# Tagged Questions

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### A method for calculating this integral hermite polynomials

I need proof this, $\int_{-\infty}^{\infty}e^{-x^2}H_n^{2}(x)x^2dx=2^nn!\sqrt{\pi}(n+\frac{1}{2})$ This is the idea: Multiply ...
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### how a discontinuous function converges to Hermite- Fourier series?

I have the proof using a text that if a function $f (x)$ is square integrable with weight function $e^{-x^2}$ and also is piecewise continuous, then $f (x)$ converges to ...
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### 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. ...
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### Local maxima of Legendre polynomials

When I plotted the (normalized) Legendre polynoials, I couldn't help noticing that all the local maxima lay on a really nice curve: What is the equation of the curve (and how can we arrive to that ...
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### Meaning of $\alpha$ in Laguerre polynomials

I found that generalized Laguerre polynomials are: $$L_n^{\alpha} (x) = \sum_{i=0}^n (-1)^i {n+\alpha \choose n-i} \frac{x^i}{i!}.$$ However, I wonder what is the meaning of $\alpha$ in this ...
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### Integral identity with square of Jacobi polynomial

This has stumped me for a while: I have a function $\zeta_k^S(x)$ that can be expressed using Jacobi polynomials $P_k^{(\alpha,\beta)}(x)$: ...
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### Irreducibility of the Gegenbauer polynomials

The Gegenbauer polynomials $C_n^{(\alpha)}(x)$ can be defined by requiring that they satisfy that $$\frac{1}{(1-2xt+t^2)^{\alpha}} = \sum_{n=0}^{\infty} C_n^{(\alpha)}(x)t^n.$$ In the cases when ...
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### Using a known recursion relation to solve an Integral

Provided that the following generating expression defines the Hermite polynomials $H_n(x)$ $$F(x,h)= e^{2hx-h^{2}} = \sum_{n=0}^{\infty} H_n(x) \frac{h^{n}}{n!}$$ Find a recursion relation and ...
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### Connection between Hermite & Legendre polynomials

Prove that $$H_n(x)= 2^{n+1}e^{x^2}\int_x^\infty e^{-t^2}t^{n+1}P_n\left(\frac{x}t\right)dt,$$ where $H_n$ is Hermite polynomial & $P_n$ is Legendre polynomial
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### Proving property of Legendre Polynomial (by induction?)

Let $P_n(x) = \dfrac{1}{2^n n!}\dfrac{d^n}{dx^n} [(x^2-1)^n]$, we know that $\sum\limits_{n=0}^{\infty} P_n(x)t^n=(1-2tx+t^2)^{-1/2}$ How can we proved $P_{2n}(0)=\dfrac{(-1)^n(2n)!}{4^n(n!)^2}$ ...
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### The relationship between Legendre Polynomials and monomial basis polynomials

I am currently doing filter designs and stumbled across this mathematical problem which I cannot understand. I was hoping for some insight from experts around this field to help me with this. ...
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### (Should be easy) Legendre polynomial integration debugging

I am trying to evaluate the following integral: $$I_n=\int\limits_{-1}^1 f(x)P_n(x)dx$$ where $f(x)=1$ for $x\in[-1,0)$ and $f(x)=-1$ for $x\in(0,1]$ and $P_n(x)$ is the Legendre polynomial of degree ...
### Is there a neat way to show $\int_{-1}^1 \frac{ U_n(z) U_n(z)}{\sqrt{1-z^2}} \mathrm{d} z = \pi (n+1)$
In answering a question on math.SE, I attempted to find integral of Fejér kernel by using $$K_n(t) = \frac{1}{n} U_{n-1}^2\left( \cos \frac{t}{2} \right)$$ where $U_n(z)$ stands for the ...