Questions tagged [orthogonal-polynomials]
Questions pertaining to certain sets of polynomials that satisfy an orthogonality criterion with respect to some specified inner product.
1
vote
0answers
15 views
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, ...
0
votes
0answers
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
$$
\sum^{\infty}_{n=0} \psi_n(x) \...
3
votes
1answer
36 views
Relation involving generalized Laguerre polynomials
Playing around with different approaches to solve the radial part of the Schrodinger equation for the hydrogen-like atom, I have obtained the following expression ($l$ and $n$ are non-negative ...
1
vote
1answer
47 views
Solve Gegenbauer integral $\int\limits_{-1}^{1} x^k \cdot (1-x^2)^{\alpha-1/2} C_n^{\alpha}(x) dx$
I am looking for an analytic solution of the integral
\begin{align}
\int\limits_{-1}^{1} x^k \cdot (1-x^2)^{\alpha-1/2} C_n^{\alpha}(x) dx
\end{align} where $C_n^{\alpha}(x)$ is a given Gegenbauer ...
0
votes
0answers
10 views
Gaussian quadrature: Orthogonal polynomial for chi distribution
I have a problem involving numerical integration of the form:
$$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{-...
0
votes
0answers
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 $$\sum_{\substack{...
1
vote
0answers
27 views
completeness of Laguerre polynomials
Can you help me to prove that system of Laguerre polynomials
$$ L_n = \dfrac{e^t}{n!}\dfrac{d^n}{dt^n} (t^n e^{-t})$$ is completeness in space $L_2((0, \infty),e^{-t}dt)$ ?
i have idea of proof:
...
1
vote
0answers
22 views
Is the following matrix defined by the roots of Chebyshev polynomial invertible?
Let $x_0, \dots , x_n$ the roots of the Chebyshev polynomial, $T_{n+1}(x)$.
We define:
$\begin{pmatrix}
\frac{1}{\sqrt2}T_0(x_0) & \cdots & \frac{1}{\sqrt2}T_0(x_0) \\
T_1(x_0) & \...
0
votes
0answers
18 views
Convergence of sum $f_n(x)=\sum_{l,k} w_{l,n} w_{l,k} x^k$ , with $w$ expansion coefs of an orthonormal system
Good day,
Let $\{P_k\}$ be a complete orthonormal system (Fourier series, Legendre-Fourier series, etc..) on interval $(a,b)$ which can be expanded into powers :
$$ P_n = \sum_{k=0}^\infty w_{n,k}x^...
0
votes
0answers
19 views
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 ...
0
votes
0answers
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 $P_m^0,~P_n^1$, $$\sum_{n=1}^{\infty}\sum_{m=0}^{\infty} \frac{1}{\sqrt{n(n+1)}} A_m\, A_n \...
1
vote
2answers
102 views
Prove that these sets of polynomials have real and distinct roots.
Can anyone tell me if the following set of polynomials have a special name?
$$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+...
1
vote
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....
1
vote
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 ...
0
votes
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 &...
4
votes
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 ...
1
vote
0answers
15 views
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 ...
1
vote
0answers
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 ...
0
votes
0answers
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}(\...
0
votes
0answers
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 ...
5
votes
0answers
64 views
What is this generalization of the Chebyshev polynomials?
For $\varepsilon>0$ consider the tridiagonal matrix
$$L_{\varepsilon}=\begin{bmatrix}
0 & 1 & \ & \ & \ & \ & \ & \ \\
1 & \varepsilon & 1 & \ & \ &...
1
vote
0answers
28 views
Integral involving the Associated Laguerre polynomials
I'm trying to solve this integral
$\int_{0}^{\infty} L^n_p L^n_{p'} e^{-x} x^{n-1} dx = \dfrac{1}{n} \dfrac{(p!)^3}{(p-n)!} \delta_{pp'}$
I started with integration by parts where
$u = $ $L^n_p L^...
0
votes
0answers
29 views
Projection on a subspace
An inner product is defined on $P_3$ such that $<f, g>$ $=$ $\int _{-1}^1\:f\left(t\right)g\left(t\right)dt$.
What is the orthogonal projection of $p(x)$ $=$ $x^3$ onto the subspace $S$ $=$ $\...
