5
votes
2answers
260 views

Curious gamma identity

I found the following curious identity for the gamma function on Wikipedia for which I'd like to know some references (proof, history, etc). The identity is as follows: $$\Gamma(t) = x^t ...
1
vote
1answer
108 views

Looking for the functions $\alpha_n(x)$ that satisfy $\sum_{n=0}^{\infty} e^{i \alpha_n(x)}\frac{H_n(x)}{\sqrt{2^n n!}}=0$

I am looking for the functions $\alpha_n(x)$ that satisfy $\sum_{n=0}^{\infty} e^{i \alpha_n(x)}\phi_n(x)=0 \text{ or } \delta(x)$ where $\phi_n(x)=\frac{H_n(x)}{\sqrt{2^n n!}}$ and $H_n(x)$ is the ...
2
votes
0answers
57 views

Renormalizing Legendre polynomials to $P_n(0)=1$

One way to define the Legendre polynomials is with the recurrence relation $$(n+1)P_{n+1} (x) = (2n+1)xP_{n} (x)-nP_{n-1} (x),$$ with $P_0(x)=1$ and $P_1(x)=x$. This standardization is normalized so ...
0
votes
1answer
564 views

Calculation of Chebyshev coefficients

The Chebyshev polynomials can be defined recursively as: $T_0(x)=1$; $T_1(x)=x$; $T_{n+1}(x)=2xT_n(x) + T_{n-1}(x)$ The coefficients of these polynomails for a function, $\space f(x)$, under ...
0
votes
0answers
23 views

Chebyshev polynomials of the third and forth kind - error in DLMF?

In many articles the definition of the Chebyshev polynomials of the third $V_n$ and forth kind $W_n$ is something like this: $$V_n(x) = ...
1
vote
1answer
1k views

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 ...
1
vote
0answers
64 views

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 ...
4
votes
1answer
107 views

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. ...
17
votes
3answers
379 views

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 ...
1
vote
1answer
269 views

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 ...
2
votes
1answer
146 views

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 $P_k^{(\alpha,\beta)}(x)$: ...
1
vote
1answer
56 views

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 ...
0
votes
1answer
68 views

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 ...
8
votes
1answer
365 views

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
1
vote
1answer
191 views

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 $P_{2n}(0)=\dfrac{(-1)^n(2n)!}{4^n(n!)^2}$ ...
1
vote
2answers
243 views

Orthogonality of Legendre Functions

The Legendre Polynomials satisfy the following orthogonality condition: The definite integral of $P(n,x) \cdot P(m,x)$ from $-1$ to $1$ equals $0$, if $m$ is not equal to $n$: $$\int_{-1}^1 P(n,x) ...
3
votes
2answers
446 views

Calculate an integral involving Hermite polynomials

I have to calculate the integral $$\frac{1}{\sqrt{2^nn!}\sqrt{2^ll!}}\frac{1}{\sqrt{\pi}}\int_{-\infty}^{+\infty}H_n(x)e^{-x^2+kx}H_l(x)\;\mathrm{d}x$$ where $H_n(x)$ is the $n^{th}$ Hermite ...
3
votes
2answers
644 views

Integral of Hermite polynomial multiplied by $\exp(-x^2/2)$

What is the value of $\int_{-\infty}^{\infty}e^{-\frac{x^2}{2}}H_n(x)dx$ where $H_n(x)$ is the $n^{\small\mbox{th}}$ Hermite Polynomial (physicist's convention)?
0
votes
1answer
178 views

Zernike and Legendre polynomials

The even and odd Zernike polynomials are defined as follows: $$Z^{m}_n(\rho,\varphi) = R^m_n(\rho)\,\cos(m\,\varphi) \!$$ and: $$Z^{-m}_n(\rho,\varphi) = R^m_n(\rho)\,\sin(m\,\varphi), \!$$ with: ...
1
vote
2answers
926 views

Problems regarding integrals involving Legendre polynomials

I am finding difficulty doing this integral involving Legendre polynomials. $$\int_{-1}^1 x^2 P_{n-1}(x)P_{n+1}(x)dx = \frac{2n(n+1)}{(2n-1)(2n+1)(2n+3)}$$ I have two strategies in my mind both of ...
1
vote
2answers
591 views

Proving a property of Legendre polynomials containing its derivatives

I am trying to prove the following property of Legendre polynomials. $$nP_n(x)=x{P_n^\prime(x)} - P^\prime_{n-1}(x)$$ My guess is that I somehow have to use the Bonnets recursion formula ...
3
votes
1answer
257 views

Chebyshev polynomial properties [duplicate]

Possible Duplicate: Chebyshev polynomial question I am trying to prove a property of Chebyshev polynomials. Given the polynomials $T_n(x), n = 0, 1, \ldots$ which are recursively defined ...
2
votes
1answer
207 views

Proving an identity involving the derivative of the Laguerre polynomials with respect to $n$

I've recently come across the following equality in a paper: suppose one defines an analytic function $L(n,x)$ which is equal to the $n$th Laguerre polynomial for $n\in\{0,1,\ldots\}$, and let* ...
3
votes
2answers
171 views

A question of the norm calculation of Hermite function.

Define the Hermite function $H_n (x)$ by $$H_n (x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2} $$ then prove that $$ \int_{\mathbb R} |H_n (x) |^2 e^{-x^2} dx = 2^n n! \sqrt{\pi}$$
1
vote
0answers
145 views

What is the correct differential equation for the Laguerre function?

