , λ1 + n − 1) . (18)

0≤k1

Proof. We deﬁne the bijection

n ϕ : {Λ | l (Λ) ≤ n} → {(k1, k2, . . . , kn) ∈ Z | 0 ≤ k1 < k2 < ··· < kn} by ϕ = T ◦ σ, where σ ∈ Sn is given by

σ (λ1, λ2, . . . , λn) = (λn, λn−1, . . . , λ1)

11 and the aﬃne transformation T is deﬁned by

T (x1, x2, . . . , xn) = (x1, x2, . . . , xn) + (0, 1, . . . , n − 1) . Note that n σ : {Λ | l (Λ) ≤ n} → {(x1, x2, . . . , xn) ∈ Z | 0 ≤ x1 ≤ x2 ≤ · · · ≤ xn} and n T : {(x1, x2, . . . , xn) ∈ Z | 0 ≤ x1 ≤ x2 ≤ · · · ≤ xn} n → {(k1, k2, . . . , kn) ∈ Z | 0 ≤ k1 < k2 < ··· < kn} since k1 = x1 ≥ 0 and

ki = xi + i − 1 ≤ xi+1 + i − 1 < xi+1 + i = ki+1, 1 ≤ i ≤ n − 1.

We can now write (16) as a sum over partitions.

Proposition 11 Let n ∈ N. Then, ∞ " n # ∆n (z) X m X Y n = z ω (λn−i+1 + i − 1) (19) (2) z m=0 Λ∈Pn(m) i=1 2 × V (λn, λn−1 + 1, . . . , λ1 + n − 1) . Proof. Using (18) in (16), we have

" n # X Y λn−i+1+i−1 ∆n (z) = ω (λn−i+1 + i − 1) z l(Λ)≤n i=1 2 × V (λn, λn−1 + 1, . . . , λ1 + n − 1) . But n n X X n (n − 1) n (λ + i − 1) = λ + = m + , n−i+1 i 2 2 i=1 i=1 and therefore " n # X Y λn−i+1+i−1 ω (λn−i+1 + i − 1) z l(Λ)≤n i=1 ∞ " n # (n)X m X Y = z 2 z ω (λn−i+1 + i − 1) .

m=0 Λ∈Pn(m) i=1

12 We have now all the elements to prove our main theorem.

Theorem 12 If Λ ∈ Pn (m) , then

V (λn, λn−1 + 1, . . . , λ1 + n − 1) = G (n) LΛΨΛ (n) , (20) where l(Λ) i−1 Y 1 Y λi L = 1 − , (21) Λ (λ )! λ + i − j i=1 i j=1 j and l(Λ) Y Ψ (n) = (n − i + 1) . (22) Λ λi i=1 Proof. Let l (Λ) = r. (i) If r = 0, then Λ = (0) and using (13), we have

V (λn, λn−1 + 1, . . . , λ1 + n − 1) = V (0, 1, 2, . . . , n − 1)

= G (n) = G (n) L(0)Ψ(0) (n) , since L(0) = Ψ(0) (n) = 1. (23) (ii) If r = 1, then Λ = (m). Using (13) and (15), we get

V (λn, λn−1 + 1, . . . , λ1 + n − 1) = V (0, 1, 2, . . . , n − 2, m + n − 1) n−1 Y Γ(m + n) = V (0, 1, 2, . . . , n − 2) (m + n − i) = G (n − 1) . Γ(m + 1) i=1 But Γ(m + n) Γ(n) Γ(m + n) (n − 1)! = = (n) , Γ(m + 1) Γ(m + 1) Γ(n) m! m and therefore (n − 1)! V (λ , λ + 1, . . . , λ + n − 1) = G (n − 1) (n) n n−1 1 m! m = G (n) L(m)Ψ(m) (n) , since 1 G (n) = (n − 1)!G (n − 1) ,L = , Ψ (n) = (n) . (m) m! (m) m 13 (iii) If r ≥ 2, then using (13) and (15), we obtain

V (λn, λn−1 + 1, . . . , λ1 + n − 1) = V (0, 1, 2, . . . , n − r − 1) n n−r Y Y ×V (λr + n − r, . . . , λ1 + n − 1) (λn−j+1 + j − i) j=n−r+1 i=1 n n−r Y Y = G (n − r − 2) V (λr + n − r, . . . , λ1 + n − 1) (λn−j+1 + j − i) . j=n−r+1 i=1

(iv) We have

n n−r r n−r Y Y YY (λn−j+1 + j − i) = (λl + n − l + 1 − i)(l = n − j + 1) j=n−r+1 i=1 l=1 i=1 r n−r−1 Y Y = (λl − l + r + 1 + s)(s = n − r − i) l=1 s=0 r Y = (λl − l + r + 1)n−r . l=1

