## Hankel hyperdeterminants and Selberg integrals

Citations: | 8 - 3 self |

### BibTeX

@MISC{Luque_hankelhyperdeterminants,

author = {Jean-gabriel Luque and Jean-yves Thibon},

title = {Hankel hyperdeterminants and Selberg integrals},

year = {}

}

### OpenURL

### Abstract

We investigate the simplest class of hyperdeterminants de ned by Cayley in the case of Hankel hypermatrices (tensors of the form A i 1 i 2 :::i k = f(i 1 +i 2 + +i k )). It is found that many classical properties of Hankel determinants can be generalized, and a connection with Selberg type integrals is established. In particular, Selberg's original formula amounts to the evaluation of all Hankel hyperdeterminants built from the moments of the Jacobi polynomials. Many higher-dimensional analogues of classical Hankel determinants are evaluated in closed form. The Toeplitz case is also briey discussed. In physical terms, both cases are related to the partition functions of one-dimensional Coulomb systems with logarithmic potential.

### Citations

390 | The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue
- Koekoek, Swarttouw
- 1998
(Show Context)
Citation Context ...re the Bell numbers). These are the moments of the discrete measure (the Poisson distribution) d a (x) = e a X k0 a k k! (x k) (42) for which the Charlier polynomials are the orthogonal system (cf. [1=-=8]-=-). The monic Charlier polynomials C (a) n (x) satisfy hC (a) n ; C (a) n i = a n n! (43) whence the classical evaluation of the Hankel determinants [38] D (1) n = a n(n 1)=2 n 1 Y j=0 j! : (44) No ana... |

121 |
Total positivity
- Karlin
- 1968
(Show Context)
Citation Context ...he monic Charlier polynomials C (a) n (x) = n!L (x n) n (a) are given by the exponential generating function X n0 C (a) n (x) t n n! = e at (1 + t) x : (115) One has the integral representation (see [=-=16]-=- p. 446) C (a) n (x) = 1 ( x) Z 1 0 e t t x 1 (t a) n dt : (116) From this, we get easily the Hankel hyperdeterminants associated to the sequence c n = C (a) n (x). The result can be cast in the form ... |

94 |
The on-line encyclopedia of integer sequences. Available online at http://www.research.att.com/njas/sequences
- Sloane
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(Show Context)
Citation Context ...) so that, for c n = 2n n , D (k) n;r = 4 kn(n 1)+nr n! n S n (r + 1 2 ; 1 2 ; k) : (34) Hankel hyperdeterminants 7 3.4. Other combinatorial sequences The Online Encyclopaedia of Integer Sequences [3=-=5]-=- provides many further examples of interesting combinatorial sequences arising from the Beta distribution. For example, it is observed in [32] that the sequence A001813 has the representation (2n)! n!... |

71 | The Calogero–Sutherland model and generalized classical polynomials - Baker, Forrester - 1997 |

56 |
The invariant theory of binary forms
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(Show Context)
Citation Context ...2k i c i c 2k i (4) whose right-hand side is well-known in classical invariant theory (it is one-half of the apolar covariant of the binary form f(x; y) = P i 2k i c i x i y 2k i with itself, see [2=-=1]-=-). The case c n = n! will be used as a running exemple throughout this paper. Using (4), we can give oursrst illustration of a higher-order determinant D (k) 2 (c) = 1 2 2k X i=0 ( 1) i (2k)! = 1 2 (2... |

49 |
polynomials associated with Selberg integrals. Siam Math analysis
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(Show Context)
Citation Context ...the result by means of Carlson's theorem (see, e.g., [30]). Hence, Selberg's integral computes precisely the Hankel hyperdeterminants D (k) n (c) when the c n are the moments of the measure d(x) = 1 [=-=0;1]-=- x a 1 (1 x) b 1 dx (the Beta distribution). The orthogonal polynomials for this measure are, up to a simple change of variables, the Jacobi polynomials P (a 1;b 1) n (1 2x). Hence, we can as well com... |

