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 ${\left\{p_n\right\}}_{n\ge 0}$ that are orthogonal with respect to this distribution, coefficients of expansion of $x^n$ in the series of $p_j,$ $j\le n$, two sequences of coefficients of the 3-term recurrence of the family of ${\left\{p_n\right\}}_{n\ge 0}$, the so called “linearization coefficients” i.e. coefficients of expansion of $$ in the series of $p_j,$ $j\le m+n\mathrm{.}$\newline Conversely, assuming knowledge of the two sequences of coefficients of the 3-term recurrence of a given family of orthogonal polynomials ${\left\{p_n\right\}}_{n\ge 0},$ we express with their help: coefficients of the power series expansion of $p_n$, coefficients of expansion of $x^n$ in the series of $p_j,$ $j\le n,$ moments of the distribution that makes polynomials ${\left\{p_n\right\}}_{n\ge 0}$ orthogonal. \newline Further having two different families of orthogonal polynomials ${\left\{p_n\right\}}_{n\ge 0}$ and $\left{ q_{n}\right} {n\geq 0}$ and knowing for each of them sequences of the 3-term recurrences, we give sequence of the so called “connection coefficients” between these two families of polynomials. That is coefficients of the expansions of $p_n$ in the series of $q_j,$ $j\le n\mathrm{.}$\newline We are able to do all this due to special approach in which we treat vector of orthogonal polynomials $\left{ p{j}\left( x)\right) \right} {j=0}^{n}$ as a linear transformation of the vector ${\left\{x^j\right\}}_{j=0}^n$ by some lower triangular $(n+1)\times (n+1)$matrix$$\mathbf{\Pi }${n}.$