Setting x = λl + 1, a = r − l and b = n − r in the identity

(x)a+b = (x)a (x + a)b , we get (λl + 1)n−l (λl − l + r + 1)n−r = . (λl + 1)r−l Hence,

r Y (λ + 1) r r l n−l Y Y(λl + 1) l=1 (λ − l + r + 1) = n−l = , l n−r r−1 (λl + 1) l=1 l=1 r−l Y (λl + 1)r−l l=1

14 since (x)0 = 1. We conclude that

r r Y Y (λ + 1) (λ + 1) n n−r l n−l l n−l Y Y l=1 l=1 (λ + j − i) = = . n−j+1 r−1r−l−1 r−1 r j=n−r+1 i=1 Y Y Y Y (λl + 1 + s) (λl + s − l) l=1 s=0 l=1s=l+1 (v) Using (12), we get

r−2 j YY V (λr + n − r, . . . , λ1 + n − 1) = (λr−j−1 − λr−i + j − i + 1) j=0 i=0 r−1r−1−l Y Y = (λl − λr−i + r − l − i)(l = r − j − 1) l=1 i=0 r−1 r Y Y = (λl − λs + s − l)(s = r − i) . l=1s=l+1

(vi) From the previous results, we have

V (λn, λn−1 + 1, . . . , λ1 + n − 1)  r  Y (λ + 1) n−r−1 !  l n−l  "r−1 r # Y  l=1  Y Y = l!   (λ − λ + s − l)  r−1 r  l s l=1 Y Y  l=1s=l+1  (λl + s − l) l=1s=l+1 " r # r−1 r Y Y Y λl + s − l − λs = (λl + 1)n−l . λl + s − l l=1 l=1s=l+1

Setting x = 1, a = λl and b = n − l in the identity (x + a) (x + b) b = a , (x)b (x)a we obtain (1) (n − l + 1) (λ + 1) = n−l λl . l n−l (1) λl

15 Thus,

r r " r #" r # (1) (n − l + 1) (n − l + 1) Y Y n−l λl Y Y λl (λl + 1)n−l = = (n − l)! (1) (λi)! l=1 l=1 λl l=1 l=1 n−1 !" r # Y Y(n − l + 1)λ = i! l (i = n − l) . (λi)! i=n−r l=1 We conclude that

V (λn, λn−1 + 1, . . . , λ1 + n − 1) "n−1 #" r # r−1 r Y Y(n − l + 1)λ Y Y λl + s − l − λs = i! l . (λi)! λl + s − l i=1 l=1 l=1s=l+1

(vii) Finally,

r−1 r r−1 r Y Y λl + s − l − λs Y Y λs = 1 − λl + s − l λl + s − l l=1s=l+1 l=1s=l+1 r s−1 YY λs = 1 − , λl + s − l s=2 l=1 since 2 ≤ l + 1 ≤ s ≤ r. But for s = 1 we get 0 Y λ1 1 − = 1 λl + 1 − l l=1 and therefore we can write r−1 r r s−1 Y Y λl + s − l − λs YY λs = 1 − . λl + s − l λl + s − l l=1s=l+1 s=2 l=1

With the help of the previous result, we obtain a new expression for ∆n (z) as a sum over partitions.

16 Corollary 13 Let n ∈ N. Then,

∞ l(Λ)  ∆n (z) X X Yω (λi + n − i) 2 = zm [L Ψ (n)] . "n−1 #  ω (n − i)  Λ Λ n Y m=0 Λ∈P(m) i=1 z(2)G2 (n) ω (i) i=0 (24)

Proof. Let Λ ∈ Pn (m) . Since

λs = 0, l (Λ) < s ≤ n, we have n n Y Y ω (λn−i+1 + i − 1) = ω (λs + n − s)(i = n − s + 1) i=1 s=1 " n # n−1 l(Λ)  n−1 Yω (λs + n − s) Y Yω (λs + n − s) Y = ω (s) = ω (s) , ω (n − s)  ω (n − s)  s=1 s=0 s=1 s=0 where we recall from (6) that ω (s) 6= 0. If l (Λ) ≤ n, then the formula follows after using (20) in (19). If l (Λ) > n, then n l(Λ) Y Y Ψ (n) = (n − i + 1) (n − i + 1) = 0, Λ λi λi i=1 i=n+1 and therefore

l(Λ)  X Yω (λi + n − i) 2 [L Ψ (n)]  ω (n − i)  Λ Λ Λ∈Pn(m) i=1

l(Λ)  X Yω (λi + n − i) 2 = [L Ψ (n)] .  ω (n − i)  Λ Λ Λ∈P(m) i=1

We now compute some particular values of LΛΨΛ (n) .

