A few remarks on orthogonal polynomials

Knowing a sequence of moments of a given, infinitely supported, distribution we obtain quickly: coefficients of the power series expansion of monic polynomials {pn}n⩾0pnn⩾0 that are orthogonal with respect to this distribution, coefficients of expansion of xnxn in the series of pj,j⩽npj,j⩽n, two sequences of coefficients of the 3-term recurrence of the family of {pn}n⩾0pnn⩾0, the so called “linearization coefficients” i.e. coefficients of expansion of pnpmpnpm in the series of pj,j⩽m+npj,j⩽m+n.Conversely, assuming knowledge of the two sequences of coefficients of the 3-term recurrence of a given family of orthogonal polynomials {pn}n⩾0pnn⩾0, we express with their help: coefficients of the power series expansion of pnpn, coefficients of expansion of xnxn in the series of pj,j⩽npj,j⩽n, moments of the distribution that makes polynomials {pn}n⩾0pnn⩾0 orthogonal.Further having two different families of orthogonal polynomials {pn}n⩾0pnn⩾0 and {qn}n⩾0qnn⩾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 pnpn in the series of qj,j⩽nqj,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)(n+1)×(n+1) matrix ΠnΠn.