Tagged Questions

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First derivative of Lagrange polynomial

Given the Lagrange basis polynomial as: $L_i(x)= \prod_{m=0, m \neq i}^n \frac{x-x_m}{x_i-x_m}$ is there a generic equation for the first derivative ${L_i}'(x)$ for any order,t hat is for any $n$?
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Polynomial Solutions for Differential Equations

Suppose we have a set of polynomials where $\deg(Q_k(x))\le k$, and consider the following differential equation, $$W:=\sum_{k=0}^n Q_k(x)\frac{d^k}{dx^k} .$$ It is known that if there is a ...
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Rolles Theorem Simple and multiple zeros

I have this problem with Legendre polinomials Use Rolle's Theorem to show that Pn cannot have multiple zeros in the open interval (-1, 1). In other words, any zeros of Pn which lie in (-1, 1) must be ...
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Help with generating functions.

Background. Let $P_0(y)=2y-3$ and define recursively $$P_{n+1}(y)=4y\cdot P_n'(y)+(5-4y)\cdot P_n(y).$$ I would like to know as many properties of $P_n$ as I can. For example, it can be shown that ...
Differentiable function and polynomials: Proof of $\phi (x)=ce^x + q(x)$ unclear to me.
I came across a proof which I can't quite understand: If $p:\mathbb{R}\to\mathbb{R}$ is a polynomial of degree $n$ and $\phi:\mathbb{R}\to\mathbb{R}$ is a differentiable function with $\phi'=\phi+p$, ...