Proposition 14 We have

L(0)Ψ(0) (n) = 1, (25)

17 m − 3 n + m − 3 L Ψ (n) = n , (26) (m−2,2) (m−2,2) 2 m − 1 and for 0 ≤ a ≤ m − 1,

m − 1n + m − 1 − a L Ψ (n) = . (27) (m−a,1,...,1) (m−a,1,...,1) a m

In particular, we have

n − 1 + m L Ψ (n) = , (28) (m) (m) m and n L Ψ (n) = . (29) (1,...,1) (1,...,1) m

Proof. Using (21) and (22), we get

1 1 2 1 m − 3 L = 1 − = (m−2,2) (m − 2)! 2! m − 2 + 1 2 (m − 1)! and (n + m − 3)! Ψ (n) = (n) (n − 1) = n . (30) (m−2,2) m−2 2 (n − 2)! Thus, n (n + m − 3)! L Ψ (n) = (m − 3) . (m−2,2) (m−2,2) 2 (m − 1)! (n − 2)! Similarly,

a+1 " i−1 # 1 Y 1 Y 1 L = 1 − 1 − (m−a,1,...,1) (m − a)! m − a + i − 1 1 + i − j i=2 j=2 a+1 1 Y 1 1 1 m − a 1 = 1 − = (m − a)! m − a + i − 1 i − 1 (m − a)! m a! i=2 and

a+1 Y (n + m − a − 1)! (n − 1)! Ψ (n) = (n) (n − i + 1) = . (m−a,1,...,1) m−a (n − 1)! (n − 1 − a)! i=2

18 Thus,

m! (n + m − a − 1)! L Ψ (n) = . (m−a,1,...,1) (m−a,1,...,1) a!(m − a − 1)!m m!(n − 1 − a)!

Formula (28) follows from (27) if we set a = 0, and (28) follows from (27) if we set a = m − 1. Using the previous result, we compute LΛΨΛ (n) for all Λ ∈ P (m) , 0 ≤ m ≤ 5.

Corollary 15 Let n ∈ N. Then, we have: (i) For Λ ∈ P (0) , L(0)Ψ(0) (n) = 1. (ii) For Λ ∈ P (1) , n L Ψ (n) = . (1) (1) 1 (iii) For Λ ∈ P (2) ,

n + 1 n L Ψ (n) = ,L Ψ (n) = . (2) (2) 2 (1,1) (1,1) 2

(iv) For Λ ∈ P (3) ,

n + 2 n + 1 L Ψ (n) = ,L Ψ (n) = 2 , (3) (3) 3 (2,1) (2,1) 3 n L Ψ (n) = . (1,1,1) (1,1,1) 3

(v) For Λ ∈ P (4) ,

n + 3 n + 2 L Ψ (n) = ,L Ψ (n) = 3 , (4) (4) 4 (3,1) (3,1) 4 nn + 1 n + 1 L Ψ (n) = ,L Ψ (n) = 3 , (2,2) (2,2) 2 3 (2,1,1) (2,1,1) 4 n L Ψ (n) = . (1,1,1,1) (1,1,1,1) 4

19 (vi) For Λ ∈ P (5) ,

n + 4 n + 3 L Ψ (n) = ,L Ψ (n) = 4 (5) (5) 5 (4,1) (4,1) 5 n + 2 n + 2 L Ψ (n) = n ,L Ψ (n) = 6 (31) (3,2) (3,2) 4 (3,1,1) (3,1,1) 5 n + 1 n + 1 L Ψ (n) = n ,L Ψ (n) = 4 (2,2,1) (2,2,1) 4 (2,1,1,1) (2,1,1,1) 5 n L Ψ (n) = . (1,1,1,1,1) (1,1,1,1,1) 5

Proof. All the results follow from Proposition 14, except for L(2,2,1)Ψ(2,2,1) (n) . In this case, we have 1 L = (2,2,1) 6 and

Ψ(2,2,1) (n) = (n)2 (n − 1)2 (n − 2)1 1 4! n + 1 = n2 (n − 1) (n − 2) (n + 1) = n . 4 4 4

Hence, n + 1 L Ψ (n) = n . (2,2,1) (2,2,1) 4

We can now an explicit formula for the ﬁrst few terms in the power series of ∆n (z) .

Corollary 16 Let gn (z) be deﬁned by ∆ (z) g (z) = n . n n−1 n Y z(2)G2 (n) ω (i) i=0 Then, ∞ X m gn (z) = 1 + rm (n) z , (32) m=1

20 where ω (n) n + 12 ω (n + 1) n2 ω (n) r (n) = n2 z, r (n) = + , 1 ω (n − 1) 2 2 ω (n − 1) 2 ω (n − 2)

n + 22 ω (n + 2) n + 12 ω (n + 1) n2 ω (n) r (n) = + 4 + , 3 3 ω (n − 1) 3 ω (n − 2) 3 ω (n − 3)

n + 32 ω (n + 3) n + 22 ω (n + 2) r (n) = + 9 4 4 ω (n − 1) 4 ω (n − 2) n2 n + 12 ω (n) ω (n + 1) + 4 3 ω (n − 1) ω (n − 2) n + 12 ω (n + 1) n2 ω (n) + 9 + , 4 ω (n − 3) 4 ω (n − 4) and n + 42 ω (n + 4) n + 32 ω (n + 3) r (n) = + 16 5 5 ω (n − 1) 5 ω (n − 2) n + 22 ω (n) ω (n + 2) n + 22 ω (n + 2) + n2 + 36 4 ω (n − 1) ω (n − 2) 5 ω (n − 3) n + 12 ω (n) ω (n + 1) + n2 4 ω (n − 1) ω (n − 3) n + 12 ω (n + 1) n2 ω (n) + 16 + . 5 ω (n − 4) 5 ω (n − 5)

