18 Orthogonal PolynomialsOther Orthogonal Polynomials18.35 Pollaczek Polynomials18.37 Classical OP’s in Two or More Variables

- §18.36(i) Jacobi-Type Polynomials
- §18.36(ii) Sobolev OP’s
- §18.36(iii) Multiple OP’s
- §18.36(iv) Orthogonal Matrix Polynomials

These are OP’s on the interval $\left(-1,1\right)$ with respect to an orthogonality measure obtained by adding constant multiples of “Dirac delta weights” at $-1$ and $1$ to the weight function for the Jacobi polynomials. For further information see Koornwinder (1984b) and Kwon et al. (2006).

Similar OP’s can also be constructed for the Laguerre polynomials; see Koornwinder (1984b, (4.8)).

These are polynomials in one variable that are orthogonal with respect to a number of different measures. They are related to Hermite-Padé approximation and can be used for proofs of irrationality or transcendence of interesting numbers. For further information see Ismail (2005, Chapter 23).

These are matrix-valued polynomials that are orthogonal with respect to a square matrix of measures on the real line. Classes of such polynomials have been found that generalize the classical OP’s in the sense that they satisfy second-order matrix differential equations with coefficients independent of the degree. For further information see Durán and Grünbaum (2005).