47,176 research outputs found
On d-symmetric classical d-orthogonal polynomials
AbstractThe d-symmetric classical d-orthogonal polynomials are an extension of the standard symmetric classical polynomials according to the Hahn property. In this work, we give some characteristic properties for these polynomials related to generating functions and recurrence-differential equations. As applications, we characterize the d-symmetric classical d-orthogonal polynomials of Boas-Buck type, we construct a (d+1)-order linear differential equation with polynomial coefficients satisfied by each polynomial of a d-symmetric classical d-orthogonal set and we show that the d-symmetric classical d-orthogonal property is preserved by the derivative operator. Some of the obtained properties appear to be new, even for the case d=1
Multiple orthogonal polynomials associated with confluent hypergeometric functions
We introduce and analyse a new family of multiple orthogonal polynomials of hypergeometric type with respect to two measures supported on the positive real line which can be described in terms
of confluent hypergeometric functions of the second kind. These two measures form a Nikishin system. Our focus is on the multiple orthogonal polynomials for indices on the step line. The sequences of the derivatives of both type I and type II polynomials with respect to these indices are again multiple orthogonal and they correspond to the original sequences with shifted parameters. For the type I polynomials, we provide a Rodrigues-type formula. We characterise the type II polynomials on the step line, also known as d-orthogonal polynomials (where d is the number of measures involved so that here d = 2), via their explicit expression as a terminating generalised hypergeometric series, as solutions to a third-order differential equation and via their recurrence relation. The latter involves recurrence coefficients which are unbounded and asymptotically periodic. Based on this information we deduce the asymptotic behaviour of the largest zeros of the type II polynomials. We also discuss limiting relations between these polynomials and the multiple orthogonal polynomials with respect to the modified Bessel weights. Particular choices on the parameters
for the 2-orthogonal polynomials under discussion correspond to the cubic components of the already known threefold symmetric Hahn-classical multiple orthogonal polynomials on star-like sets
Polynomial-Time Algorithms for Quadratic Isomorphism of Polynomials: The Regular Case
Let and be
two sets of nonlinear polynomials over
( being a field). We consider the computational problem of finding
-- if any -- an invertible transformation on the variables mapping
to . The corresponding equivalence problem is known as {\tt
Isomorphism of Polynomials with one Secret} ({\tt IP1S}) and is a fundamental
problem in multivariate cryptography. The main result is a randomized
polynomial-time algorithm for solving {\tt IP1S} for quadratic instances, a
particular case of importance in cryptography and somewhat justifying {\it a
posteriori} the fact that {\it Graph Isomorphism} reduces to only cubic
instances of {\tt IP1S} (Agrawal and Saxena). To this end, we show that {\tt
IP1S} for quadratic polynomials can be reduced to a variant of the classical
module isomorphism problem in representation theory, which involves to test the
orthogonal simultaneous conjugacy of symmetric matrices. We show that we can
essentially {\it linearize} the problem by reducing quadratic-{\tt IP1S} to
test the orthogonal simultaneous similarity of symmetric matrices; this latter
problem was shown by Chistov, Ivanyos and Karpinski to be equivalent to finding
an invertible matrix in the linear space of matrices over and to compute the square root in a matrix
algebra. While computing square roots of matrices can be done efficiently using
numerical methods, it seems difficult to control the bit complexity of such
methods. However, we present exact and polynomial-time algorithms for computing
the square root in for various fields (including
finite fields). We then consider \\#{\tt IP1S}, the counting version of {\tt
IP1S} for quadratic instances. In particular, we provide a (complete)
characterization of the automorphism group of homogeneous quadratic
polynomials. Finally, we also consider the more general {\it Isomorphism of
Polynomials} ({\tt IP}) problem where we allow an invertible linear
transformation on the variables \emph{and} on the set of polynomials. A
randomized polynomial-time algorithm for solving {\tt IP} when
is presented. From an algorithmic point
of view, the problem boils down to factoring the determinant of a linear matrix
(\emph{i.e.}\ a matrix whose components are linear polynomials). This extends
to {\tt IP} a result of Kayal obtained for {\tt PolyProj}.Comment: Published in Journal of Complexity, Elsevier, 2015, pp.3
Characters of classical groups, Schur-type functions, and discrete splines
We study a spectral problem related to the finite-dimensional characters of
the groups , , and , which form the classical series
, , and , respectively. The irreducible characters of these three
series are given by -variate symmetric polynomials. The spectral problem in
question consists in the decomposition of the characters after their
restriction to the subgroups of the same type but smaller rank . The main
result of the paper is the derivation of explicit determinantal formulas for
the coefficients in this decomposition.
In fact, we first compute these coefficients in a greater generality -- for
the multivariate symmetric Jacobi polynomials depending on two continuous
parameters. Next, we show that the formulas can be drastically simplified for
the three special cases of Jacobi polynomials corresponding to the --
characters. In particular, we show that then the coefficients are given by
piecewise polynomial functions. This is where a link with discrete splines
arises.
In type (that is, for the characters of the unitary groups ),
similar results were earlier obtained by Alexei Borodin and the author [Adv.
Math., 2012], and then reproved by another method by Leonid Petrov [Moscow
Math. J., 2014]. The case of the symplectic and orthogonal characters is more
intricate.Comment: Accepted in Sbornik: Mathematic
A Bochner Theorem for Dunkl Polynomials
We establish an analogue of the Bochner theorem for first order operators of
Dunkl type, that is we classify all such operators having polynomial solutions.
Under natural conditions it is seen that the only families of orthogonal
polynomials in this category are limits of little and big -Jacobi
polynomials as
A "missing" family of classical orthogonal polynomials
We study a family of "classical" orthogonal polynomials which satisfy (apart
from a 3-term recurrence relation) an eigenvalue problem with a differential
operator of Dunkl-type. These polynomials can be obtained from the little
-Jacobi polynomials in the limit . We also show that these polynomials
provide a nontrivial realization of the Askey-Wilson algebra for .Comment: 20 page
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