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Questions tagged [orthogonal-polynomials]

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

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Indefinite integral involving the product of two generalized Laguerre polynomials

I am trying to find the indefinite integral \begin{align} \int{x^{\alpha +1}e^{-x}\left(L_{m}^{\alpha}(x)\right)^{2}dx} \end{align} where $L_{m}^{\alpha}(x)$ is the generalized Laguerre Polynomial, ...
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11 views

Completeness relation for Jacobi Polynomials

I was wondering if there exists a completeness relation for Jacobi Polynomials, $P^{\alpha,\beta}_{n}(x)$ as in the case of Hermite polynomials, $H_{n}(x)$ such that $$ \sum^{\infty}_{n=0} \psi_n(x) \...
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1answer
36 views

Relation involving generalized Laguerre polynomials

Playing around with different approaches to solve the radial part of the Schrodinger equation for the hydrogen-like atom, I have obtained the following expression ($l$ and $n$ are non-negative ...
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1answer
47 views

Solve Gegenbauer integral $\int\limits_{-1}^{1} x^k \cdot (1-x^2)^{\alpha-1/2} C_n^{\alpha}(x) dx$

I am looking for an analytic solution of the integral \begin{align} \int\limits_{-1}^{1} x^k \cdot (1-x^2)^{\alpha-1/2} C_n^{\alpha}(x) dx \end{align} where $C_n^{\alpha}(x)$ is a given Gegenbauer ...
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10 views

Gaussian quadrature: Orthogonal polynomial for chi distribution

I have a problem involving numerical integration of the form: $$I = \int_0^\infty \!dx \, w(x) f(x)$$ where the weighting function is a chi distribution of degree 2, i.e., $$w(x) = x \, e^{\frac{-...
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30 views

Finding $\int_{0}^{1} \frac{T_{i}^{*''}(x) T_{j}^{*}(x)}{\sqrt{x-x^2}} dx$

Shifted Chebyshev polynomials $$T_{i}^{*}(x) = \cos(i \arccos(2x-1))$$ We want to calculate $$I=\int_{0}^{1} \frac{T_{i}^{*''}(x) T_{j}^{*}(x)}{\sqrt{x-x^2}} dx$$ Which is equal to $$\sum_{\substack{...
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27 views

completeness of Laguerre polynomials

Can you help me to prove that system of Laguerre polynomials $$ L_n = \dfrac{e^t}{n!}\dfrac{d^n}{dt^n} (t^n e^{-t})$$ is completeness in space $L_2((0, \infty),e^{-t}dt)$ ? i have idea of proof: ...
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Is the following matrix defined by the roots of Chebyshev polynomial invertible?

Let $x_0, \dots , x_n$ the roots of the Chebyshev polynomial, $T_{n+1}(x)$. We define: $\begin{pmatrix} \frac{1}{\sqrt2}T_0(x_0) & \cdots & \frac{1}{\sqrt2}T_0(x_0) \\ T_1(x_0) & \...
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18 views

Convergence of sum $f_n(x)=\sum_{l,k} w_{l,n} w_{l,k} x^k$ , with $w$ expansion coefs of an orthonormal system

Good day, Let $\{P_k\}$ be a complete orthonormal system (Fourier series, Legendre-Fourier series, etc..) on interval $(a,b)$ which can be expanded into powers : $$ P_n = \sum_{k=0}^\infty w_{n,k}x^...
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Derive 3-term recursion of polynomials from some recursion OR are my polynomials orthogonal

considering a sequence of polynomials $(Q_n)$ on $\mathbb{R}$ given by $$(q_1 - x)Q_0 + p_1 Q_1 =0$$ and for $n\geq 2$ by $$(q_n-x)Q_{n} +p_n Q_{n+1}+\sum_{i=0}^{n-1} c_{n,i} Q_i =0$$ where all ...
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63 views

Sum and Integration with Legendre Polynomial

What is the value of the following infinite sum after integrating the product of two Legendre Polynomials $P_m^0,~P_n^1$, $$\sum_{n=1}^{\infty}\sum_{m=0}^{\infty} \frac{1}{\sqrt{n(n+1)}} A_m\, A_n \...
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2answers
102 views

Prove that these sets of polynomials have real and distinct roots.

