Tagged Questions

Questions pertaining to certain sets of polynomials that satisfy an orthogonality criterion with respect to some specified inner product.

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2
votes
2answers
70 views

An infinite series involving Legendre polynomials

For $x \in [-1,1]$ and $0 \le g < 1$, consider the convergent series $$ H(x,g) = \sum_{k = 0}^\infty (2k+1) g^k P_k(x)^2 $$ where $P_k$ is the $k$-th Legendre polynomial. Then $H(1,g) = ...
0
votes
0answers
20 views

Orthonormal Basis for $[5,1+t]^{\perp}$

Consider the vector space $\Bbb{V}=P_3(\Bbb{R})$ of the real polynomials of degree less or equal 3, with the inner product given by $$\langle f,g\rangle=\int_0^1f(t)g(t)dt,\forall f,g\in\Bbb{V}.$$ ...
1
vote
2answers
25 views

Matching polynomial of a complete bipartite graph is a generalized Laguerre polynomial.

Consider a graph $G$ with $n$ vertices. Let $m_k$ be the number of $k$-edge matchings. (A matching in a graph is a set of edges without common vertices.) Several different types of matching ...
3
votes
1answer
42 views

Show that Polynomials Are Complete on the Real Line

Consider the Hilbert Space of weighted-square-integrable functions f(x): $$ \begin{equation} \int_{-\infty}^{\infty}\frac{f(x)^2}{e^{x}+e^{-x}}dx<\infty. \end{equation} $$ Note this integral is ...
0
votes
1answer
33 views

Can anybody explain orthogonal basis for constant polynomials?

I have a problem. $(f,g)= \int_{1}^{3} f(x)g(x) \, dx$ and $f(x)=\dfrac{1}{x}$ I have to find out a nearest constant polynomials to $f$. So, I assumed $a$ to be a orthonormal basis for constant ...
0
votes
0answers
32 views

LCM of two polynmials when they are represented as point-value.

I`m wondering if we can obtain least common multiple of two polynomial when each polynomial represented as point-value. To be more clear, can we do any computation on these point-values and obtain ...
0
votes
1answer
30 views

Why are non-polynomial Legendre functions and Legendre polynomials not orthogonal?

The eigenfunctions of distinct eigenvalues for a Hermitian operator are proved to be orthogonal. Why does the same not apply to Legendre polynomials and functions that have different eigenvalues ? ...
0
votes
0answers
30 views

Iterated integrals of Legendre polynomials

Let $P_n$ denote the $n$th order Legendre polynomial. It is known that for $n\not= 0$ we have $\int P_n(x)\, \mathrm{d}x = \frac{1}{1+2n} (P_{n+1}(x) - P_{n-1}(x))+ C $. Setting all integration ...
0
votes
0answers
17 views

How can I expand a zonal polynomial in a sum of two matrices? C(A+B) = C(A)*C(B)

I am trying to solve an integral involving zonal polynomials. TO do that I need to somehow separate out the zonal polynomial C(A+B) where my A varies but B is constant. They're all square matrices ...
0
votes
0answers
23 views

Example of incomplete orthogonal system in L2 space

I was asked to provide an example of an incomplete orthogonal in any L2 space and was wondering if this is a valid answer. Take some known complete system, for example the Hermite polynomials over ...
3
votes
0answers
42 views

Gaps between roots of trigonometric polynomials?

Given a polynomial in $e^{\mathrm{i}k t}$ of the form $$ p(t) = \sum_{-n\leq k\leq n} c_k e^{\mathrm{i}k t} $$ with $\bar c_{-k} = c_k$, is there a good way of characterising how close its roots can ...
5
votes
1answer
50 views

Asymptotics bound on Jacobi polynomials in the complex plane and for large $n$

Dear mathematicians and theoretical physicists, I am a theoretical physicist and I am bothering to you since I need to know some asymptotic and analytical properties of Jacobi polynomials ...
1
vote
2answers
48 views

Least squares approximation: Legendre polynomial

Find the best quadratic least squares approximation to $f(x)=e^x$ on $[-1,1]$ with respect to the inner product $\langle f(x),g(x) \rangle = \displaystyle\int_{-1}^1 f(x)g(x)dx$. I cannot figure out ...
1
vote
1answer
38 views

Proof that Qm(x) and Pn(x) have the same sign

I have difficulty in this exercise, is from the book Apostol Calculus Vol 2, I really want to know is there is a technique that can show that the two polynomials share the same roots. Thanks.
6
votes
0answers
84 views

Proving that two functions involving integrals with Legendre polynomials are equal

I have two functions that I expect to be equal (where $P_{2l}$ are the even Legendre Polynomials): $$F_{2l}(x)=x\, \tanh(\pi x/2)\left|\int_0^1 u^{i x-1}P_{2l}(u)\,du\right|^2$$ ...
5
votes
2answers
268 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
28 views

A sum of Laguerre polynomials

I'm looking to find a closed-form expression for the sum $$S = \sum_{n=0}^N e^{-x/2} L_n^{0}(x),$$ where $L_n^{0}$ is the $n$th Laguerre polynomial. Using the formula $$L_n^{\alpha}(x) = \sum_{m=0}^n ...
1
vote
1answer
183 views

How does one prove Rodrigues' formula for Legendre Polynomials?