2
votes
0answers
27 views
An closed-form expression of an integral of Chebyshev series and exponential function
Does the following integral has a closed-form expression?
$\int_{-1}^{1} T_n(x)\exp(i\pi x)dx,$
where, $T_n$ is the Chebyshev polynomial of degree n.
2
votes
2answers
126 views
Orthogonal Projection on a Polynomial Space
An inner product is defined on $P3$ such that $<f, g>$ $=$ $\int _{-1}^1\:f\left(t\right)g\left(t\right)dt$.
What is the orthogonal projection of $p(x)$ $=$ $x^3$ onto $P2$?
So I got that $f_1\...
0
votes
0answers
21 views
Identifying the orthogonal polynomial from the recurrence relation
I am trying to see if this recursion relation can be solved in terms of standard orthogonal polynomials :
$$
p_{n+1}(x)=2 x p_{n}(x)-2(n+1) p_{n-1}(x),
$$
with $p_0(x)=1$ and $p_1(x)=2x$. These ...
0
votes
1answer
23 views
The generalized Laguerre polynomials: Are there any expressions valid for any case?
There are general expressions of the generalized Laguerre polynomials.
For example:
$$
L_n^{(\alpha)}(x) = {}_{n+\alpha}\mathrm{C}_n\ {}_1F_1(-n, \alpha+1, x),
\hspace{20pt}(1)
$$
$$
L_n^{(\alpha)}(x)...
0
votes
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 $w(x)=(1-x^2)^\frac{1}{2}$.
...
2
votes
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 ...
0
votes
0answers
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?
4
votes
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 ...
3
votes
0answers
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 $c, d$:
$$\frac{1}{\...
0
votes
0answers
21 views
Orthogonal polynomials as eigenfunctions of a second-order difference operator
I am Reading the theorem 6.1.3 of this book https://books.google.com.mx/books?id=RusIDAAAQBAJ&pg=PA146&lpg=PA146&dq=up+to+normalization,+the+charlier,+krawtchouk,+meixner,+and+chebyshev%E2%...
6
votes
3answers
130 views
Weird Integral Involving Hermite Polynomials
I have stumbled upon the following integral involving the Hermite polynomials:
$$ 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\} \...
2
votes
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 ...
1
vote
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 ...
3
votes
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 ...
4
votes
0answers
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:
$$\int_0^\infty \exp \left(-\frac{a+b+c}{2}x \right) L_j (a x) L_k (b x) ...
0
votes
1answer
63 views
How to prove two function $\phi_m$ and $\phi_n$ are orthogonal?
A while back I found a specific proof for the orthogonality of two functions but I can't seem to find it online anywhere. I just want to make sure the proof I've given is defined exactly, which is:
"...
3
votes
0answers
42 views
The Distribution of Abscissae and Sums of Weights in Gaussian Quadrature
Let $n$ be a positive integer, and for $i = 1, 2, \ldots, n$, let
$x_i$ be the $i^\text{th}$ abscissa for $n$-point Gaussian
quadrature, and $w_i$ the associated weight, so that $-1 < x_1 <
x_2 &...
1
vote
0answers
68 views
Express these polynomials in terms of orthogonal ones
In a problem in QM I faced with this polynomials, sadly they are not orthogonal. I was wondering if someone else knows these polynomials, I was looking up and didn't find anything about them but they ...
2
votes
0answers
25 views
Help finding source papers for questions concerning orthogonal polynomials.
I solved some problems concerning orthogonal polynomials from chapter four of the book "Classical and Quantum Orthogonal Polynomials in One Variable" by Mourad E. H. Ismail. I would like to find the ...
2
votes
0answers
49 views
I want to prove that $ \int_{-1}^{+1}xp_{s}(x)p_{r}(x)dx=\delta_{r,s+1}\frac{2r}{(2r+1)(2r-1)}+\delta_{s,r+1}\frac{2s}{(2s+1)(2s-1)}. $
Please make an illustration to me in proving of the following.
Problem: Assume that
$ p_{n}(x)$ is a Legendre polynomials.
I want to prove that
$$ \int_{-1}^{+1}xp_{s}(x)p_{r}(x)dx=\delta_{r,...
1
vote
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 $...
0
votes
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?
0
votes
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
votes
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
votes
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 ...
0
votes
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
votes
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 ...