Can anyone tell me if the following set of polynomials have a special name? $$P_{0}(x)=1,P_{1}(x)=x$$ $$P_{n}(x)=xP_{n-1}-P_{n-2}$$ The above gives: $$P_{2}(x)=x^2-1;P_{3}(x)=x^3-2x;P_{4}(x)=x^4-3x^2+...
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1answer
34 views

On the construction of orthogonal polynomials

In the following proof, argument goes on based on considering $C_n$ to be nonzero then it finishes the proof for $C_n=0$ : Also if we set $C_n=0$ in Eq. (6.10) then must $m=0,1,2,...,n- 2$ in Eq. (6....
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1answer
54 views

Can I form a complete set of functions using $e^{-nx}$?

If I start with the set of functions $e^{-nx}$ for all integers $n>1$, can I use them as basis to create a complete set of orthogonal functions on the interval $(0,+\infty)$? By complete I mean ...
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1answer
26 views

Let $S$ be the subspace of $\Bbb R^3$ spanned by the vector $x= (x_1,x_2,x_3)^T$ and $y= (y_1,y_2,y_3)^T$

(a) Let $S$ be the subspace of $\Bbb R^3$ Spanned by the vector $x= (x_1,x_2,x_3)^T$ and $y= (y_1,y_2,y_3)^T$, let $A =\begin{bmatrix}x_1 &x_2 & x_3 \\ y_1 &y_2 &...
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1answer
52 views

Can we refine this asymptotic for Laguerre polynomials?

I just found an interesting and useful limit for Laguerre polynomials: $$\lim_{n \to \infty} L_n \left( \frac{2r}{n+1/2} \right)=J_0(2 \sqrt{2r})$$ I'm using specifically this form of the argument ...
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Orthogonal polynomials on triangles: Connection with quadrature rules?

In a 1D interval $[a, b]$, quadrature rules and orthogonal polynomials are tightly interconnected. For example, given the $n$ roots $t_j$ of the $n$th orthogonal polynomial $p_n$, one has the ...
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25 views

Evaluating an integral using Gegenbauer polynomials

I want to evaluate the following integral $$\int \frac{(r-r'\cos\theta')^2r'^2\,dr'\sin\theta'\,d\theta'\,d\phi'}{(r^2+r'^2-2rr'\cos\theta')^{3/2}}$$ Working that a little bit i end up with this ...
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23 views

Orthogonality of generalized Newton symbol

Consider the functions $P_{n}(x)={x \choose n}.$ My question is, if there exists a measure $\mu$ with support being a subset of $(0,\infty)$ such that the family $\{P_{n}\}$ is orthogonal in $L^{2}(\...
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74 views

Define on $P3$ the inner product $<f,g>=\int_{-1}^1 f(t)g(t)dt$, find orthogonal projection

Define on $P3$ the inner product $\langle f,g \rangle=\int_{-1}^1 f(t)g(t)dt$. a) find the orthogonal projection of $p(x)=x^3$ onto $P2$ I know the orthogonal projection formula, but how do I solve ...
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64 views

What is this generalization of the Chebyshev polynomials?

For $\varepsilon>0$ consider the tridiagonal matrix $$L_{\varepsilon}=\begin{bmatrix} 0 & 1 & \ & \ & \ & \ & \ & \ \\ 1 & \varepsilon & 1 & \ & \ &...
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28 views

Integral involving the Associated Laguerre polynomials

I'm trying to solve this integral $\int_{0}^{\infty} L^n_p L^n_{p'} e^{-x} x^{n-1} dx = \dfrac{1}{n} \dfrac{(p!)^3}{(p-n)!} \delta_{pp'}$ I started with integration by parts where $u = $ $L^n_p L^...
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29 views

Projection on a subspace

An inner product is defined on $P_3$ such that $<f, g>$ $=$ $\int _{-1}^1\:f\left(t\right)g\left(t\right)dt$. What is the orthogonal projection of $p(x)$ $=$ $x^3$ onto the subspace $S$ $=$ $\...
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27 views

An closed-form expression of an integral of Chebyshev series and exponential function