I would like to derive the correct Laguerre function from the differential equation but the differential equations seems different from the original one. What is the correct differential equation and ...
0
votes
1answer
167 views

Seeking for some neat function for Hermite polynomial

Let us define $$F_n=\int f(z) |He_n(z)|^2 \, dz \, dz^*$$ is there any type of function $f$ could make that $F_n=0$ for $n\geq 2$ and $F_n>0$ for $n<2?$ ...
1
vote
1answer
240 views

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?
3
votes
0answers
160 views

Common zeros of associated Legendre functions

Suppose that $x_{0}$ is a zero of the associated Legendre function $P_{n}^{m}(x)$ (the degree $n$ is a positive integer while the order $m$ is an integer in the range from $0$ to $n$). If there exist ...
2
votes
2answers
631 views

Why are the Gegenbauer polynomials called “ultraspherical”?

There has to be a good reason why the Gegenbauer polynomials were also named "ultraspherical" polynomials. I am aware that when $\alpha=\frac{1}{2}$, the Gegenbauer polynomials reduce to the Legendre ...
1
vote
0answers
600 views

How to prove that Legendre polynomials form a complete basis using functional analysis

I would like to prove that the Legendre polynomials form a complete basis on the interval [-1, 1] using functional analysis. Here is what I came up with so far. Legendre polynomials $P_n(x)$ are ...
3
votes
0answers
134 views

Satisfying a Differential Equation and complex Laguerre

I have the following problem Show that $$L_n(x)=\frac{e^x}{2 \pi i}\oint \frac{t^n e^{-t}}{(t-x)^{n+1}}dt$$ satisfies $$x\, L_n^{\prime\prime}+(1-x)L_n^\prime+n\, L_n=0$$ where the contour is ...
6
votes
1answer
1k views

Connection between Legendre polynomial and Bessel function

In Abramovitz and Stegun (Eq. 9.1.71) I found this curious relation $$\lim_{\nu\to\infty} \left[ \nu^\mu P_\nu^{-\mu}\left(\cos \frac{x}{\nu} \right) \right]= J_\mu(x) \qquad(1)$$ valid for $x>0$. ...
4
votes
2answers
343 views

Finding some orthogonality in a convolution-like integral over Legendre polynomials

I encountered the following integral in my (physics) research, and I've yet to find an analytic solution: $$I(n_1,n_2,n_3) = \int_{-1}^{1} d(\cos\theta_1) \int_{-1}^{1} d(\cos\theta_2) ...
3
votes
1answer
270 views

Solving Shallow water Equations with Hermite polynomials

I have problem with solving the shallow water equations near beaches to achieve the wave run-up over the shore line. The main equation is $$\frac{d^2\eta}{dt^2} + ...
7
votes
2answers
780 views

The roots of Hermite polynomials are all real?

The Hermite polynomials are defined as $$H_n(x)=(-1)^n e^{x^2}\dfrac{d^n}{dx^n}e^{-x^2}.$$ How does one prove that all the roots of the Hermite polynomial are real?
1
vote
1answer
737 views

Associated Legendre polynomials of fractional order

My question concerns the associated or generalized Legendre polynomials. They are labeled by two numbers $m$ and $l$, i.e $P_l^m(x)$, for $x \in [-1,1]$. Usually one assumes that $m$ and $l$ are both ...
5
votes
1answer
498 views

On the completeness of the generalized Laguerre polynomials

I am trying to prove that the generalized Laguerre polynomials form a basis in the Hilbert space $L^2(\mathbb{R})$. 1. Orthonormality \begin{equation} \int_0^{\infty} ...
4
votes
1answer
252 views

Orthogonality of the Gegenbauer Polynomials

Typically the orthogonality relation for the Gegenbauer polynomials is given as: $$ ...
2
votes
2answers
504 views

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. ...
1
vote
4answers
402 views

(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 ...
8
votes
2answers
150 views

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 ...
3
votes
1answer
757 views

Differentiation of generating function of Hermite's polynomials

The generating function of Hermite's polynomials is given by $G(x,t)=e^{2xt-t^2}$ for $x, t \in \mathbf{R}$. It is known that $\displaystyle G(x,t)=\sum_{n=0}^\infty H_n(x) \frac{t^n}{n!}$ for $x, t ...
4
votes
2answers
1k views

Integrating Legendre Polynomials over half range

Solving for the potential of a conducting sphere with hemispheres at opposite potentials, (not using Green's function) I am stuck at this point: $$I_l = V_1 \int_0^1 P_l(x)dx+V_2 \int_{-1}^0 ...
0
votes
2answers
406 views

Proving a Laguerre polynomial integral

After a fair bit of effort, I managed to prove that $$\int_0^\infty t^\alpha \exp(-t) L_n^{\alpha+1}(t)\mathrm dt=\Gamma(\alpha+1)$$ where $L_n^\alpha (t)$ is a generalized Laguerre polynomial, with ...
5
votes
2answers
931 views

Integral Representations of Hermite Polynomial?

One of my former students asked me how to go from one presentation of the Hermite Polynomial to another. And I'm embarassed to say, I've been trying and failing miserably. (I'm guessing this is a ...
16
votes
5answers
4k views

Roots of Legendre Polynomial

I was wondering if the following properties of the Legendre polynomials are true in general. They hold for the first ten or fifteen polynomials. Are the roots always simple (i.e., multiplicity $1$)? ...