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

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 […]

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 […]

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 […]

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 […]

This question How to work out orthogonal polynomials for regression model and the answer https://math.stackexchange.com/a/354807/51020 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., $z = f(x, y)$)?

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 $[−1, 1]$. […]

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 $H_n (x)$ 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 […]

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 […]

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