A ‘missing’ family of classical orthogonal polynomials

Abstract

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 q-Jacobi polynomials in the limit q = −1. We also show that these polynomials provide a nontrivial realization of the Askey-Wilson algebra for q = −1.

A limit $q=-1$ for the big $q$-Jacobi polynomials

“…which corresponds to the weight function of the little −1 Jacobi polynomials [20]. So far, we considered the case 0 < c < 0.…”

“…We constructed in [20] a system of "classical" orthogonal polynomials P n (x) containing two real parameters α, β and corresponding to the limit q → −1 of the little q-Jacobi polynomials. By "classical" we mean that these polynomials satisfy (apart from a 3-term recurrence relation) a nontrivial eigenvalue equation of the form (1.1) LP n (x) = λ n P n (x).…”

“…Guided by the q → −1 limit of the little q-Jacobi polynomials, we derived in [20] an explicit expression of the polynomials P n (x) in terms of Gauss' hypergeometric functions. We also found explicitly the recurrence coefficients and showed that the polynomials P n (x) are orthogonal on the interval [−1, 1] with a weight function related to the weight function of the generalized Jacobi polynomials [7].…”

The Bannai-Ito polynomials as Racah coefficients of the $sl_{-1}(2)$ algebra

“…Recently, a series of orthogonal polynomials corresponding to limits q → −1 of q-polynomials of the Askey scheme were discovered [19,22,25,26]. These polynomials are eigenfunctions of operators of Dunkl type, which involve the reflection operator [7,24].…”

Measuring Rice Yield from Space

“…Our land cover classification uses a random forest classifier algorithm (Breiman 2001), which has been widely tested and proved robust and efficient in the classification of remote sensing images (Hansen et al 2000;Pal 2005;Zhu, Fu, Woodcock, Olofsson, Vogelmann, Holden, and Yu 2016;. We test four types of input datasets for the classification (Table 4).…”