27 |
Bemerkninger om et multiplet integral
- Selberg
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(Show Context)
Citation Context ...x n ) : (10) When k = 1, this is a well-known formula due to Heine [14]. For arbitrary k, the integral can be evaluated in closed form in many interesting cases by means of Selberg's integral formula =-=-=-[34] which gives, for S n (a; b; k) = Z 1 0 Z 1 0 j(x)j 2k n Y i=1 x a 1 i (1 x i ) b 1 dx i (11) the value S n (a; b; k) = n 1 Y j=0 (a+ jk)(b + jk)((j + 1)k + 1) (a+ b + (n + j 1)k)(k + 1) : (12)... |

25 |
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(Show Context)
Citation Context ...ated to sequences of the form c n = Q n (x), where (Q n ) is a family of orthogonal polynomials, have been called Turanians by Karlin and Szego, who computed their values for the classical families [1=-=7-=-]. Recent references on this subject can be found in [26], where these results have been generalized by a dierent method based on a little-known determinantal identity due to Turnbull. In this section... |

22 |
Enumeration of certain Young tableaux with bounded height
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(Show Context)
Citation Context ...tegral D (k) n;r (c) = 2 2kn(n 1)+n(2r+1) n! n S n (r + 1 2 ; 3 2 ; k) (32) In the case k = 1, the Hankel determinants of shifted Catalan numbers have been computed by Desainte-Catherine and Viennot [=-=9]-=- with the aim to enumerate the Young tableaux with even columns and height at most 2n. It would be interesting tosnd a combinatorial interpretation of the hyperdeterminants of shifted Catalan numbers,... |

20 |
Polynômes de Laguerre généralisés
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(Show Context)
Citation Context ...: (72) This formula is needed to calculate the degenerate cases of Kaneko's integral corresponding to Laguerre and Hermite polynomials. The generalized Laguerre polynomials L a (y; ) are dened by [23] (see also [2]) L a (y 1 ; : : : ; y r ; ) = lim b!1 P (a;b) y 1 b ; : : : ; y r b ; (73) (we use there the convention of [42]). Let LS n (a; c) denote the Laguerre-Selberg integral (26). O... |

18 |
Polynômes de Jacobi généralisés
- Lassalle
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(Show Context)
Citation Context ...ll have to rely upon one of its extensions, which is due to Kaneko [15]. 5.1. Kaneko's integral and its variants The required integral formula involves the generalized Jacobi polynomials p ;;s (y) [19=-=, 41, 8, 2-=-2], which are the symmetric polynomials in r variables (y 1 ; : : : ; y r ) obtained by applying the Gram-Schmidt process to the basis of monomial symmetric functions m (y) (ordered by the condition ... |

17 |
Formulas for elementary spherical functions and generalized Jacobi polynomials
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(Show Context)
Citation Context ...ll have to rely upon one of its extensions, which is due to Kaneko [15]. 5.1. Kaneko's integral and its variants The required integral formula involves the generalized Jacobi polynomials p ;;s (y) [19=-=, 41, 8, 2-=-2], which are the symmetric polynomials in r variables (y 1 ; : : : ; y r ) obtained by applying the Gram-Schmidt process to the basis of monomial symmetric functions m (y) (ordered by the condition ... |

14 |
Calcul effectif de certains déterminants de
- Radoux
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(Show Context)
Citation Context ...er polynomials are the orthogonal system (cf. [18]). The monic Charlier polynomials C (a) n (x) satisfy (37) 〈C (a) n , C(a) n 〉 = ann! (39) whence the classical evaluation of the Hankel determinants =-=[37]-=- D (1) n = an(n−1)/2 n−1 ∏ j! . (40) j=0 However, no analogue of Selberg’s integral is known for the measure dµa. So, the best that we can do is to evaluate the fourth-order hyperdeterminants by means... |

