Gómez-Ullate et al., “Exceptional orthogonal polynomials”

Mathematics is full of surprises. Orthogonal polynomials and their properties appear in many important questions in mathematics, in physics, and in chemistry. It was thought that the classical orthogonal polynomials, which are captured by Bochner’s celebrated 1928 theorem, covered all complete orthogonal systems defined by Sturm-Liouville problems. It turns out that this is not the case and that these classical polynomials admit an unexpected and fruitful generalization, the exceptional orthogonal polynomials discovered by Gómez-Ullate et al. There are a number of exciting questions and generalizations to pursue: applications to quantum mechanics, integrable systems, and exact solutions of other equations of mathematical physics. Many of these questions are being investigated by researchers from different groups. From the interview to the second author of the paper, “Niky Kamran Discusses Orthogonal Polynomials,” New Hot Paper Commentary, November 2010. More technical information in David Gomez-Ullate, Niky Kamran, Robert Milson, “Exceptional orthogonal polynomials and the Darboux transformation,” J. Phys. A 43: 434016, 2010 [ArXiv, 13 Feb 2010] y David Gomez-Ullate, Niky Kamran, Robert Milson, “An extended class of orthogonal polynomials defined by a Sturm-Liouville problem,” J. Math. Anal. Appl. 359: 352-367, 2009 [ArXiv, 24 Jul 2008].

Bochner’s theorem (1929) states that if an infinite sequence of polynomials \{P_n(z)\}_{n=0}^\infty satisfies a second order eigenvalue equation of the form

p(x)P_n''(x) + q(x) P_n'(x) + r(x) P_n(x)=\lambda_n P_n(x),\qquad n=0,1,2,\dots

then p(x), q(x) and r(x) must be polynomials of degree 2, 1 and 0 respectively. In addition, if the \{P_n(x)\}_{n=0}^\infty sequence is an orthogonal polynomial system, then it has to be (up to an affine transformation of z) one of the classical orthogonal polynomial systems of Jacobi, Laguerre or Hermite. Gómez-Ullate et al. have shown that there exist complete orthogonal polynomial systems, defined by Sturm-Liouville problems, that extend beyond the classical families of orthogonal polynomials if it is allowed that the first eigenpolynomial of the sequence need not be of degree zero, i.e., \{P_n(z)\}_{n=m}^\infty with m\geq 1. The class can be extended to zero degree, but the initial polynomials \{P_n(z)\}_{n=0}^{m-1} are not square integrable.

For example, let us consider the family of (exceptional) X_1 Laguerre polynomials, denoted by \hat{L}^{(k)}_n(x), n=0,1,2,\ldots, whose first members are

\hat{L}^{(k)}_1(x) = -x-(1+k), \qquad \hat{L}^{(k)}_2(x) = x^2-k(k+2),

\hat{L}^{(k)}_3(x) = -\frac{1}{2} x^3 + \frac{k+3}{2} x^2 + \frac{k(k+3)}{2} x - \frac{k}{2} (3 + 4 k + k^2).

for real k>0. These polynomials are orthogonali with respect to the weight

W_k(x) dx = \frac{\displaystyle e^{-x} x^k}{\displaystyle (x+k)^2}\,dx,

i.e., by the inner product

\langle f, g\rangle_k := \int^\infty_{0} f(x) g(x)\,W_k(x) dx.

Clearly, the first member of the X_1 Laguerre polynomials has degree 1. Another family of exceptional orthogonal polynomials is X_1 Jacobi polynomials (see the papers above for their mathematical definition).

Gómez-Ullate et al. have shown that if the sequence \{P_n\}_{n=m}^\infty is allowed to start with a degree $m\geq 1$ polynomial, then there exist complete sequences of polynomial eigenfunctions that obey differential equations of the form

p(x)P_n''(x) + q(x) P_n'(x) + r(x) P_n(x)=\lambda_n P_n(x),\qquad n=0,1,2,\dots

where p,q and r are polynomials of degrees m+2, m+1 and m, respectively.

Applications of exeptional orthogonal polynomials are starting to arise in several fields, like quantum exactly solvable systems. Long life orthogonal polynomials.

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