Does the following integral has a closed-form expression? $\int_{-1}^{1} T_n(x)\exp(i\pi x)dx,$ where, $T_n$ is the Chebyshev polynomial of degree n.
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126 views

Orthogonal Projection on a Polynomial Space

An inner product is defined on $P3$ such that $<f, g>$ $=$ $\int _{-1}^1\:f\left(t\right)g\left(t\right)dt$. What is the orthogonal projection of $p(x)$ $=$ $x^3$ onto $P2$? So I got that $f_1\...
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Identifying the orthogonal polynomial from the recurrence relation

I am trying to see if this recursion relation can be solved in terms of standard orthogonal polynomials : $$ p_{n+1}(x)=2 x p_{n}(x)-2(n+1) p_{n-1}(x), $$ with $p_0(x)=1$ and $p_1(x)=2x$. These ...
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1answer
23 views

The generalized Laguerre polynomials: Are there any expressions valid for any case?

There are general expressions of the generalized Laguerre polynomials. For example: $$ L_n^{(\alpha)}(x) = {}_{n+\alpha}\mathrm{C}_n\ {}_1F_1(-n, \alpha+1, x), \hspace{20pt}(1) $$ $$ L_n^{(\alpha)}(x)...
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1answer
32 views

Use Gram-Schmidt process on Chebychev polynomials

I need help using the Gram-Schmidt orthogonalization process to derive the first four orthonormal Chebychev polynomials. Using the range $[-1,1]$ and the weight function $w(x)=(1-x^2)^\frac{1}{2}$. ...
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1answer
85 views

Orthogonal polynomials with respect to discrete probability distributions

The polynomials from the Askey scheme are orthogonal with respect to a standard probability distribution. For example, Hermite polynomials (in a suitable form) are orthogonal with respect to a ...
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8 views

Polynomials orthogonal wrt Rayleigh distribution

As per the title, have the class of polynomials that are orthogonal with respect to Rayleigh distribution been studied? Do they have a name?
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1answer
51 views

Algorithm for orthogonalizing polynomials with specific inner product

I am attempting to generate a as big as possible collection of orthogonal polynomials $p_1, p_2, ..., p_n$, $\left\langle p_i, q_i\right \rangle = \delta_{ij}$ where the inner product is with respect ...
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40 views

$L_2$ scalar product between Hermite polynomials

I am trying to compute the $L_2$ scalar product between (probabilists’) Hermite polynomials (defined as in Wiki) with Gaussian weight and different scales, i.e. for some constants $c, d$: $$\frac{1}{\...
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Orthogonal polynomials as eigenfunctions of a second-order difference operator

I am Reading the theorem 6.1.3 of this book https://books.google.com.mx/books?id=RusIDAAAQBAJ&pg=PA146&lpg=PA146&dq=up+to+normalization,+the+charlier,+krawtchouk,+meixner,+and+chebyshev%E2%...
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3answers
130 views

Weird Integral Involving Hermite Polynomials

I have stumbled upon the following integral involving the Hermite polynomials: $$ I(m) = \int_\mathbb{R} e^{i m x} \left[ e^{-\frac{x^2}{2}} H_m(x) \right] dx \, , \quad m \in \mathbb{N} \cup \{0\} \...
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1answer
61 views

An Addition formula for Hermite polynomials

My question concerns an addition formula that can be found on the Wikipedia page of Hermite Polynomials but I can't find it anywhere else. The well-known formula that can be found in many books is the ...
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1answer
39 views

Sum involving Hermite polynomials

I am wondering if there is a simpler form of the following summation involving the (physicists') Hermite polynomials: $$\sum_{k=0}^{n}\frac{H_k(x)}{(2i)^k},$$ where $i=\sqrt{-1}$ is the imaginary ...
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1answer
138 views

Find the Fourier-Bessel Series for $f(x)$ With Respect to the Orthogonal Set: How Was $w(x)$ Found?

