A few remarks on orthogonal polynomials

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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≥ 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 { pₙ} ₙ≥ 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 { pₙ} ₙ≥ 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 { pₙ} ₙ≥ ₀ orthogonal. Further having two different families of orthogonal polynomials { pₙ} ₙ≥ ₀ and { qₙ} ₙ≥ 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 { xʲ} ⱼ₌₀ⁿ by some lower triangular (n+1)× (n+1) matrix Πₙ.