§18.36(i) Jacobi-Type Polynomials

These are OP’s on the interval $(-1,1)$ 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)).

§18.36(ii) Sobolev OP’s

Sobolev OP’s are orthogonal with respect to an inner product involving derivatives. For an introductory survey to this subject, see Marcellán et al. (1993). Other relevant references include Iserles et al. (1991) and Koekoek et al. (1998).

§18.36(iii) Multiple OP’s

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).

§18.36(iv) Orthogonal Matrix Polynomials

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).