# Questions tagged [orthogonal-polynomials]

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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### Are there extensions of Hilb's and Laplace's formulas to Jacobi polynomials with $\alpha,\beta\le-1$?

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### Transformation which “opens up” an arc

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### Higher-order inner products of an orthonormal basis

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### Integral involving associated Laguerre polynomial and Bessel function

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### Gegenbauer's addition theorem for Jacobi polynomials

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### Chebyshev polynomials from roots [closed]

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### Integration on sphere $\mathbb{S}^{d-1}$ for $d$ large — Change of variables

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### mollifier satisfying moment conditions

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### Convergence of gPC expansions for random variables in the total variation distance

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### Convergence of Gaussian quadrature rules for integration

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### Recurrence involving families of orthogonal polynomials

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### About a family of orthogonal polynoms satisfying a recurrence relation

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### Upper bound over $[0,1] $ for strange family of polynomials

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### Riemann-Hilbert and Orthogonal polynomials

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### Evaluating an integral with Jacobi and Legendre polynomials

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### An asymptotic behavior of a sequence of special polynomials

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### Is this Hermite polynomial identity known?

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### Orthogonal Polynomials and Sturm Liouville operators

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### Determinants associated to orthogonal polynomials

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### Orthogonal basis of polynomials?

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### Function approximation via an orthonormal basis (with singular weight)

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### building set of 2D orthogonal polynomials with minimum crossed terms

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### Closed form of :$\int_{-1}^1 x^{2k} (\operatorname{erf}(x))^k \,dx $ for $ k$ is even integer and :$\int _{0}^{t}\exp(-x^2 \operatorname{erf}(x))dx$

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### Gaps between roots of consecutive Hermite polynomials

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### Jacobi polynomials with negative integer parameters

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### Existence of moment-constrained maximum entropy distribution with support $[0,1]^n$

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### Rate of convergence of generalized polynomial chaos

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### Reverse Markov-Bernstein inequality for trigonometric polynomials

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### Lower $L^1$ norm estimates of null average trigonometric polynomials depending on the order of the polynomial

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### Questions about generalized Polynomial Chaos, book by Dongbin Xiu

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### Identities for Chebyshev polynomials of the second kind

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### Polynomials for which $f''$ divides $f$

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### What are the orthogonal polynomials with respect to the weight $2\cosh(\beta x)e^{-x^2}$?

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### How to use this generalised 'generating function' for the Gegenbauer polynomials

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### Proof Reference - Polynomial interpolation at quadrature points

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### Closure of polynomials in $L^2_w$ with log-normal weight function

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### Orthogonal polynomials with quadratic recurrence coefficients

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### Clenshaw-Curtis integration without Fourier

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### A generalization of binary Krawtchouk polynomials

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### Is there an explicit expression for Chebyshev polynomials modulo $x^r-1$?

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### Determining the Associated Sequence If Sheffer Conditions are not Met

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### Completeness of the solutions to the Schrödinger Hydrogen Atom

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### Orthogonal polynomial expansion for bivariate noncentral chi-square and bi-variate noncentral student t distribution

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### Bounds on Legendre polynomials on the complex plane

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### Holonomic generating function

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### Rational generating function and recursion

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### Expand the pdf of Wishart distribution into power series via orthogonal polynomials

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### Relation between Legendre and Chebyshev polynomials

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### need help on the proof of the Christoffel–Darboux formula of Laguerre Polynomial

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