Thursday, July 11, 2013 - 11:00
Hankel determinants, continued fractions, and orthogonal polynomials
There is a close connection between continued fractions, Hankel determinants, and orthogonal polynomials. Suppose that a sequence of polynomials is orthogonal with respect to a linear functional L. Then the numbers L(x^n) are called the moments of the orthogonal polynomial sequence. For many classical orthogonal polynomials these moments are sequences of combinatorial interest, such as Catalan numbers, factorials, Bell numbers, Euler numbers, and Genocchi numbers. Any sequence of orthogonal polynomials satisfies a three-term recurrence relation, in which, for classical orthogonal polynomials, the coefficents are given by simple formulas. The continued fraction for the generating function for the moments can be expressed simply in terms of these coefficients, as can the Hankel determinants of the moments. I'll talk about these ideas and describe some interesting examples.