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

This question is related to my question here such that i want to find a closed form of $\int_{-1}^1 x^{2k} (\operatorname{erf}(x))^k \,dx$ , for $k$ is even integer because for odd integer is $0$ as ...
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### Gaps between roots of consecutive Hermite polynomials

Let $H_k(x)$ be (probabilists' or physicists', does not matter for this question) Hermite polynomials. It is well-known that all the gaps between consecutive roots of $H_k(x)$ are at least a multiple ...
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### Lower $L^1$ norm estimates of null average trigonometric polynomials depending on the order of the polynomial

Let $p(x)=\sum_{k=1}^m [a_k\cos(n_kx)+b_k\sin(n_kx)]$ be a null average trigonometric polynomial (null average means that is $\int_\mathbb T p =0$ or, equivalently, there are no $a_0$ and $b_0$). ...
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### Questions about generalized Polynomial Chaos, book by Dongbin Xiu

I have some questions about Chapter 5 from the book Numerical Methods for Stochastic Computations, by Dongbin Xiu. Theorem 5.7: Let $Y$ be a random variable and $\mathbb{E}[Y^2]<\infty$. Let $Z$ ...
While calculating an integral in a quantum mechanical problem by two different methods, I came across the following identity $$\sum_{k=0}^n\sum_{m=0}^{2k}(-2)^m\binom{2(n-k)}{n-k}\binom{2k}{k}\binom{... 1answer 431 views ### Polynomials for which f'' divides f Let n \geq 2 and let a < b be real numbers. Then it is easy to see that there is a unique up to scale polynomial f(x) of degree n such that$$f(x) = \frac{(x-a)(x-b)}{n(n-1)} f''(x).$$... 0answers 234 views ### What are the orthogonal polynomials with respect to the weight 2\cosh(\beta x)e^{-x^2}? In the study of a statistical physics problem, I need to know the orthogonal polynomials with respect to the weight$$2\cosh(\beta x)e^{-x^2},$$where \beta \in \mathbb{R}^+. Is this already known? ... 0answers 120 views ### How to use this generalised 'generating function' for the Gegenbauer polynomials Cohl (2011) gives a generalisation of the standard generating function for the Gegenbauer polynomials C_n^\mu(x): (1 -2tx + t^2)^{-\nu} = A_{\mu,\nu} \frac{(1-t^2)^{-\nu+\mu+1/2}}{t^{\mu+1/2}} \... 1answer 87 views ### Proof Reference - Polynomial interpolation at quadrature points If \left( p_n \right)_{n=0}^{\infty} is a family of orthogonal polynoamials with respect to a measure \mu on [-1,1], and \left( x_j, w_j \right) are the quadrature points and weights for the ... 1answer 103 views ### Closure of polynomials in L^2_w with log-normal weight function Consider the Hilbert space L^2_w with scalar product \langle f,g\rangle_w =\int_0^\infty f(x)g(x)w(x)dx where the weight w is the density function of a log-normal distribution$$ w(x)=\frac{1}{\...
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)=...