A few remarks on orthogonal polynomials

作者：Szablowski Pawel J^*

来源：Applied Mathematics and Computation, 2015, 252: 215-228.

DOI：10.1016/j.amc.2014.11.112

• Moment problem
• Moment matrix
• Cholesky decomposition
• Hankel matrices
• Connection coefficients
• Linearization coefficients

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) x (n + 1) matrix Pi(n).

• 出版日期2015-2-1