Symmetric orthogonal polynomials

In this talk I will discuss symmetric orthogonal polynomials on the real line. Symmetric polynomials give rise to orthogonal systems which have important applications in spectral methods, with several important advantages if their differentiation matrix is skew-symmetric and highly structured. The symmetric orthogonal polynomials discussed will include a modified Hermite weight, which arises in random matrix theory, and generelised Freud weights, which arise in the matrix model in two-dimensional quantum gravity. It is well-known that orthogonal polynomials satisfy a three-term recurrence relation. I will show that for modified Hermite weight the coefficients in the recurrence relation are expressed in terms of special function solutions of the fifth Painleve equation and for the generelised Freud weights in terms of solutions of Wronskians of generalised hypergeometric functions. Joint work with Kerstin Jordaan (University of South Africa) and Ana Loureiro (University of Kent, UK).