Some uses of orthogonal polynomials

October 6, 2004
Richard Askey | University of Wisconsin Madison

One thing which will be explained is the evaluation of a determinant originally stated by Sylvester and rediscovered by Mark Kac. The determinant is tridiagonal with x on the diagonal, 1,2,…,N above the diagonal, and N,N-1,…,1 below the diagonal. The polynomials responsible for this evaluation are Krawtchouk, although Kac was unaware that orthogonal polynomials were behind this and Sylvester just stated the values of this determinant for N=1,2,3,4,5,6 and said that the obvious rule continued. There are a number of other determinants which can be found and evaluated using other orthogonal polynomials. A summary of what some of these polynomials are and how they arise in many different areas will be sketched.

Speaker Details

Richard Askey started out as a classical analyst proving norm inequalities for series of the classical orthogonal polynomials. Next came a period of proving certain series are positive, which was followed by a period of evaluating some sums and integrals or showing that two seemingly different series are equal. Some of the work was heavily influenced by series and integrals Ramanujan considered. More recently school mathematics has been a focus. Almost all of this work was motivated by the desire to understand and use some beautiful formulas.