11 | Some aspects of hankel matrices in coding theory and combinatorics
- Tamm
(Show Context)
Citation Context ... 2 + 9j 3 ) Q k(n+m 1) 1 j=0 (4j 2 j) R (k) n ( 2; 11; 18; 9; 0; 1; 4) (B.5) and it remains tosnd a closed form for R (k) n (a 1 ; a 2 ; a 3 ; a 4 ; b 1 ; b 2 ; b 3 ). When k = 1, the result is known =-=-=-[39]. If c n = (2n)! we have D (k) n (c) = 1 2 n Z 1 0 Z 1 0 (x) 2k n Y i=1 exp( p x i ) p x i dx i (B.6) = 2 kn(n 1) n Y m=0 (mk 1)! km 1 Y j=0 (2j 1)R (k) n (0; 2; 4; 0) (B.7) Let us give now som... |

11 | Determinants and their applications in mathematical physics
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(Show Context)
Citation Context ...times the 4-fold hyperdeterminant. 3. Examples derived from Selberg's integral The evaluation of Hankel determinants built on classical sequences of combinatorial numbers arises in many contexts (see =-=[20, 40]-=- and references therein). Recent work on the theory of coherent states has led to the discovery of integral representations of many such sequences [31, 32]. For the sequences proportional to moments o... |

10 |
Note on the theory of permutations
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(Show Context)
Citation Context ...ong before Cayley raised the question of extending the notion of determinant to higher-dimensional arrays (e.g., cubic matrices A ijk ), and proposed several answers, under the name hyperdeterminants =-=[6, 7-=-]. The most sophisticated notion of hyperdeterminant has been the object of recent investigations, summarized in the book [11]. However, the simplest possible generalization of the determinant, dened ... |

9 |
Spatial matrices and their applications (in
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Citation Context ...y the k-tuple alternating sum (which vanishes for odd k) Det k (A) = 1 n! X 1 ;; k 2Sn ( 1 ) ( k ) n Y i=1 A 1 (i) k (i) (1) has been almost forgotten. After the book by Sokolov [36] which contains an exhaustive list of references up to 1960, we have found only [37, 13, 4, 12]. These references contain evaluations of a few higher dimensional analogues of some classical determinan... |

8 |
Hua-type integrals, hypergeometric functions and symmetric polynomials
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(Show Context)
Citation Context ...ll have to rely upon one of its extensions, which is due to Kaneko [15]. 5.1. Kaneko's integral and its variants The required integral formula involves the generalized Jacobi polynomials p ;;s (y) [19=-=, 41, 8, 2-=-2], which are the symmetric polynomials in r variables (y 1 ; : : : ; y r ) obtained by applying the Gram-Schmidt process to the basis of monomial symmetric functions m (y) (ordered by the condition ... |

8 |
Advanced determinant calculus, in The Andrews Festschrift, Séminaire Lotharingien de Combinatoire 42
- Krattenthaler
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(Show Context)
Citation Context ...times the 4-fold hyperdeterminant. 3. Examples derived from Selberg's integral The evaluation of Hankel determinants built on classical sequences of combinatorial numbers arises in many contexts (see =-=[20, 40]-=- and references therein). Recent work on the theory of coherent states has led to the discovery of integral representations of many such sequences [31, 32]. For the sequences proportional to moments o... |

7 |
Introduction to the theory of multidimensional matrices
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(Show Context)
Citation Context ...; k 2Sn ( 1 ) ( k ) n Y i=1 A 1 (i) k (i) (1) has been almost forgotten. After the book by Sokolov [36] which contains an exhaustive list of references up to 1960, we have found only [37, 13, 4, 1=-=2]-=-. These references contain evaluations of a few higher dimensional analogues of some classical determinants (Vandermonde, Smith, . . . ). However, the analogues of Hankel determinants do not seem to h... |

6 | Polynômes de Jacobi généralisés et intégrales de Selberg, Electron - Barsky, Carpentier - 1996 |