2.3 Examples In this section we apply the previous results to some speciﬁc families of orthogonal polynomials.

Example 17 Charlier polynomials. If 1 ω (k) = , (33) k! then the polynomials orthogonal with respect to the linear functional (5) are known as Charlier polynomials [22, 18.19]. The monic Charlier polynomials

21 have the hypergeometric representation [22, 18.20.7]

−n, −x P (x) = (−z)n F ; −z−1 , n 2 0 − where ∞ k a1, . . . , ap X(a1)k ··· (ap)k z pFq ; z = b1, . . . , bq (b1) ··· (bq) k! k=0 k k denotes the generalized hypergeometric function [22, Chapter 16]. When z > 0, the monic Charlier polynomials satisfy the orthogonality relation

∞ X zk P (k) P (k) = n!znezδ . (34) n m k! n,m k=0 If we use (33) in (32), we obtain

5 X zk g (z) = nk + O z6 . n k! k=0 In the next section, we will prove that

nz gn (z) = e .

Example 18 Meixner polynomials. If

(a) ω (n) = n , (35) n! then the polynomials orthogonal with respect to the linear functional (5) are known as Meixner polynomials [22, 18.19]. The monic Meixner polynomials can be represented by [22, 18.20.7] −n −n, −x M (x) = (a) 1 − z−1 F ; 1 − z−1 , n n 2 1 a where a > 0 and 0 < z < 1. They satisfy the orthogonality relation

∞ X (a) −a−2n M (k) M (k) k zk = n!(a) zn (1 − z) δ . (36) n m k! n n,m k=0

22 If we use (35) in (32), we obtain

5 X zk g (z) = (n (n + a − 1)) + O z6 . n k k! k=0 In the next section, we will prove that

−n(a+n−1) gn (z) = (1 − z) .

Example 19 Generalized Charlier polynomials. If 1 1 ω (k) = , b > −1, (37) (b + 1)k k! then the polynomials orthogonal with respect to the linear functional (5) are known as Generalized Charlier polynomials [27]. If we use (37) in (32), we obtain 1 n2 + bn − 1 r1 (n) = , r2 (n) = , n + b (n + b − 1)3 n4 + 2bn3 + (b2 − 5) n2 − 3bn + 4 r3 (n) = , (n + b − 2)5 and

1 7 6 2 5 2 4 r4 (n) = n + 4bn + 6b − 14 n + 2b 2b − 17 n (n + b)(n + b − 3)7 + b4 − 26b2 + 49 n3 − 2b 3b2 − 31 n2 + 19b2 − 36 n − 30b .

It is of course possible to compute higher terms, but we haven’t found a convenient way to represent them, and the expressions become increasingly cumbersome.

Example 20 In general, let ξ (k) ω (k) = . (38) k! We have ω (n − i + λ ) ξ (n − i + λ ) (n − i)! ξ (n − i + λ ) 1 i = i = i . ω (n − i) ξ (n − i) (n − i + λ )! ξ (n − i) (n − i + 1) i λi

23 Thus, l(Λ) l(Λ) Yω (n − i + λi) 1 Yξ (n − i + λi) = . ω (n − i) Ψ (n) ξ (n − i) i=1 Λ i=1 From (14), we get

n−1 n−1 Y Y G2 (n) ω (i) = G (n) ξ (i) . i=0 i=0 Hence, replacing in (24) we obtain

∞ l(Λ)  ∆n (z) X X Yξ (n − i + λi) = zm L2 Ψ (n) . (39) n−1  ξ (n − i)  Λ Λ n Y m=0 Λ∈P(m) i=1 z(2)G (n) ξ (i) i=0 All families of discrete semiclassical orthogonal polynomials have weight functions like (38) [8]. They include:

Polynomials ξ (k) Charlier 1

Meixner (a)k , a > 0 Generalized Charlier 1 , b > −1 (b+1)k (a) Generalized Meixner k , a > 0, b > −1 (b+1)k Generalized Krawtchouk (a)k (−N)k , a > 0,N ∈ N (a1)k(a2)k Generalized Hahn , a1, a2 > 0, b > −1. (b+1)k 3 Combinatorial interpretation

3.1 Young tableaux In this section, we will relate our previous results to some combinatorial objects. We begin with some deﬁnitions and examples, to aid those reader who are not experts in these topics.

