Questions tagged [orthogonal-polynomials]
Questions pertaining to certain sets of polynomials that satisfy an orthogonality criterion with respect to some specified inner product.
3
votes
1answer
27 views
Associated Laguerre polynomials of half-integer parameters
It appears that associated Laguerre polynomials $L_n^{(\alpha)}(x^2)$ for half integer parameters in $n$ and $\alpha$ may be expressed in terms of an exponential function, an imaginary error function ...
1
vote
2answers
32 views
Integral involving a generalized Laguerre Polynomial
I want to evaluate the following integral:
$$ \int_0^\infty z^{1/2}e^{-a\space z}L_{m}^{1/2}(z)\space dz, $$
where $L_{m}^{1/2}$ is a generalized Laguerre polynomial.
I found a certain indefinite ...
0
votes
0answers
43 views
Density of polynomials in $L^2$ on the unit ball
Are polynomials dense in the Lebesgue space $L^2$ of the unit disk?
References are very welcome, thank you.
1
vote
1answer
17 views
Demonstration problem on orthogonal polynomials
Consider the set of polynomials {${\phi_n}$}built using the Gramm-Schmidt method with respect to the inner product:
$$(f,g) = \int_{a}^{b} f(x)g(x)w(x)dx$$
with $w(x)>0$.
Prove that if $x_0$ is ...
0
votes
0answers
38 views
Integrals involving Legendre polynomials and exponentials
I found the following integrals in the article here:
$$I_1(x,\lambda)=\int_{-1}^{1}P_n(x')\,\mathrm{e}^{-(x-x')\,\lambda}\,\mathrm{d}x=\mathrm{e}^{-\lambda\,x}\sum_{k=0}^n\frac{(n+k)!\,(-1)^n}{\...
1
vote
0answers
11 views
Development of a 2-dimensional function on $[0,\infty)\times[0,2\pi]$ in a complete set
Actually I would like to develop function $g(r,\phi)$ on $[0,\infty)\times[0,2\pi]$ in a complete set of functions. This does not seem to very difficult:
$$g(r,\phi) = \sum_{m=-\infty}^{\infty} R_m(...
1
vote
0answers
37 views
Szegő's method of finding the generating function of the Jacobi polynomials
In Orthogonal Polynomials (4th ed., Amer. Math. Soc. Colloq. Publ., vol. 23, 1975), Szegő starts off section 4.4 by giving the following integral representation of the Jacobi polynomials:
$$P_n^{(\...
0
votes
0answers
28 views
Are there degrees or categories of orthogonality?
The vectors $\begin{bmatrix}a \\ 0 \\ 0\end{bmatrix}$, $\begin{bmatrix}0 \\ b \\ 0\end{bmatrix}$, and $\begin{bmatrix}0 \\ 0 \\ c\end{bmatrix}$ are orthogonal under the dot product.
The vectors $1$, $...
2
votes
0answers
34 views
Jacobi polynomials and Gram determinants
On page 294, Andrews, Askey and Roy - Special functions. For sequences of (independent) functions $\lbrace \phi(x) \rbrace_{n=0}^{\infty}$ and $\lbrace \psi(x) \rbrace_{n=0}^{\infty}$, a sequence $\...
1
vote
1answer
24 views
Do orthogonal polynomials determine the moments of their orthogonality measure?
I am currently learning about the inverse problem for orthogonal polynomials for orthogonality measures supported on the real line.
My question is not about finding the orthogonality measure from the ...
1
vote
0answers
45 views
Chebyshev Polynomials of the Second Kind from Orthogonality
I am tasked with finding the degree 5 Chebyshev-II polynomial, using the fact that it's orthogonal to those preceding it w.r.t the Chebyshev-II inner product. I am told to use the normalisation that ...
0
votes
1answer
23 views
Orthogonal polynomials in the complex domain
A source tells me that a set $P_n: \mathbb{C} \to \mathbb{C}$ of polynomials are orthogonal over a contour if
$$
\frac{1}{2\pi} \int_{\Gamma} P_n(z) \overline{P_m(z)}h(z)d|z|=\delta_{nm},
$$
where h ...