I am trying to prove that $\frac{1}{n!\space2^n}\frac{d^n}{dx^n}\{(x^2-1)^n\}=P_n(x)$, where $P_n(x)$ is the Legendre Polynomial of order n. I've been told that the proof uses complex analysis, of ...
0
votes
0answers
13 views

Best uniforme approximation of nule function in the meaning of Tchebychev

I would be interest to know , why exactly approximate a nule function and it is in the same time nule ? I would be like someone give me enough (papers, link ...) about "The best uniforme ...
1
vote
2answers
32 views

not complete polynomial and roots

It might be very simple but I need a formal proof for accepting or rejecting the idea below. Let g be a polynomial of the order of n given below $$ g(L)=1-\theta_1 L-\theta_2 L^2- \ldots -\theta_n ...
11
votes
1answer
163 views

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 ...
0
votes
2answers
198 views

Legendre polynomial expansion of the unit step function.

The problem is to determine the expansion of the unit step function in terms of Legendre polynomials on the interval $[-1,1]$. Here the Legendre polynomials are the family of orthogonal polynomials ...
0
votes
0answers
22 views

Orthogonal polynomials induction proof

I tried writing this all out but cannot seem to get anything sensible. Basically I want to prove that assuming w(x) is the weight function of a Gram Schmidt orthogonalization process and w is an ...
0
votes
0answers
45 views

Laplace transform and Laguerre Polynomials

let be the Laplace transform of a function $$ F(s)= \int_{0}^{\infty}dtf(t)e^{-st} $$ then if the function satisfies that for every integer 'n' $ D^{n}f(x)=0 $ then $$ s^{n}F(s)= ...
1
vote
0answers
17 views

orthogonal polynomials and quadrature formulae

is it true that for every set of orthogonal polynomials $$ \int_{a}^{b} w(x)P_{n}(x)P_{m}(y) =a_{m,n}\delta _{m}^{n}$$ with respect a measure $ w(x)$ can we always find a quadrature formula for ...
1
vote
0answers
24 views

Finding another solution in certain form (confluent hypergeometric series)

Consider the differential equation $$xf''(x)+(\alpha+1-x)f'(x)+nf(x)=0 ..........(*)$$, we know that the Laguerre Polynomial ...
1
vote
0answers
24 views

Checking a solution of differential equation

I am trying to prove that the Laguerre polynomial $$L=L_n^{\alpha}(x)=\frac{x^{-\alpha}e^{-x}}{n!}\frac{d^n}{dx^n}(e^{-x}x^{n+\alpha})= ...
-1
votes
1answer
68 views

Fermat Last theorem on Poly-Euler numbers

The poly-Euler numbers, denoted as $E_{n}^{(k)}$, are defined by the following generating functions :$${2\operatorname{Li}_k(1-e^{-x}) \over 1+e^{-x}}=\sum_{n=0}^\infty E_n^{(k)}{x^n\over n!}$$ The ...
0
votes
1answer
41 views

Use the zeroes of T3 to construct an interpolating polynomial

Use the zeroes of T3 to construct an interpolating polynomial of degree two for the function x^3 on the interval [-1,1] Okay, so I have been looking at Finding the zeroes using Chebyshev polynomials ...
4
votes
1answer
89 views

Numerical evaluation of polynomials in Chebyshev basis

I have high order (15 and higher) polynomials defined in Chebyshev basis and need to evaluate them (for plotting) on some intervals inside the canonical interval $[1,\,-1]$. A good accuracy near 1 and ...
2
votes
1answer
42 views

Orthogonality of Lagrange Polynomials in Hermite Inner Product

My question is as shown above. I have churned through the first part, but I am stuck on showing the orthogonality of the Lagrange polynomials. My first hope was to use the fact that the Lagrange ...
4
votes
1answer
240 views

functions orthogonal to the exponential Bell polynomials

Consider the single variable Bell polynomials $\phi_{n}(x)$ given by: $$\phi_{n}(x)=e^{-x}\sum_{k=0}^{\infty}\frac{k^{n}x^{k}}{k!}$$ I am looking for a set of functions $\tilde{\phi}_{n}(x)$ such ...
1
vote
0answers
44 views

Turning a summation into an integral

I have a summation of the form: $$y(x) = \sum\limits_{h=-L}^L\frac{A(h)\cdot R(h)^2}{((x-h)^2+R(h)^2)^{3/2}}$$ Where I wish to solve/optimise $R(h)$ (leaving $A(h) = const/h$) or $R(h)$ and $A(h)$ ...
0
votes
1answer
40 views

How could I prove that $P_n (1)=1=-1$ for the Legendre polynomials?