Deﬁnition 21 The (Ferrers or Young) diagram DΛ of the partition Λ with

24 l (Λ) = τ is the set of boxes

λ1 boxes ···

λ2 boxes ··· . .

λτ boxes ··· .

In other words (with coordinates representing boxes)

DΛ = {(i, j) : 1 ≤ i ≤ l (Λ) , 1 ≤ j ≤ λi} .

Example 22 The diagrams for the partitions Λ ∈ P (5) are

, , , , , , .

Deﬁnition 23 A Young tableau T (Λ) (of shape Λ) is a Young diagram DΛ for a partition Λ ∈ P (n) and a bijective assignment of the numbers (1, ..., n) to the n boxes of the diagram DΛ. A Young tableau T (Λ) is called standard if and only if the numbers in- crease in each row and each column.

Example 24 The standard Young tableaux for Λ = (3, 2) are

1 2 3 1 2 4 1 2 5 1 3 4 1 3 5 4 5 , 3 5 , 3 4 , 2 5 , 2 4 .

Deﬁnition 25 Let Λ be a partition. We deﬁne

f (Λ) = number of standard Young tableaux T (Λ) . (40)

Example 26 For m = 5, we have

f (1, 1, 1, 1, 1) = 1, f (2, 1, 1, 1) = 4, f (2, 2, 1) = 5, (41) f (3, 1, 1) = 6, f (3, 2) = 5, f (4, 1) = 4, f (5) = 1.

25 The next result shows that the values of f (Λ) for Λ ∈ P (m) also give the dimensions of the irreducible representations of the symmetric group Sm [26, Chapter V]. Theorem 27 We have X f 2 (Λ) = m!. Λ∈P(m) Proof. See [26, VI.2.2.]. Next, we relate the function LΛ deﬁned in the previous section with f (Λ) . Theorem 28 Let Λ ∈ P (m) with l (Λ) = τ and

ti = λi + τ − i, 1 ≤ i ≤ τ. Then, Y (ti − tk) 1≤i

Corollary 29 Let Λ be a partition. The function LΛ deﬁned in (21) can be written as f (Λ) L = . Λ (kΛk)! Proof. Let Λ be a partition with l (Λ) = τ. From (42), we get τ Y Y (λi − λk + k − i) (λi − λk + k − i) τ f (Λ) 1≤i

j=1 j=λi+1 j=1 τ Y = (λi)! (λi + k − i)(j = k − i) . k=i+1

26 Thus, τ τ τ τ f (Λ) Y 1 Y (λi − λk + k − i) Y 1 Y λk = = 1 − . m! (λi)! (λi + k − i) (λi)! λi + k − i i=1 k=i+1 i=1 k=i+1 But τ τ τ k−1 Y Y λk YY λk 1 − = 1 − , λi + k − i λi + k − i i=1k=i+1 k=1i=1 and therefore τ k−1 f (Λ) Y 1 Y λk = 1 − = LΛ. m! (λk)! λi + k − i k=1 i=1

We now ﬁnd a connection between the function ΨΛ (n) deﬁned in the previous section, and some properties of the diagram DΛ.

Definition 30 Let Λ be a partition with diagram DΛ. 1) We define the hook length hi,j (Λ) of Λ at (i, j) ∈ DΛ as the number of boxes directly to the right or directly below the box located at (i, j) , counting (i, j) itself once. 2) We define the content ci,j (Λ) of Λ at (i, j) ∈ DΛ by

ci,j (Λ) = j − i. (43) Example 31 The hook lengths of Λ = (3, 2) are

4 3 1 2 1 . (44) The contents of Λ = (3, 2) are

0 1 2 − 1 0 . (45)

Remark 32 Let Λ be a partition. Then,

X 2 2 2 hi,j (Λ) − ci,j (Λ) = kΛk .

(i,j)∈DΛ For a proof see [20, I.1. Example 5.].

27 Proposition 33 Let Λ = (λ1, λ2, . . . , λr) and

ti = λi + r − i, 1 ≤ i ≤ r.

Then, r Y (ti)! Y i=1 hi,j (Λ) = Y . (t − t ) (i,j)∈DΛ i k 1≤i

Proof. See [20, I.1. Example 1.].

Corollary 34 Hook length formula. Let Λ be a partition. Then,

(kΛk)! f (Λ) = Y . (46) hi,j (Λ)

(i,j)∈DΛ

Example 35 Let Λ = (3, 2) . Using (44) in (46), we obtain

5! f (3, 2) = = 5, 4 × 3 × 1 × 2 × 1 in agreement with (41).