1
vote
0answers
34 views
Indefinite integral involving the product of two generalized Laguerre polynomials
I am trying to find the indefinite integral
\begin{align}
\int{x^{\alpha +1}e^{-x}\left(L_{m}^{\alpha}(x)\right)^{2}dx}
\end{align}
where $L_{m}^{\alpha}(x)$ is the generalized Laguerre Polynomial, ...
0
votes
0answers
16 views
Completeness relation for Jacobi Polynomials
I was wondering if there exists a completeness relation for Jacobi Polynomials, $P^{\alpha,\beta}_{n}(x)$ as in the case of Hermite polynomials, $H_{n}(x)$ such that
$$
\sum^{\infty}_{n=0} \psi_n(x) \...
3
votes
1answer
42 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
57 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 ...
1
vote
1answer
44 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.,
$$w(x) = x \, e^{\frac{-...
0
votes
0answers
33 views
Finding $\int_{0}^{1} \frac{T_{i}^{*''}(x) T_{j}^{*}(x)}{\sqrt{x-x^2}} dx$
Shifted Chebyshev polynomials $$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{...
2
votes
0answers
34 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
26 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
19 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 :
$$ P_n = \sum_{k=0}^\infty w_{n,k}x^...
0
votes
0answers
29 views
Derive 3-term recursion of polynomials from some recursion OR are my polynomials orthogonal
considering a sequence of polynomials $(Q_n)$ on $\mathbb{R}$ given by $$(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$$ where all ...
0
votes
0answers
68 views
Sum and Integration with Legendre Polynomial
What is the value of the following infinite sum after integrating the product of two Legendre Polynomials $P_m^0,~P_n^1$, $$\sum_{n=1}^{\infty}\sum_{m=0}^{\infty} \frac{1}{\sqrt{n(n+1)}} A_m\, A_n \...
1
vote
2answers
108 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:
$$P_{2}(x)=x^2-1;P_{3}(x)=x^3-2x;P_{4}(x)=x^4-3x^2+...
1
vote
1answer
36 views
On the construction of orthogonal polynomials
In the following proof, argument goes on based on considering $C_n$ to be nonzero then it finishes the proof for $C_n=0$ :
Also if we set $C_n=0$ in Eq. (6.10) then must $m=0,1,2,...,n- 2$ in Eq. (6....
1
vote
1answer
54 views
Can I form a complete set of functions using $e^{-nx}$?
If I start with the set of functions $e^{-nx}$ for all integers $n>1$, can I use them as basis to create a complete set of orthogonal functions on the interval $(0,+\infty)$? By complete I mean ...
0
votes
1answer
31 views
Let $S$ be the subspace of $\Bbb R^3$ spanned by the vector $x= (x_1,x_2,x_3)^T$ and $y= (y_1,y_2,y_3)^T$
(a)
Let $S$ be the subspace of $\Bbb R^3$ Spanned by the vector $x= (x_1,x_2,x_3)^T$ and $y= (y_1,y_2,y_3)^T$, let $A =\begin{bmatrix}x_1 &x_2 & x_3 \\
y_1 &y_2 &...
4
votes
1answer
59 views
Can we refine this asymptotic for Laguerre polynomials?
I just found an interesting and useful limit for Laguerre polynomials:
$$\lim_{n \to \infty} L_n \left( \frac{2r}{n+1/2} \right)=J_0(2 \sqrt{2r})$$
I'm using specifically this form of the argument ...
1
vote
0answers
20 views
Orthogonal polynomials on triangles: Connection with quadrature rules?
In a 1D interval $[a, b]$, quadrature rules and orthogonal polynomials are tightly interconnected. For example, given the $n$ roots $t_j$ of the $n$th orthogonal polynomial $p_n$, one has the ...
1
vote
0answers
27 views
Evaluating an integral using Gegenbauer polynomials
I want to evaluate the following integral
$$\int \frac{(r-r'\cos\theta')^2r'^2\,dr'\sin\theta'\,d\theta'\,d\phi'}{(r^2+r'^2-2rr'\cos\theta')^{3/2}}$$
Working that a little bit i end up with this ...
