Articles of orthogonal polynomials

Equivalent definitions of Hermite polynomials

The most common definition of the (physicists’) Hermite polynomials that I have found in the literature is the following: $$ H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2} $$ Now, Wikipedia also gives the equivalent definition $$ H_n(x) = \left( 2x – \frac{d}{dx} \right)^n(1) $$ where the operator above is being applied to the constant function $g(x) […]

Complex hermite polynomials as an eigenfunction of an operator?

The real Hermite polynomials are given by $$H_n(x)=(-1)^n e^{x^2}\frac{d^n}{dx^n}e^{-x^2}$$ for $x\in \mathbb R$ and $n=0, 1, 2 …$. The Hermite function $H_n$ of order $n$ is an eigenfunction of the harmonic oscillator $\Delta=-\frac{\partial^2}{\partial x^2}+x^2$ corresponding to the eigenvalue $2n+1$, i.e., $$\Delta H_n (x)=(2n+1) H_n(x) .$$ I would like to know, what happens for the complex […]

Is the orthogonality between Associated Legendre polynomials preserved on an interval

So I am aware of the orthogonality between the Associated Legendre polynomials on the interval $[-1,1]$, that is: \begin{equation} \int_{-1}^{1}P^m_kP^m_ldx\propto\delta_{k,l} \end{equation} where $\delta_{k,l}$ is the kronecker delta function (I am only interested in the case where the upper indices of the Legendre polynomials are equal, but feel free to also discuss the opposite case as […]

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 noticed that it does not match the ones I did analytically for $n=0$, $m=1$; and some other values, so, there must be […]

Show that Polynomials Are Complete on the Real Line

Consider the Hilbert Space of weighted-square-integrable functions f(x): $$ \begin{equation} \int_{-\infty}^{\infty}\frac{f(x)^2}{e^{x}+e^{-x}}dx<\infty. \end{equation} $$ Note this integral is taken over the real line in both directions, and the weighting is roughly exponential. Hilbert Polynomials are complete using a different weighting, while Laguerre polynomials use a similar exponential weighting, but only on the positive real line. What […]

Why the sum of the squares of the roots of the $n$th Hermite polynomial is equal to $n(n-1)/2$?

How to prove that the sum of the squares of the roots of the $n$th Hermite polynomial is $\frac{n(n-1)}{2}$? I tried with Vieta formulas, but it’s hard. I appreciate a proof or reference to it. An idea is to use the definition of sum of Hermite polynomials, but do not know.

find linear polynomial g that is closest to f, where $f(x) = e^x$ and the distance between the two

In the real linear space [-1,1] with inner product $\int^1_{-1} f(x)g(x)\,dx$. Find the linear polynomial $g$ nearest to $f$ and find $||g -f||^2$ for this $g$. My problem is that I simply don’t know how to do this. All the examples that involve projections (maybe even Graham-Schmidt here?) were all vectors. So I don’t know […]

Proving that Legendre Polynomial is orthogonal

This is from here $$\int^1_{-1}f_n(x)P_n(x)dx = 2(-1)^n\frac{a_n}{2^n}\int^1_0(x^2-1)^ndx=2(-1)^n\frac{a_n}{2^n}.I_n$$……..(6) I don’t understand as in shouldnt it be like this, $$\int^1_{-1}f_n(x)P_n(x)dx = (-1)^n\frac{a_n}{2^n}\int^1_{-1}(x^2-1)^ndx=0$$ as they should cancel out even if the integral is non-zero. Edited: Lastly, how does $\int^1_{-1}f_n(x)P_n(x)dx$ shows orthogonality?

Operator for Laguerre polynomial

Is there any operator that could truncate Laguerre polynomial so that the polynomial is only left with the highest order term?

How to work out orthogonal polynomials for regression model

I put this question here as it has a pure maths element to it, even though it has a statistical twist. Basically, I have been given the table of data: $$\begin{matrix} i & \mathrm{Response} \, y_i & \mathrm{Covariate} \, x_i \\ 1 & -1 & -0.5 \\ 2 & -0.25 &-0.25 \\ 3 & 1 […]