Deﬁnition 36 Let Λ be a partition. We deﬁne the content polynomial of Λ by [20, I.1 Example 11] Y CΛ (x) = [x + ci,j (Λ)] . (47)

(i,j)∈DΛ

Example 37 Let Λ = (3, 2) . Using (45) in (47), we get

2 C(3,2) (x) = (x − 1) x (x + 1) (x + 2) .

Note that from (30) we have

(n + 5 − 3)! Ψ (n) = n = n2 (n − 1) (n + 2) (n + 1) = C (n) . (3,2) (n − 2)! (3,2)

28 Proposition 38 Let Λ = (λ1, λ2, . . . , λr). Then,

CΛ (x) = ΨΛ (x) .

Proof. Using (43) and (47), we have

r λ r λ −1 Y YYi Y Yi [x + ci,j (Λ)] = (x + j − i) = (x − i + 1 + j)

(i,j)∈DΛ i=1 j=1 i=1 j=0 r Y = (x − i + 1) = Ψ (x) . λi Λ i=1

3.2 Schur polynomials In this section now introduce the concept of Schur polynomials, which will be essential in ﬁnding closed-form formulas for the Charlier and Meixner polynomials.

Deﬁnition 39 Let n ∈ N and Λ be a partition. We deﬁne the Schur polynomial sΛ by [20, I.3]

sΛ (x1, . . . , xn) = 0, l (Λ) > n and n−j+λj det xi 1≤i,j≤n s (x , . . . , x ) = , l (Λ) ≤ n. Λ 1 n n−j det xi 1≤i,j≤n

Example 40 Let n = 3 and m = 5. We have

2 s(1,1,1,1,1) = 0, s(2,1,1,1) = 0, s(2,2,1) = e2e3, s(3,1,1) = e1 − e2 e3, 2 2 2 s(3,2) = e1e2 − e1 + e2 e3, s(4,1) = (e3 − e1e2) 2e2 − e1 5 3 2 2 s(5) = e1 − 4e1e2 + 3e3e1 + 3e1e2 − 2e3e2, where ek denotes the elementary symmetric polynomials deﬁned by

n n Y X k n−k (t − xk) = (−1) ek (x1, . . . , xn) t . k=1 k=0

29 The following is known as the principal specialization of Schur polynomials.

Theorem 41 Let n ∈ N and Λ be a partition with l (Λ) ≤ n. Then,

n Y n + ci,j (Λ) sΛ (1 ) = , hi,j (Λ) (i,j)∈DΛ where   n sΛ (1 ) = sΛ 1, 1,..., 1 . | {z } n times

Proof. See [20, I.3. Example 4]. We have now all the elements to represent LΛΨΛ (n) and ∆n (z) in terms of Schur polynomials.

Corollary 42 Let n ∈ N and Λ ∈ Pn (m) . Then, f (Λ) s (1n) = C (n) = L Ψ (n) (48) Λ m! Λ Λ Λ and

∞ m " n # ∆n (z) X z X Yξ (n − i + λi) = f (Λ) s (1n) . "n−1 # m! ξ (n − i) Λ n Y m=0 Λ∈P (m) i=1 z(2)G (n) ξ (i) n i=0 (49)

The next result will be needed in ﬁnding the Hankel determinants for the Charlier polynomials.

Proposition 43 Let n, m ∈ N. Then,

X m m f (Λ) sΛ (x1, . . . , xn) = (x1 + ··· + xn) = e1 . (50)

Λ∈Pn(m)

Proof. See [20, I.4. Example 3].

30 Example 44 Let n = 3 and m = 5. We have

X 2 f (Λ) sΛ (x1, x2, x3) = 5e2e3 + 6 e1 − e2 e3

Λ∈P3(5) 2 2 2 +5 e1e2 − e1 + e2 e3 + 4 (e3 − e1e2) 2e2 − e1 5 3 2 2 5 +e1 − 4e1e2 + 3e3e1 + 3e1e2 − 2e3e2 = e1.

Corollary 45 Charlier polynomials. Let ξ (k) = 1. Then,

(n) nz ∆n (z) = z 2 G (n) e .

Proof. Setting x1 = x2 = ··· = xn = 1 in (50), we have X n m f (Λ) sΛ (1 ) = n . (51) Λ∈P(m) Thus, from (49) we get

∞ m ∆n (z) X z = nm = enz. (n) m! z 2 G (n) m=0

The next result will be essential in ﬁnding the Hankel determinants for the Meixner polynomials.

Theorem 46 Let n1, n2 ∈ N. Then,

∞ n1 n2 X m X YY −1 z sΛ (x1, . . . , xn1 ) sΛ (y1, . . . , yn2 ) = (1 − xiyjz) . (52) m=0 Λ∈P(m) i=1 j=1

Proof. See [20, I.4. ].