6 |
Random matrices, log–gases and the Calogero–Sutherland models
- Forrester
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Citation Context ... measure can be written in the form d(x) = Ce V (x) dx, the integral (10) represents the partition function of a one-component logpotential Coulomb system on the line, evaluated ats= 2k (see, e.g., [1=-=0]-=-). It is a common feature of most of these systems that the partition function may be evaluated in closed form only fors= 1; 2; 4 (the cases= 1 can be formulated in terms of Pfaans of bimoments of ske... |

6 |
Handbuch der Kugelfunktionen
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Citation Context ...aster algorithm than the denition. Now, if is a measure on the real line, then D (k) n (c) = 1 n! Z R n 2k (x)d(x 1 ) d(x n ) : (10) When k = 1, this is a well-known formula due to Heine [14]. For arbitrary k, the integral can be evaluated in closed form in many interesting cases by means of Selberg's integral formula [34] which gives, for S n (a; b; k) = Z 1 0 Z 1 0 j(x)j 2k n Y i... |

6 |
On certain formulas of Karlin and Szegö, Séminaire Lotharingien de Combinatoire 41
- Leclerc
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Citation Context ...) is a family of orthogonal polynomials, have been called Turanians by Karlin and Szego, who computed their values for the classical families [17]. Recent references on this subject can be found in [2=-=6-=-], where these results have been generalized by a dierent method based on a little-known determinantal identity due to Turnbull. In this section, we will calculate the hyperdeterminantal analogues of ... |

5 |
On the theory of determinants, Trans
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Citation Context ...tential. 1. Introduction Although determinants have been in use since the mid-eighteenth century, it took almost one hundred years before the modern notation as square arrays was introduced by Cayley =-=[5]-=-. Then, it was not long before Cayley raised the question of extending the notion of determinant to higher-dimensional arrays (e.g., cubic matrices A ijk ), and proposed several answers, under the nam... |

4 | and J-Y Thibon, Pfaffian and Hafnian identities in shuffle algebras - Luque |

3 |
Higher-dimensional GCD matrices, Linear Algebra and its Applications 170
- Haukkanen
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Citation Context ...; k 2Sn ( 1 ) ( k ) n Y i=1 A 1 (i) k (i) (1) has been almost forgotten. After the book by Sokolov [36] which contains an exhaustive list of references up to 1960, we have found only [37, 13, 4, 1=-=2]-=-. These references contain evaluations of a few higher dimensional analogues of some classical determinants (Vandermonde, Smith, . . . ). However, the analogues of Hankel determinants do not seem to h... |

2 |
Selberg integrals and hypergeometric fuctions associated with Jack polynomials
- Kaneko
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(Show Context)
Citation Context ...the classical orthogonal polynomials. Interestingly enough, Selberg's formula will not be sucient to deal with these cases, and we will have to rely upon one of its extensions, which is due to Kaneko =-=[-=-15]. 5.1. Kaneko's integral and its variants The required integral formula involves the generalized Jacobi polynomials p ;;s (y) [19, 41, 8, 22], which are the symmetric polynomials in r variables (y ... |

2 | Polynômes de Hermite généraliés - Lassalle - 1991 |

2 |
Calcul eectif de certains determinants de
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(Show Context)
Citation Context ...lier polynomials are the orthogonal system (cf. [18]). The monic Charlier polynomials C (a) n (x) satisfy hC (a) n ; C (a) n i = a n n! (43) whence the classical evaluation of the Hankel determinants =-=[38-=-] D (1) n = a n(n 1)=2 n 1 Y j=0 j! : (44) No analogue of Selberg's integral is known for the measure d a , andsnding a closed form for the hyperdeterminants would amount tosnd such a generalization. ... |