I could "prove" that $P_n(1)=1=-1$. But I'm not sure where is my mistake. If I use the generating function: $$\frac{1}{\sqrt{1-2z+z^2}}=\sum_{n=0}^{\infty}P_n(1)z^n$$ Since: $\sqrt{1-2z+z^2 } = ...
0
votes
0answers
25 views

a question about Jacobi polynomials

Imagine if I have a defined function $\omega(\alpha, \beta, \gamma)$, where $0<\alpha<2\pi$, $0<\beta<\pi$, and $0<\gamma<2\pi$. I can then expand this function into series just like ...
0
votes
1answer
27 views

Expanding unity in terms of orthogonal functions cos( alpha(i) * y)

It is written in the book I am reading without proof that if we expand unity in terms of orthogonal functions $cos( \alpha_i y)$, we get: (Please check this link) ...
1
vote
1answer
679 views

bijection between lattice path and permutations

I am studying Viennot's combinatorial model for the Laguerre polynomials this semester under the guidance of my math professor. If I understand correctly, a bijection exists between the number of ...
0
votes
1answer
43 views

Calculating Fourier expansion using Legendre Polynomials

I'm trying to write any function of the type $t^m$ using Legendre polynomials $P_n(t)$ . That means: $$t^m=\sum_{n=0}^\infty\langle P_n,t_m\rangle P_n =\sum_{n=0}^\infty a_{mn}P_n$$ Where I have to ...
1
vote
0answers
26 views

The coefficients of multivariate hermite polynomials

Can someone please help me with finding a table that contains the coefficients of the multivariate hermite polynomials? I have a really hard time understanding the subject and I hope such a table ...
0
votes
1answer
21 views

On Jacobi Polynomial

Prove that The $n$-th jacobi polynomial $P_n^{(\alpha, \beta)}(x)$ with parameter $(\alpha, \beta)$ defined by $P_n^{(\alpha, ...
3
votes
1answer
31 views

Constructing sequence of orthogonal polynomials for arbitrary scalarproduct?

I need to construct a sequence of orthogonal polynomial $(P_i)_{i=0}^{\infty}$ for a family of scalarproducts. I want to look at different scalar products $\langle P_n(x),P_m(x) ...
1
vote
0answers
60 views

Problems in Orthogonal Polynomials

Let $P_n (x) , n=0,1,...$ be the monic orthogonal polynomials associated to the weight function $w(x)$. Let $Q(x), n=0,1,2,...$ be the monic orthogonal polynomials associated to the weight function ...
1
vote
1answer
138 views

Hermite Polynomials Triple Product

Similar to the question Legendre Polynomials Triple Product, I would like to ask whether there are any explicit formulas for the inner product of the Hermite polynomial triple product \begin{align} ...
0
votes
0answers
24 views

Recurrence formula for orthogonal polynomials

Consider the recurrence formula: $P_n(x)=(x-c_n)P_{n-1}(x)-\lambda_n P_{n-2}(x)$ The problem consist on showing that $\xi_1<c_n<\eta_1$ where $[\xi,\eta]$ is the true interval of orthogonality ...
2
votes
1answer
123 views

Lie algebra (su(1,1)) from legendre polynomials; question regarding http://arxiv.org/abs/1205.6353

Apologies if this question is a duplicate. OK, so my question heavily involves the paper http://arxiv.org/abs/1205.6353 which nicely details the Lie algebra su(1,1) coming from the Laguerre $L_n$, ...
2
votes
1answer
60 views

Intuition behind the weight function

The inner product in a $L^2$ space can be defined as: $$\langle f,g\rangle =\int_a^b \bar{f}(x)g(x)w(x)dx$$ For Legendre polynomials, we define it as: $$\langle P_m,P_n\rangle =\int_0^1 ...
2
votes
0answers
88 views

Derive Differentiation Identity for Gegenbauer Polynomial

For $\lambda > 0$ and $n = 0,1,2,3,\ldots$, the following is an n-th order polynomial: $$ P_{\lambda,n}(x)=(1-x^{2})^{-\lambda+1/2}\frac{d^{n}}{dx^{n}}(1-x^{2})^{n+\lambda-1/2} $$ ...
1
vote
1answer
66 views

Equivalent definitions of Hermite polynomials

The most common definition of the (physicists') Hermite polynomials that I have found in the literature is the following: $$ H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2} $$ Now, Wikipedia also ...
1
vote
0answers
13 views

Polynomials/Function family that remains within interval bounds

Is there such a thing as a family of orthogonal functions defined over an interval $x\in[a,b]$, where the output is the interval $f(x)=y\in[c,d]$ (e.g. for simplicity $[0,1] \rightarrow [0,1]$). I ...
3
votes
1answer
64 views

Why should we care about orthogonal polynomials?

I know the mathematical definition but I'm having a hard time understanding the utility of orthogonal polynomials. I'm not saying they are useless, far from that! It is just that I like understanding ...