Articles of orthogonal polynomials

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 […]

Find some kind of generating functions for odd Hermite polynomials

Question: I am looking at the following series: $$\sum_{n=0}^\infty t^n \cdot \frac{(2n-1)!!}{(2n+1)!}\cdot H_{2n+1}(x), $$ which can also be written as $$\sum_{n=0}^\infty u^n \cdot \frac{1}{(2n+1)!}\cdot H_{2n+1}(x)H_{2n}(0)$$ where $H_n$ are the Hermite polynomials and $t,u$ are some complex numbers. My goal is to somehow simplify it. Ideally, it should look similar to the usual type of genereting […]

Legendre polynomials, Laguerre polynomials: Basic concept

I am asking a simple conceptual question. I saw in many Mathematics and Mathematical physics text books that the Legendre polynomials and Laguerre polynomials “falling from the sky”! I mean, I didn’t get the concept behind and how those polynomials derived. The textbooks rather says a second order differential equation and says the solution is […]

Will this sequence of polynomials converge to a Hermite polynomial pointwise?

While trying to solve this question my testing lead to an observation that I found interesting in its own right. Consider the linear transformation $L:P\to P$ from the space of polynomial functions $p\in\Bbb{R}[x]$ to itself defined by setting $L(p)=p+p’$, and its iterates $$ L^n(p)=\sum_{i=0}^\infty\binom{n}{i}p^{(i)}. $$ Write $G_{m,n}(x):=L^n(x^m)$. These polynomials are monic of degree $m$. If […]

Two dimensional (discrete) orthogonal polynomials for regression

This question How to work out orthogonal polynomials for regression model and the answer explain how to build orthogonal polynomials for regression. However they only consider one dimensional functions. How can we use (discrete) orthogonal polynomials for regression with two dimensional functions (i.e., $z = f(x, y)$)?

Deriving the Normalization formula for Associated Legendre functions: Stage $4$ of $4$

The question that follows is the final stage of the previous $3$ stages found here: Stage 1, Stage 2 and Stage 3 which are needed as part of a derivation of the Associated Legendre Functions Normalization Formula: $$\color{blue}{\displaystyle\int_{x=-1}^{1}[{P_{L}}^m(x)]^2\,\mathrm{d}x=\left(\frac{2}{2L+1}\right)\frac{(L+m)!}{(L-m)!}}\tag{1}$$ where for each $m$, the functions $${P_L}^m(x)=\frac{1}{2^LL!}\left(1-x^2\right)^{m/2}\frac{\mathrm{d}^{L+m}}{\mathrm{d}x^{L+m}}\left(x^2-1\right)^L\tag{2}$$ are a set of Associated Legendre functions on $[−1, 1]$. […]

Is there a representation of an inner product where monomials are orthogonal?

There are plenty of examples of inner products on special sequences of polynomials such that they are orthogonal. I can’t quite wrap my head around the inner product s.t. monomials are orthogonal. Say we have polynomials defined on the unit interval $[0, 1]$. I can define an inner product by stating: $$\langle x^m, x^n\rangle = […]

Hermite polynomials recurrence relation

Hermite polynomials $H_n (x)$ can be obtained using the recurrence relation $$H_{n+1} (x)=2xH_n (x)-2nH_{n-1} (x).$$ To prove this, I started by calculating the first derivative of the Hermite’s Rodrigues formula $H_n (x)=(-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2 } $. The process goes like this: $$ \frac{d}{dx}H_n (x)=(-1)^n 2xe^{x^2}\frac{d^n}{dx^n} e^{-x^2 }+(-1)^n e^{x^2} \frac{d^{n+1}}{dx^{n+1}}e^{-x^2 } $$ Rearranging the terms […]

Chebyshev polynomial question

Consider the Chebyshev polynomials $T_n(x), n = 0, 1, \ldots$ which are recursively defined by $$T_0(x) = 1; \quad T_1(x) = x; \quad T_n(x) = 2x\cdot T_{n−1}(x) − T_{n−2}(x)$$ for $n = 2, 3, \ldots$. I need to show that $$T_n(x) = 2^{n−1}\cdot (x − x_0)(x − x_1) . . . (x − x_{n−1})$$ I […]