2 |
Vilenkin and A U Klimyk. Representation of Lie Groups and Special Functions: Recent Advances, volume 316
- Ja
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(Show Context)
Citation Context ...he generalized Laguerre polynomials L a (y; ) are dened by [23] (see also [2]) L a (y 1 ; : : : ; y r ; ) = lim b!1 P (a;b) y 1 b ; : : : ; y r b ; (73) (we use there the convention of [42]). Let LS n (a; c) denote the Laguerre-Selberg integral (26). One can deduce from (67) the Laguerre version of Kaneko's integral. Indeed, Kaneko's formula can also be written as [15] Z [0;1] n R(x; y)... |

1 |
M Kapranov and A
- Gelfand, M
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(Show Context)
Citation Context ... A ijk ), and proposed several answers, under the name hyperdeterminants [6, 7]. The most sophisticated notion of hyperdeterminant has been the object of recent investigations, summarized in the book =-=[11-=-]. However, the simplest possible generalization of the determinant, dened for a kth order tensor on an n-dimensional space by the k-tuple alternating sum (which vanishes for odd k) Det k (A) = 1 n! X... |

1 |
Osservazioni sugli iperdeterminanti
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(Show Context)
Citation Context ...; k 2Sn ( 1 ) ( k ) n Y i=1 A 1 (i) k (i) (1) has been almost forgotten. After the book by Sokolov [36] which contains an exhaustive list of references up to 1960, we have found only [37, 13, 4, 1=-=2]-=-. These references contain evaluations of a few higher dimensional analogues of some classical determinants (Vandermonde, Smith, . . . ). However, the analogues of Hankel determinants do not seem to h... |

1 |
The function P 1 n=1 n r z n and associated polynomials
- Lawden
- 1951
(Show Context)
Citation Context ...f the Hankel determinants D (1) n (c) associated to the sequence c n = d n dt n f(t) with f(t) = e t 1 e t x (54) whose particular case x = 1 gives back one of the determinants computed by Lawden [2=-=5-=-]. To investigate the hyperdeterminantal analogues, we willsnd it convenient to Hankel hyperdeterminants 10 make the substitution t ! i t, and to assume atsrst that x = N is a positive integer. Up to ... |

1 | and J-Y Thibon, Pfaan and Hafnian identities in shue algebras - Luque |

1 | A Penson and J-M Sixdeniers, Integral representations of Catalan and related numbers - unknown authors |

1 |
A Penson and A I Solomon, Coherent states from combinatorial sequences, preprint arXiv:quant-ph/0111151
- unknown authors
- 2001
(Show Context)
Citation Context ...atorial numbers arises in many contexts (see [20, 40] and references therein). Recent work on the theory of coherent states has led to the discovery of integral representations of many such sequences =-=[31, 32]-=-. For the sequences proportional to moments of a Beta distribution, the Hankel hyperdeterminants can be evaluated immediately. We have B(a + n; b) = B(a; b) (a) n (a + b) n (24) so that it is sucient ... |

1 |
J-Y Thibon and B G Wybourne, Powers of the Vandermonde determinant and the quantum Hall eect
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- 1994
(Show Context)
Citation Context ...dermonde determinant, a dicult problem which has been thoroughly discussed in recent years, mainly in view of its potential applications to Laughlin's theory of the fractional quantum Hall eect (see [=-=33]-=- and references therein). Since D (k) n (h) is a homogeneous polynomial of degree n in the h i , its Schur expansion will only involve partitions of length at most n. We can therefore assume that x = ... |

1 |
The function ∑∞ n=1 nrzn and associated polynomials
- Lawden
- 1951
(Show Context)
Citation Context ...inds the Hankel determinants D (1) n (c) associated to the sequence cn = dn ( t e dtnf(t) with f(t) = 1 − et ) x (73) whose particular case x = 1 gives back one of the determinants computed by Lawden =-=[25]-=-. To investigate the hyperdeterminantal analogues, we will find it convenient to make the substitution t → iπ − t, and to assume at first that −x = N is a positive integer. Up to a trivial sign, we ca... |