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Results tagged with orthogonal-polynomials
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user 56553
A familly of orthogonal polynomials is a sequence of polynomials in one variable, one in each degree, such that any two of them are orthogonal with respect to some fixed scalar product on the space of polynomials. They are closely related to continued fractions and useful in harmonic analysis. There are many different families of orthogonal polynomials, among which one can cite Hermite polynomials, Laguerre polynomials, and Jacobi polynomials.
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Orthogonal polynomials with quadratic recurrence coefficients
Consider the monic orthogonal polynomials determined by the recurrence
$$p_{n+1}(x)=(x-n(n+b))p_{n}(x)-n(n+a)p_{n-1}(x), \quad n\in\mathbb{N},$$
with the initial conditions $p_{-1}(x)=0$ and $p_{0}(x) …
- 2,188
3
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Polynomials orthogonal w.r.t. the logarithmic weight
Recently, I have encountered the family of orthogonal polynomials $p_{n}(x)$ which is orthogonal w.r.t. the function $-\ln(x)$ on $(0,1)$. This means we have
$$\int_{0}^{1}p_{n}(x)p_{m}(x)\ln(1/x)dx=\ …
- 2,188
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An inverse spectral problem for Jacobi matrices (or orthogonal polynomials)
I will formulate this question in the language of Jacobi operators and spectral measures although it could be entirely rewritten in terms of orthogonal polynomials and measures of orthogonality.
Obj …
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An asymptotic behavior of a sequence of special polynomials
For $n\to\infty$, I would like to know the asymptotic behavior of the polynomials defined in terms of the Gauss hypergeometric series:
$$
p_{n}(z):={}_{2}F_{1}(-n,-nz+\alpha;1;\beta),
$$
where $\alpha …
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