Proposition 47 Let m ∈ N0 and x, y be indeterminate variables. Then, X (xy) L2 Ψ (x)Ψ (y) = m . (53) Λ Λ Λ m! Λ∈P(m)

31 Proof. Setting

xi = yj = 1, 1 ≤ i ≤ n1, 1 ≤ j ≤ n2 in (52), we get

∞ n1 n2 X m X n1 n2 YY −1 z sΛ (1 ) sΛ (1 ) = (1 − z) m=0 Λ∈P(m) i=1 j=1 ∞ −n n X (n1n2) = (1 − z) 1 2 = m zm. m! m=0 Hence, from (48) we obtain

X X (n1n2) L2 Ψ (n )Ψ (n ) = s (1n1 ) s (1n2 ) = m . Λ Λ 1 Λ 2 Λ Λ m! Λ∈P(m) Λ∈P(m)

But this is an identity between polynomials of degree m, and since it’s valid for all n1, n2 ∈ N it must be valid for indeterminate variables x, y.

Example 48 Let m = 5. Using (31), we have

2 X 2 (5!) LΛΨΛ (x)ΨΛ (y) = (x − 4)5 (y − 4)5 + 16 (x − 3)5 (y − 3)5 Λ∈P(5)

+25xy (x − 2)4 (y − 2)4 + 36 (x − 2)5 (y − 2)5 + 25xy (x − 1)4 (y − 1)4

+16 (x − 1)5 (y − 1)5 + (x)5 (y)5 = 5! (xy)5 .

Corollary 49 Meixner polynomials. Let ξ (k) = (a)k . Then,

"n−1 # n Y −n(a+n−1) (2) ∆n (z) = z G (n) (a)i (1 − z) . i=0

Proof. Replacing

l(Λ) l(Λ) Yξ (n − i + λi) Y = (a + n − i) = ΨΛ (a + n − 1) ξ (n − i) λi i=1 i=1

32 in (39), we have

∞ ∆n (z) X X = zm L2 Ψ (n)Ψ (a + n − 1) , n−1 Λ Λ Λ n Y m=0 (2) Λ∈P(m) z G (n) (a)i i=0 and using (53), we get

∞ ∆n (z) X (n (a + n − 1)) −n(a+n−1) = zm m = (1 − z) . n−1 m! n Y m=0 (2) z G (n) (a)i i=0

We can also use the previous results to ﬁnd a general identity for ΨΛ (x) , corresponding to the Charlier weight.

Remark 50 From (53), we see that ∞ X X −xy xy zm L2 Ψ (x)Ψ (y) = (1 − z) = F ; z . Λ Λ Λ 1 0 − m=0 Λ∈P(m) Thus, ∞ m X z X 2 lim LΛΨΛ (x)ΨΛ (y) y→∞ y m=0 Λ∈P(m) xy z xz = lim 1F0 ; = e . y→∞ − y

But since ΨΛ (y) is a monic polynomial with

deg ΨΛ (y) = m, Λ ∈ P (m) , we have Ψ (y) lim Λ = 1, y→∞ ym and using Tannery’s theorem [2], we obtain

∞ X m X 2 xz z LΛΨΛ (x) = e , m=0 Λ∈P(m)

33 or X xm L2 Ψ (x) = Λ Λ m! Λ∈P(m) which is the extension of (51) for an indeterminate variable.

Finally, for weight functions corresponding to discrete semiclassical orthogonal polynomials, we have the following.

Example 51 In general, let

(a ) ··· (a ) ξ (k) = 1 k p k . (b1 + 1)k ··· (bq + 1)k Then,

ξ (n − i + λ ) (a1 + n − i) ··· (ap + n − i) i = λi λi , ξ (n − i) (b + 1 + n − i) ··· (b + 1 + n − i) 1 λi q λi and therefore

l(Λ) Yξ (n − i + λi) ΨΛ (a1 + n − 1) ··· ΨΛ (ap + n − 1) = . ξ (n − i) Ψ (b + n) ··· Ψ (b + n) i=1 Λ 1 Λ q Replacing in (39), we conclude that

"n−1 # (n) Y (a1)k ··· (ap)k ∆n (z) = z 2 G (n) (b1 + 1) ··· (bq + 1) k=0 k k ∞ X m X 2 ΨΛ (a1 + n − 1) ··· ΨΛ (ap + n − 1) × z LΛΨΛ (n) . ΨΛ (b1 + n) ··· ΨΛ (bq + n) m=0 Λ∈P(m)

Unfortunately, we haven’t been able to ﬁnd any identities for Schur polynomials that will allow the computation of X 2 ΨΛ (a1 + n − 1) ··· ΨΛ (ap + n − 1) LΛΨΛ (n) ΨΛ (b1 + n) ··· ΨΛ (bq + n) Λ∈P(m) for p > 1 and q > 0.

