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 p n n ? 0 that are orthogonal with respect to this distribution, coefficients of expansion of x n in the series of p j , j ≤ n , two sequences of coefficients of the 3-term recurrence of the family of p n n ? 0 , the so called "linearization coefficients" i.e. coefficients of expansion of p n p m in the series of p j , j ≤ m + n .Conversely, assuming knowledge of the two sequences of coefficients of the 3-term recurrence of a given family of orthogonal polynomials p n n ? 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 ≤ n , moments of the distribution that makes polynomials p n n ? 0 orthogonal.Further having two different families of orthogonal polynomials p n n ? 0 and q n 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 p n in the series of q j , j ≤ n .We are able to do all this due to special approach in which we treat vector of orthogonal polynomials p j ( x ) j = 0 n as a linear transformation of the vector x j j = 0 n by some lower triangular ( n + 1 ) i? ( n + 1 ) matrix ? n .