A few remarks on orthogonal polynomials (1303.0627v4)

Abstract: Knowing a sequence of moments of a given, infinitely supported, distribution we obtain quickly: coefficients of the power series expansion of monic polynomials $\left{ p_{n}\right} {n\geq 0}$ that are orthogonal with respect to this distribution, coefficients of expansion of $x^{n}$ in the series of $p{j},$ $j\leq n$, two sequences of coefficients of the 3-term recurrence of the family of $\left{ p_{n}\right} {n\geq 0}$, the so called "linearization coefficients" i.e. coefficients of expansion of $% p{n}p_{m}$ in the series of $p_{j},$ $j\leq m+n.$\newline Conversely, assuming knowledge of the two sequences of coefficients of the 3-term recurrence of a given family of orthogonal polynomials $\left{ p_{n}\right} {n\geq 0},$ we express with their help: coefficients of the power series expansion of $p{n}$, coefficients of expansion of $x^{n}$ in the series of $p_{j},$ $j\leq n,$ moments of the distribution that makes polynomials $\left{ p_{n}\right} {n\geq 0}$ orthogonal. \newline Further having two different families of orthogonal polynomials $\left{ p{n}\right} {n\geq 0}$ and $\left{ q{n}\right} {n\geq 0}$ 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 $p{n}$ in the series of $q_{j},$ $j\leq n.$\newline We are able to do all this due to special approach in which we treat vector of orthogonal polynomials $\left{ p_{j}\left( x)\right) \right} {j=0}^{n}$ as a linear transformation of the vector $\left{ x^{j}\right} _{j=0}^{n}$ by some lower triangular $(n+1)\times (n+1)$ matrix $\mathbf{\Pi }{n}.$