In this contribution we deal with sequences of polynomials orthogonal with respect to a Sobolev type inner product. A banded symmetric operator is associated with such a sequence of polynomials ...according to the higher order difference equation they satisfy. Taking into account the Darboux transformation of the corresponding matrix we deduce the connection with a sequence of orthogonal polynomials associated with a Christoffel perturbation of the measure involved in the standard part of the Sobolev inner product. A connection with matrix orthogonal polynomials is stated. The Laguerre-Sobolev type case is studied as an illustrative example. Finally, the bispectrality of such matrix orthogonal polynomials is pointed out.
The classical 2-orthogonal polynomials share the so-called Hahn property, this means that they are 2-orthogonal polynomials whose the sequences of their derivatives are also 2-orthogonal polynomials. ...Based only on this property, a new class of classical 2-orthogonal polynomials is obtained as particular solution of the non-linear system governing the coefficients involved in the recurrence relation fulfilled by these polynomials. A differential-recurrence relation as well as a third-order differential equation satisfied by the resulting polynomials are given. Many interesting subcases are highlighted and explicitly presented with special reference to some connected results that exist in the literature. The integral representations of their associated linear functionals will be exhaustively discussed in a forthcoming publication.
We generalize the representations of X1 exceptional orthogonal polynomials through determinants of matrices that have certain adjusted moments as entries. We start out directly from the Darboux ...transformation, allowing for a universal perspective, rather than one dependent upon the particular system (Jacobi or Type of Laguerre polynomials). We include a recursion formula for the adjusted moments and provide the initial adjusted moments for each system. Throughout we relate to the various examples of X1 exceptional orthogonal polynomials. We especially focus on and provide complete proofs for the Jacobi and the Type III Laguerre case, as they are less prevalent in literature. Lastly, we include a preliminary discussion explaining that the higher codimension setting becomes more involved. The number of possibilities and choices is exemplified, but only starts, with the lack of a canonical flag.
Polynomials are incredibly useful mathematical tools that have a wide array of applications. This book provides a comprehensive overview of polynomials and recent developments in the field. It ...includes ten chapters that address such topics as polynomials-based cyclic coding, Hermite polynomials, Routh polynomials, fitting parametric polynomials with control point coefficients, the thermoelastic wave model, and much more.
In 10 we constructed new examples of exceptional Jacobi polynomials. The difference with the so called standard families is that the examples constructed in 10 depend on an arbitrary number of ...continuous parameters. Besides these continuous parameters, denoted by M={M0,M1,…}, each one of the new families of exceptional Jacobi polynomials is associated to two negative integers a≤b≤−1 and a finite set of positive integers F satisfying that {−b,…,−a−b−1}⊂F. In this paper we prove that when F={u,u+1,…,−a−b−1}, with 1≤u≤−b, the exceptional Jacobi polynomials are deformations of Jacobi polynomials (when the continuous parameters Mi→1). Otherwise, the exceptional Jacobi polynomials are deformations of standard exceptional Jacobi polynomials (again when the continuous parameters Mi→1).
A new interpretation and applications of the 'Diophantine' and factorization properties of finite orthogonal polynomials in the Askey scheme are explored. The corresponding twelve polynomials are the ...(q-)Racah, (dual, q-)Hahn, Krawtchouk and five types of q-Krawtchouk. These (q-)hypergeometric polynomials, defined only for the degrees of
$ 0,1,\ldots,N $
0
,
1
,
...
,
N
, constitute the main part of the eigenvectors of N + 1-dimensional tri-diagonal real symmetric matrices, which correspond to the difference equations governing the polynomials. The monic versions of these polynomials all exhibit the 'Diophantine' and factorization properties at higher degrees than N. This simply means that these higher degree polynomials are zero-norm 'eigenvectors' of the N + 1-dimensional tri-diagonal real symmetric matrices. A new type of multi-indexed orthogonal polynomials belonging to these twelve polynomials could be introduced by using the higher degree polynomials as the seed solutions of the multiple Darboux transformations for the corresponding matrix eigenvalue problems. The shape-invariance properties of the simplest type of the multi-indexed polynomials are demonstrated. The explicit transformation formulas are presented.
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