Tagged Questions

Questions pertaining to certain sets of polynomials that satisfy an orthogonality criterion with respect to some specified inner product.

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creating a loop and maybe a potential loop within a loop in matlab [on hold]

I'm extremely new at Matlab so would like some help. I'm trying to create a loop and maybe a loop within a loop, I'm not too sure at the moment but here's what I'm trying to achieve. I began with ...
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Express Jacobi polynomial $P_n^{\alpha-1,\beta-1}$ into the sum of $P_m^{\alpha,\beta}$

I want to express Jacobi polynomial $P_n^{\alpha-1,\beta-1}$ into the sum of $P_m^{\alpha,\beta}$， where $m$ is $n,n-1,n+1...$. Can this be done? Equation (20,21) in this link might be useful.
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Polynomial Chaos with Beta Distribution - Standard Beta Random Variable, Transformation of Beta Random Variable

Background: I am dealing with a non-intrusive polynomial chaos expansion (e.g. here [Hosder,Walters;2010]). This means I want to represent an uncertain output $U(\xi)$, dependent on a vector of ...
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Calculating Gauss Jacobi Lobatto quadrature nodes and weights

Ke!san online calculator http://keisan.casio.com/exec/system/1329114617 has the nodes and weights quadrature calculation facility to calculate lobatto, legendre, jacobi, etc. quadratures. i want to ...
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Help computing this product through the laguerre polynomial generating function

I don't understand generating functions very well and I was told in one of my questions that the coefficient of $t^N$ in the product between $$\frac{1}{(1-t)^2}\,\exp\left(-\frac{tx}{1-t}\right)$$ and ...
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Two dimensional (discrete) orthogonal polynomials for regression

This question How to work out orthogonal polynomials for regression model and the answer http://math.stackexchange.com/a/354807/51020 explain how to build orthogonal polynomials for regression. ...
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Simplifying this series of Laguerre polynomials

I trying to figure out whether a simpler form of this series exists. $$\sum_{i=0}^{n-2}\frac{L_{i+1}(-x)-L_{i}(-x)}{i+2}\left(\sum_{k=0}^{n-2-i} L_k(x)\right)$$ $L_n(x)$ is the $n$th Laguerre ...
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polynomial chaos expansion: linear combination properties?

I'm dealing with polynomial chaos expansion, for finite support specifically. Assume $X$ and $Y$ are r.v.'s whose the inverse CDFs expressed as $$F^{-1}_X(x) = \sum_{j=0}^{N} s_j^{(X)} \psi(\xi)$$ ...
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I have stumbled upon the following sum over $x,y$ for non-negative integers $\kappa,\lambda$: $$\sum_{x=0}^{\kappa}\sum_{y=0}^{\lambda}\left(-\right)^{x+y}{2\kappa \choose 2x}{2\lambda \choose 2y}\... 0answers 22 views Prove that the special Hermite functions are eigenfunctions of R_{x,y}? How to prove that the special Hermite functions are eigenfunctions of the rotation operators$$R_{x,y} = x\frac{\partial}{\partial y}- y\frac{\partial}{\partial x}.$$Where the special Hermite ... 0answers 24 views Refences to Sturm–Liouville theory with a singular weight function. For \alpha,\beta, nonpositive integers at least one of which is non-zero, define \omega\colon (-1,1)\to \mathbb{R} as \omega(x) = (1-x)^\alpha(1+x)^\beta. Then \omega blows at at x=-1 or x=... 1answer 34 views How can I show that this Jacobi polynomial can be expressed as the sum of these two Legendre polynomials? Let n\in \mathbb{N}^+ be a positive integer. Let L_n\colon \mathbb{R}\to \mathbb{R} be the n'th order Legendre polynomial. Let J_n^{(\alpha,\beta)}\colon \mathbb{R} \to \mathbb{R} be the n'... 0answers 19 views Expansions and Approximations of Functions When looking at series expansions of functions, the main cases I've worked with have been Taylor series and Fourier series. Whilst the latter can arguably be viewed as a subset of the former (e.g. ... 0answers 71 views Inner Product of Chebyshev polynomials of the second kind with x as weighting I have tried to solve the integral$${\int_0^1 U_n (x) U_m (x)x dx }, where ${U_n (x) }$ denotes Chebyshev polynomial of the second kind. Solving the integration and checking the result, I ...
I'm struggling with this question, which says "given the generating funcion $g(x,z)=e^{-z^2 + 2xz} = \sum_{n=0}^{\infty}H_n(x) \frac{z^n}{n!}$ prove that the Hermite polynomials satisfy the Hermite ...