34 4 Conclusions

We have obtained a Taylor series

"n−1 # ∞ (n) 2 Y X m ∆n (z) = z 2 G (n) ω (i) rm (n) z , i=0 m=0 for the Hankel determinant

∞ ! X i+j k ∆n (z) = det k ω (k) z , 0≤i,j≤n−1 k=0 where ω (k) is a known nonzero function, G (n) is deﬁned by

n−1 Y G (n) = (i!) , i=0 and the coeﬃcients rm (n) are given by

l(Λ)  X Yω (λi + n − i) 2 r (n) = [L Ψ (n)] , m  ω (n − i)  Λ Λ Λ∈P(m) i=1 with l(Λ) i−1 Y 1 Y λi L = 1 − , Λ (λ )! λ + i − j i=1 i j=1 j and l(Λ) Y Ψ (x) = (x − i + 1) . Λ λi i=1 In particular, when ω (k) is the weight function associated with discrete semiclassical orthogonal polynomials (a ) ··· (a ) 1 ω (k) = 1 k p k , (b1 + 1)k ··· (bq + 1)k k! we have

X 2 ΨΛ (a1 + n − 1) ··· ΨΛ (ap + n − 1) rm (n) = LΛΨΛ (n) . ΨΛ (b1 + n) ··· ΨΛ (bq + n) Λ∈P(m)

35 Using Schur polynomials, we showed that X xm L2 Ψ (x) = Λ Λ m! Λ∈P(m) and X (xy) L2 Ψ (x)Ψ (y) = m . Λ Λ Λ m! Λ∈P(m) From these identities, we obtained exact evaluations for the Hankel determinants of the Charlier polynomials

∞ k ! X i+j z n nz det k = z(2)G (n) e 0≤i,j≤n−1 k! k=0 and Meixner polynomials

∞ k ! "n−1 # X i+j z n Y −n(a+n−1) det k (a) = z(2)G (n) (a) (1 − z) . 0≤i,j≤n−1 k k! i k=0 i=0 We plan to extend these results and study the multivariate rational functions

R (t, x1, . . . , xp, y1, . . . , yq)

X 2 ΨΛ (a1 + x1 − 1) ··· ΨΛ (ap + xp − 1) = LΛΨΛ (t) , ΨΛ (b1 + y1) ··· ΨΛ (bq + yq) Λ∈P(m) in order to obtain closed-form formulas for the Hankel determinants of other families of discrete semiclassical orthogonal polynomials.

Acknowledgement 52 This work was done while visiting the Johannes Kepler Universit¨atLinz. It was supported by the Austrian Science Fund (FWF) under grant SFB F50-07 and by the strategic program ”Innovatives OO–¨ 2010 plus” from the Upper Austrian Government.

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[8] D. Dominici and F. Marcell´an.Discrete semiclassical orthogonal polynomials of class one. Paciﬁc J. Math., 268(2):389–411, 2014.

[9] A. J. Dur´an.Wronskian type determinants of orthogonal polynomials, Selberg type formulas and constant term identities. J. Combin. Theory Ser. A, 124:57–96, 2014.

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39 Technical Reports of the Doctoral Program “Computational Mathematics”

2018

2018-01 D. Dominici: Laguerre-Freud equations for Generalized Hahn polynomials of type I Jan 2018. Eds.: P. Paule, M. Kauers 2018-02 C. Hofer, U. Langer, M. Neumüller: Robust Preconditioning for Space-Time Isogeometric Analysis of Parabolic Evolution Problems Feb 2018. Eds.: U. Langer, B. Jüttler 2018-03 A. Jiménez-Pastor, V. Pillwein: Algorithmic Arithmetics with DD-Finite Functions Feb 2018. Eds.: P. Paule, M. Kauers 2018-04 S. Hubmer, R. Ramlau: Nesterov’s Accelerated Gradient Method for Nonlinear Ill-Posed Problems with a Locally Convex Residual Functional March 2018. Eds.: U. Langer, R. Ramlau 2018-05 S. Hubmer, E. Sherina, A. Neubauer, O. Scherzer: LaméParameter Estimation from Static Displacement Field Measurements in the Framework of Nonlinear Inverse Problems March 2018. Eds.: U. Langer, R. Ramlau 2018-06 D. Dominici: A note on a formula of Krattenthaler March 2018. Eds.: P. Paule, V. Pillwein 2018-07 C. Hofer, S. Takacs: A parallel multigrid solver for multi-patch Isogeometric Analysis April 2018. Eds.: U. Langer, B. Jüttler 2018-08 D. Dominci: Power series expansion of a Hankel determinant June 2018. Eds.: P. Paule, M. Kauers

The complete list since 2009 can be found at https://www.dk-compmath.jku.at/publications/ Doctoral Program “Computational Mathematics”

Director: Dr. Veronika Pillwein Research Institute for Symbolic Computation

Deputy Director: Prof. Dr. Bert J¨uttler Institute of Applied Geometry

Address: Johannes Kepler University Linz Doctoral Program “Computational Mathematics” Altenbergerstr. 69 A-4040 Linz Austria Tel.: ++43 732-2468-6840

E-Mail: oﬃ[email protected]

Homepage: http://www.dk-compmath.jku.at

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