scholarly journals Hankel determinants of linear combinations of moments of orthogonal polynomials, II

2021 ◽  
Author(s):  
C. Krattenthaler
Keyword(s):  
Orthogonal Polynomials ◽  
Matrix Theory ◽  
Binomial Coefficients ◽  
Linear Recurrence ◽  
Hankel Determinants ◽  
Motzkin Numbers ◽  
Schröder Numbers ◽  
Delannoy Numbers ◽  
Dodgson Condensation ◽  
Full Proof

AbstractWe present a formula that expresses the Hankel determinants of a linear combination of length $$d+1$$ d + 1 of moments of orthogonal polynomials in terms of a $$d\times d$$ d × d determinant of the orthogonal polynomials. This formula exists somehow hidden in the folklore of the theory of orthogonal polynomials but deserves to be better known, and be presented correctly and with full proof. We present four fundamentally different proofs, one that uses classical formulae from the theory of orthogonal polynomials, one that uses a vanishing argument and is due to Elouafi (J Math Anal Appl 431:1253–1274, 2015) (but given in an incomplete form there), one that is inspired by random matrix theory and is due to Brézin and Hikami (Commun Math Phys 214:111–135, 2000), and one that uses (Dodgson) condensation. We give two applications of the formula. In the first application, we explain how to compute such Hankel determinants in a singular case. The second application concerns the linear recurrence of such Hankel determinants for a certain class of moments that covers numerous classical combinatorial sequences, including Catalan numbers, Motzkin numbers, central binomial coefficients, central trinomial coefficients, central Delannoy numbers, Schröder numbers, Riordan numbers, and Fine numbers.

Elementary Notes on Simple Summations and Summations of Products

1936 ◽  
Vol 15 (1) ◽  
pp. 141-176
Author(s):  
Duncan C. Fraser
Keyword(s):  
Orthogonal Polynomials ◽  
Least Squares ◽  
Divided Differences ◽  
Binomial Coefficients ◽  
Special Point

SynopsisThe paper is intended as an elementary introduction and companion to the paper on “Orthogonal Polynomials,” by G. J. Lidstone, J.I.A., vol. briv., p. 128, and the paper on the “Sum and Integral of the Product of Two Functions,” by A. W. Joseph, J.I.A., vol. lxiv., p. 329; and also to Dr. Aitken's paper on the “Graduation of Data by the Orthogonal Polynomials of Least Squares,” Proc. Roy. Soc. Edin., vol. liii., p. 54.Following Dr. Aitken Σux is defined for the immediate purpose to be u0+…+ux−1.The scheme of successive summations is set out in the form of a difference diagram and is extended to negative arguments. The special point to which attention is drawn is the existence of a wedge of zeros between the sums for positive arguments and those for negative arguments.The rest of the paper is for the greater part a study of the table of binomial coefficients for positive and for negative arguments. The Tchebychef polynomials are simple functions of the binomial coefficients, and after a description of a particular example and of its properties general methods are given of forming the polynomials by means of tables of differences. These tables furnish examples of simple, differences, of divided differences, of adjusted differences, and of a system of special adjusted differences which gives a very easy scheme for the formation of the Tchebychef polynomials.

scholarly journals On Delannoy numbers and Schröder numbers

2011 ◽  
Vol 131 (12) ◽  
pp. 2387-2397 ◽  
Author(s):  
Zhi-Wei Sun
Keyword(s):  
Schröder Numbers ◽  
Delannoy Numbers

Time-and-band limiting for matrix orthogonal polynomials of Jacobi type

2017 ◽  
Vol 06 (04) ◽  
pp. 1740001 ◽  
Author(s):  
M. Castro ◽  
F. A. Grünbaum
Keyword(s):  
Signal Processing ◽  
Differential Operator ◽  
Orthogonal Polynomials ◽  
Random Matrix Theory ◽  
Random Matrix ◽  
Jacobi Polynomials ◽  
Matrix Theory ◽  
Matrix Orthogonal Polynomials

We extend to a situation involving matrix-valued orthogonal polynomials a scalar result that plays an important role in Random Matrix Theory and a few other areas of mathe-matics and signal processing. We consider a case of matrix-valued Jacobi polynomials which arises from the study of representations of [Formula: see text], a group that plays an important role in Random Matrix Theory. We show that in this case an algebraic miracle, namely the existence of a differential operator that commutes with a naturally arising integral one, extends to this matrix-valued situation.

Some congruences on Delannoy numbers and Schröder numbers

2018 ◽  
Vol 14 (07) ◽  
pp. 2035-2041 ◽  
Author(s):  
Ji-Cai Liu ◽  
Long Li ◽  
Su-Dan Wang
Keyword(s):  
Schröder Numbers ◽  
Delannoy Numbers

The Delannoy numbers and Schröder numbers are given by [Formula: see text] respectively. We mainly prove that for any prime [Formula: see text], [Formula: see text] which was originally conjectured by Sun in 2011. A related congruence on the Delannoy numbers is also proved.

scholarly journals Hankel determinants of sums of consecutive weighted Schröder numbers

2012 ◽  
Vol 437 (9) ◽  
pp. 2285-2299 ◽  
Author(s):  
Sen-Peng Eu ◽  
Tsai-Lien Wong ◽  
Pei-Lan Yen
Keyword(s):  
Hankel Determinants ◽  
Schröder Numbers

Random matrix theory and integrable systems

2018 ◽  
Author(s):  
Alexander R. Its
Keyword(s):  
Orthogonal Polynomials ◽  
Matrix Models ◽  
Random Matrix Theory ◽  
Random Matrix ◽  
Integrable Systems ◽  
Matrix Theory ◽  
Kp Hierarchy ◽  
Brownian Motions ◽  
Airy Process ◽  
Virasoro Constraints

This article discusses the interaction between random matrix theory (RMT) and integrable theory, leading to ordinary and partial differential equations (PDEs) for the eigenvalue distribution of random matrix models of size n and the transition probabilities of non-intersecting Brownian motion models, for finite n and for n → ∞. It first provides an overview of the connection between the theory of orthogonal polynomials and the KP-hierarchy in integrable systems before examining matrix models and the Virasoro constraints. It then considers multiple orthogonal polynomials, taking into account non-intersecting Brownian motions on ℝ (Dyson’s Brownian motions), a moment matrix for several weights, Virasoro constraints, and a PDE for non-intersecting Brownian motions. It also analyses critical diffusions, with particular emphasis on the Airy process, the Pearcey process, and Airy process with wanderers. Finally, it describes the Tacnode process, along with kernels and p-reduced KP-hierarchy.

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