0
votes
0answers
24 views
Orthogonality of generalized Newton symbol
Consider the functions $P_{n}(x)={x \choose n}.$ My question is, if there exists a measure $\mu$ with support being a subset of $(0,\infty)$ such that the family $\{P_{n}\}$ is orthogonal in $L^{2}(\...
0
votes
0answers
119 views
Define on $P3$ the inner product $<f,g>=\int_{-1}^1 f(t)g(t)dt$, find orthogonal projection
Define on $P3$ the inner product $\langle f,g \rangle=\int_{-1}^1 f(t)g(t)dt$.
a) find the orthogonal projection of $p(x)=x^3$ onto $P2$
I know the orthogonal projection formula, but how do I solve ...
5
votes
0answers
64 views
What is this generalization of the Chebyshev polynomials?
For $\varepsilon>0$ consider the tridiagonal matrix
$$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
30 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 $S$ $=$ $\...
1
vote
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
290 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
24 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
27 views
The generalized Laguerre polynomials: Are there any expressions valid for any case?
There are general expressions of the generalized Laguerre polynomials.
For example:
$$
L_n^{(\alpha)}(x) = {}_{n+\alpha}\mathrm{C}_n\ {}_1F_1(-n, \alpha+1, x),
\hspace{20pt}(1)
$$
$$
L_n^{(\alpha)}(x)...
0
votes
1answer
37 views
Use Gram-Schmidt process on Chebychev polynomials
I need help using the Gram-Schmidt orthogonalization process to derive the first four orthonormal Chebychev polynomials. Using the range $[-1,1]$ and the weight function $w(x)=(1-x^2)^\frac{1}{2}$.
...
2
votes
1answer
107 views
Orthogonal polynomials with respect to discrete probability distributions
The polynomials from the Askey scheme are orthogonal with respect to a standard probability distribution. For example, Hermite polynomials (in a suitable form) are orthogonal with respect to a ...
0
votes
0answers
10 views
Polynomials orthogonal wrt Rayleigh distribution
As per the title, have the class of polynomials that are orthogonal with respect to Rayleigh distribution been studied? Do they have a name?
4
votes
1answer
63 views
Algorithm for orthogonalizing polynomials with specific inner product
I am attempting to generate a as big as possible collection of orthogonal polynomials $p_1, p_2, ..., p_n$, $\left\langle p_i, q_i\right \rangle = \delta_{ij}$ where the inner product is with respect ...
3
votes
0answers
44 views
$L_2$ scalar product between Hermite polynomials
I am trying to compute the $L_2$ scalar product between (probabilists’) Hermite polynomials (defined as in Wiki) with Gaussian weight and different scales, i.e. for some constants $c, d$:
$$\frac{1}{\...
0
votes
0answers
27 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
142 views
Weird Integral Involving Hermite Polynomials
I have stumbled upon the following integral involving the Hermite polynomials:
$$ I(m) = \int_\mathbb{R} e^{i m x} \left[ e^{-\frac{x^2}{2}} H_m(x) \right] dx \, , \quad m \in \mathbb{N} \cup \{0\} \...
2
votes
1answer
79 views
An Addition formula for Hermite polynomials
My question concerns an addition formula that can be found on the Wikipedia page of Hermite Polynomials but I can't find it anywhere else. The well-known formula that can be found in many books is the ...
1
vote
1answer
45 views
Sum involving Hermite polynomials
I am wondering if there is a simpler form of the following summation involving the (physicists') Hermite polynomials:
$$\sum_{k=0}^{n}\frac{H_k(x)}{(2i)^k},$$
where $i=\sqrt{-1}$ is the imaginary ...
3
votes
1answer
152 views
Find the Fourier-Bessel Series for $f(x)$ With Respect to the Orthogonal Set: How Was $w(x)$ Found?
I have the following problem:
If $f(x) = x$, $0 < x < 2$, find the Fourier-Bessel series for $f(x)$ with respect to the orthogonal set $\{ J_1 (k_n x) \}$, where $k_n$ is the $n$th positive ...
4
votes
0answers
81 views
Integral of a triple product of Laguerre polynomials
I would like to know if there's an exact expression for this integral in terms of known elementary or special functions:
$$\int_0^\infty \exp \left(-\frac{a+b+c}{2}x \right) L_j (a x) L_k (b x) ...