I have the following problem: If $f(x) = x$, $0 < x < 2$, find the Fourier-Bessel series for $f(x)$ with respect to the orthogonal set $\{ J_1 (k_n x) \}$, where $k_n$ is the $n$th positive ...
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65 views

Integral of a triple product of Laguerre polynomials

I would like to know if there's an exact expression for this integral in terms of known elementary or special functions: $$\int_0^\infty \exp \left(-\frac{a+b+c}{2}x \right) L_j (a x) L_k (b x) ...
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1answer
63 views

How to prove two function $\phi_m$ and $\phi_n$ are orthogonal?

A while back I found a specific proof for the orthogonality of two functions but I can't seem to find it online anywhere. I just want to make sure the proof I've given is defined exactly, which is: "...
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0answers
42 views

The Distribution of Abscissae and Sums of Weights in Gaussian Quadrature

Let $n$ be a positive integer, and for $i = 1, 2, \ldots, n$, let $x_i$ be the $i^\text{th}$ abscissa for $n$-point Gaussian quadrature, and $w_i$ the associated weight, so that $-1 < x_1 < x_2 &...
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0answers
68 views

Express these polynomials in terms of orthogonal ones

In a problem in QM I faced with this polynomials, sadly they are not orthogonal. I was wondering if someone else knows these polynomials, I was looking up and didn't find anything about them but they ...
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25 views

Help finding source papers for questions concerning orthogonal polynomials.

I solved some problems concerning orthogonal polynomials from chapter four of the book "Classical and Quantum Orthogonal Polynomials in One Variable" by Mourad E. H. Ismail. I would like to find the ...
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0answers
49 views

I want to prove that $ \int_{-1}^{+1}xp_{s}(x)p_{r}(x)dx=\delta_{r,s+1}\frac{2r}{(2r+1)(2r-1)}+\delta_{s,r+1}\frac{2s}{(2s+1)(2s-1)}. $

Please make an illustration to me in proving of the following. Problem: Assume that $ p_{n}(x)$ is a Legendre polynomials. I want to prove that $$ \int_{-1}^{+1}xp_{s}(x)p_{r}(x)dx=\delta_{r,...
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1answer
50 views

Determining Rodriguez formula for Legendre polynomials

While proving Rodriguez formula for Legendre classical orthogonal polynomials I found a part I cannot prove. Namely, if we define $Q_n(x) = \frac{(-1)^n}{2^n n!}((1-x^2)^n)^{(n)}$ and further observe $...
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0answers
19 views

Is there a Grassman equivalent of orthogonal functions?

Orthogonal functions such as the Hermite functions work with commutative variables. Is there a similar thing that works with Grassman variables?
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1answer
100 views

Use Gram-Schmidt orthogonalisation to orthogonalise the system of vectors

I have been working on this problem, we are given the below system of vectors $f_{1} = x, f_{2} = \cos(x), f_{3}= \sin(x)$ from the inner product of $C_{\mathbb{R}}[-1,1]$ and we have to ...
2
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0answers
65 views

How to construct an orthogonal, complete set of functions on the interval $[0, L]$ with given conditions at $x=0,L$?

The functions must be at least $C^2$, preferably $C^\infty$. Completeness for square-integrable functions is sufficient. The conditions at the points $x=0,L$ prescribe the values of the function and ...
3
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0answers
43 views

Is there a name for “almost-subspaces” $U$ that do not satisfy $cu \in U, \forall u \in U$? If so do they have any properties?

I am working with a 'near-subspace' in which the subspace is defined by $$ U := \{p(x) \in \mathbb{P}^3 \mid p(x)> 0, 0\le x \le 1 \lor p(x) \equiv 0 \} $$ This satisfies that $0\in U$ and it ...
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0answers
41 views

Is any algebraic number a root of a given orthogonal polynomial?

Let $S=\left\{P_n(x)\right\}_{n=0}^{\infty}$ be any sequence of $n$-degree polynomials which are orthogonal in the interval $(a,b)\in\mathbb{R}$. If $\xi$ is an arbitrary algebraic number such that $a&...
3
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1answer
50 views

Basis functions for a Galerkin procedure

For a Galerkin procedure, I am trying to construct a set of linerly independent functions $\{\varphi_n\}_{n = 1}^N$ satisfying $$ \varphi_n(0) = 0, ~~ \varphi_n'(1) = 0, $$ for all $n \geq 1$. A ...