A few remarks on orthogonal polynomials

摘要

Knowing a sequence of moments of a given, infinitely supported, distribution we obtain quickly: coefficients of the power series expansion of monic polynomials pnn⩾0 that are orthogonal with respect to this distribution, coefficients of expansion of xn in the series of pj,j⩽n, two sequences of coefficients of the 3-term recurrence of the family of pnn⩾0, the so called “linearization coefficients” i.e. coefficients of expansion of pnpm in the series of pj,j⩽m+n.Conversely, assuming knowledge of the two sequences of coefficients of the 3-term recurrence of a given family of orthogonal polynomials pnn⩾0, we express with their help: coefficients of the power series expansion of pn, coefficients of expansion of xn in the series of pj,j⩽n, moments of the distribution that makes polynomials pnn⩾0 orthogonal.Further having two different families of orthogonal polynomials pnn⩾0 and qnn⩾0 and knowing for each of them sequences of the 3-term recurrences, we give sequence of the so called “connection coefficients” between these two families of polynomials. That is coefficients of the expansions of pn in the series of qj,j⩽n.We are able to do all this due to special approach in which we treat vector of orthogonal polynomials pj(x)j=0n as a linear transformation of the vector xjj=0n by some lower triangular (n+1)×(n+1) matrix Πn.