Jacobi polynomials from compatibility conditions

Bulletin des Sciences Mathématiques, 2003

For every value of the parameters α, β > −1 we find a matrix valued weight whose orthogonal polynomials satisfy an explicit differential equation of Jacobi type.  2003 Éditions scientifiques et médicales Elsevier SAS. All rights reserved.

In this letter we present a brief description of a ladder operator formalism applied to orthogonal polynomials with discontinuous weights. The two coefficient functions, A n (z) and B n (z), appearing in the ladder operators satisfy the two fundamental compatibility conditions previously derived for smooth weights. If the weight is a product of an absolutely continuous reference weight w 0 and a standard jump function, then A n (z) and B n (z) have apparent simple poles at these jumps. We exemplify the approach by taking w 0 to be the Hermite weight. For this simpler case we derive, without using the compatibility conditions, a pair of difference equations satisfied by the diagonal and off-diagonal recurrence coefficients for a fixed location of the jump. We also derive a pair of Toda evolution equations for the recurrence coefficients which, when combined with the difference equations, yields a particular Painlevé IV.

Journal of Approximation Theory, 2010

Big q-Jacobi polynomials {P n (•; a, b, c; q)} ∞ n=0 are classically defined for 0 < a < q −1 , 0 < b < q −1 and c < 0. For the family of little q-Jacobi polynomials { p n (•; a, b|q)} ∞ n=0 , classical considerations restrict the parameters imposing 0 < a < q −1 and b < q −1. In this work we extend both families in such a way that wider sets of parameters are allowed, and we establish orthogonality conditions for those cases for which Favard's theorem does not work. As a by-product, we obtain similar results for the families of big and little q-Laguerre polynomials.

Indagationes Mathematicae, 2003

The main purpose of this paper is to present new families of Jacobi type matrix valued orthogonal polynomials.

Filomat, 2003

Given any sequence of orthogonal polynomials, satisfying the three term recurrence relation xpn(x) = ?n+1pn+1(x) + ?npn(x) + ?npn-1(x), p-1(x)=0 po(x)=1 with ?n ? 0, n ? N, ?0 = 1, an infinite Jacobi matrix can be associated in the following way ? ? ??0 ?1 0...? ? ? ??1 ?1 ?2...? ? ? J = ?0 ?2 ?2...? ?.... ? ?.... ? ?....? ? ? In the general case if the sequences {?n} or {?n} are complex the associated Jacobi matrix is complex. Under the condition that both sequences {?n}and {?n} are uniformly bounded, the associated Jacobi matrix can be understood as a linear operator J acting on ?2, the space of all complex square-summable sequences, where the value of the operator J at the vector x is a product of an infinite vector x and an infinite matrix J in the matrix sense. The case when the sequences {?n} and {?n} are not uniformly bounded, an operator acting on ?2 can not be defined that easily. Additional properties of the sequence of orthogonal polynomials are needed in order to be able...

In this letter we present a brief description of a ladder operator formalism applied to orthogonal polynomials with discontinuous weights. The two coefficient functions, A n (z) and B n (z), appearing in the ladder operators satisfy the two fundamental compatibility conditions previously derived for smooth weights. If the weight is a product of an absolutely continuous reference weight w 0 and a standard jump function, then A n (z) and B n (z) have apparent simple poles at these jumps. We exemplify the approach by taking w 0 to be the Hermite weight. For this simpler case we derive, without using the compatibility conditions, a pair of difference equations satisfied by the diagonal and off-diagonal recurrence coefficients for a fixed location of the jump. We also derive a pair of Toda evolution equations for the recurrence coefficients which, when combined with the difference equations, yields a particular Painlevé IV.

East Journal on Approximations, 2007

First we give a compact treatment of the Jacobi polynomials on a simplex in IR d which exploits and emphasizes the symmetries that exist. This includes the various ways that they can be defined: via orthogonality conditions, as a hypergeometric series, as eigenfunctions of an elliptic pde, as eigenfunctions of a positive linear operator, and through conditions on the Bernstein-Bézier form. We then consider all aspects of the limiting case when the parameters µ = (µ 0 ,. .. , µ d) of the Jacobi polynomials approach −1, and the weight becomes singular. We show that the orthogonal projection of a continuous function onto the Jacobi polynomials of degree n has a limit as the µ j → −1, and give an explicit formula for the corresponding 'orthogonal' expansion. It turns out that this expansion is closely related to the limit of the eigenfunction expansion of the Bernstein operator and a new mean value interpolant.

1999

Abstract. Under some integrability conditions we derive raising and lowering differential recurrence relations for polynomials orthogonal with respect to a weight function supported in the real line. We also derive a second-order differential equation satisfied by these polynomials. We discuss the Lie algebra generated by the generalized creation and annihilation operators. From the differential equations, Plancherel–Rotach type asymptotics are derived. Under certain conditions, stated in the text, an Airy function emerges.

2017

In this contribution we deal with classical Jacobi polynomials orthogonal with respect to different weight functions, their special cases - classical Legendre polynomials and generalized brothers of them. We derive expressions of generalized Legendre polynomials and generalized ultraspherical polynomials by means of classical Jacobi polynomials.

Annali di Matematica Pura ed Applicata, 2012

In this paper, we consider the second-order differential expression