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- W2008850316 abstract "Knowing a sequence of moments of a given, infinitely supported, distribution we obtain quickly: coefficients of the power series expansion of monic polynomials $left{ p_{n}right} _{ngeq 0}$ that are orthogonal with respect to this distribution, coefficients of expansion of $x^{n}$ in the series of $p_{j},$ $jleq n$, two sequences of coefficients of the 3-term recurrence of the family of $left{ p_{n}right} _{ngeq 0}$, the so called linearization coefficients i.e. coefficients of expansion of $% p_{n}p_{m}$ in the series of $p_{j},$ $jleq m+n.$newline Conversely, assuming knowledge of the two sequences of coefficients of the 3-term recurrence of a given family of orthogonal polynomials $left{ p_{n}right} _{ngeq 0},$ we express with their help: coefficients of the power series expansion of $p_{n}$, coefficients of expansion of $x^{n}$ in the series of $p_{j},$ $jleq n,$ moments of the distribution that makes polynomials $left{ p_{n}right} _{ngeq 0}$ orthogonal. newline Further having two different families of orthogonal polynomials $left{ p_{n}right} _{ngeq 0}$ and $left{ q_{n}right} _{ngeq 0}$ 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 $q_{j},$ $jleq n.$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 $mathbf{Pi }_{n}.$" @default.
- W2008850316 created "2016-06-24" @default.
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- W2008850316 date "2015-02-01" @default.
- W2008850316 modified "2023-10-14" @default.
- W2008850316 title "A few remarks on orthogonal polynomials" @default.
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- W2008850316 doi "https://doi.org/10.1016/j.amc.2014.11.112" @default.
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