Some uses of orthogonal polynomials

Опубликовано 6 сентября 2016, 5:07

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.