Questions tagged [orthogonal-polynomials]
A family of orthogonal polynomials is a sequence of polynomials in one variable, one in each degree, such that any two of them are orthogonal with respect to some fixed scalar product on the space of polynomials. They are closely related to continued fractions and useful in harmonic analysis. There are many different families of orthogonal polynomials, among which one can cite Hermite polynomials, Laguerre polynomials, and Jacobi polynomials.
231 questions
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Polynomial filter on symmetric tridiagonal matrices
During my research, I encountered a problem involving symmetric tridiagonal matrices and their eigenvectors, which I have been attempting to solve since then. So far, I have only managed to obtain a ...
- 61
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80
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Orthogonal feature functions for model fitting on unbounded intervals (like Chebyshev polynomials for bounded intervals)
This question is rummaging around in my head for quite some time. I will start with exposition on "model fitting" and then explain how Chebyshev polynomials are perfect on bounded intervals ...
- 343
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Roots of derivatives of the polynomial $(x+1)^m(x-1)^m$
My research lead me to the following question:
Given the polynomial $$p(x)=(x+1)^m(x-1)^m,$$ where $m=\frac{nd}{2}$ with $n,d$ being the parameters of the problem, I want to find the largest root of $...
- 61
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2
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224
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Weighted integral of product of Laguerre polynomials
I'm trying to compute the following integral
\begin{equation}
\int_0^\infty dx \, x^{2\Delta}e^{-x} \left(L_n^{2\Delta-1}(x)\right)^2,
\end{equation}
where $L_n^{2\Delta-1}(x)$ is the generalized ...
- 19
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Why do orthogonal bases of zonal functions on projective spaces transform according to the Jacobi polynomials?
My reference for this question is Section 2 of Riesz and Green energy on projective spaces, Anderson et al, 2022. The exposition of this paper is pretty self-contained, except for the content of my ...
- 51
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0
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83
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Properties/roots “almost” Chebyshev polynomials
Does anyone know if the polynomials, defined by the recurrence
$P_n(X)=2XP_{n-1}(X)-\alpha(2X+\beta)P_{n-2}(X)$
with $\alpha$ and $\beta$ given real numbers are known and already analysed? In ...
- 11
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200
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A closed formula for a sum involving hypergeometric functions
Can we find a closed formula for this sum:
$$\sum_{p,q\geq 0} (p+q+1)r^{p+q} \frac{{}_1F_1(1+p;2+p+q;r^2)}{{}_1F_1(1+p;2+p+q;1)}$$
where
$$_1F_1(a;c;z) = \sum_{n=0}^{\infty} \frac{(a)_n}{(c)_n n!} z^...
- 571
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152
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Linearization coefficients for Jacobi polynomials
In general, for families of polynomials $\{ Q_n\}, \{ R_n\},\{S_n\}$, there exist linearization coefficients such that one may write the product $Q_m R_n = \sum_k c_{m,n}^k S_k$.
Let $P^{\alpha,\beta}...
- 177
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158
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Deeper reason for why classical orthogonal polynomials have simple generating functions?
Is there a known reason why all classical families of orthogonal polynomials have simple generating functions? I was wondering whether one could get an explanation using the connection with Sturm-...
- 464
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How to efficiently replace the Paterson-Stockmeyer algorithm and reduce non-scalar multiplications
Paterson-Stockmeyer algorithm
If we need to compute a high-degree polynomial expression, such as:
$$
P(y) = \sum_{k=0}^{B} a_k y^k
$$
the Paterson-Stockmeyer algorithm can process the powers in ...
- 27
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vote
0
answers
98
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Asymptotically small submatrices of random matrices
Consider an ensemble of $N \times N$ random Hermitian matrices distributed according to some unitarily invariant measure
$$P(M) \mathrm{d}M = \frac{1}{Z_{N}} e^{-\mathrm{tr}[ Q(M)]}\mathrm{d}M,$$
for ...
- 131
1
vote
1
answer
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Closed formula / asymptotics for a generating function involving Gegenbauer / ultraspherical polynomials
Are there asymptotics, or even a closed form, for the following series
$$ \sum_{k = 0}^\infty e^{2 \pi i \sqrt{k^2 + (d-1) k} } \left( \binom{d+k}{k} - \binom{d+k-2}{k-2} \right) G_{\frac{d-1}{2},k}(t)...
- 507
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answer
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Can the Chebyshev polynomials be constructed from the extremal property?
It is a known fact that the Chebyshev polynomials, defined as $T_n := \cos(n \arccos x)$, have the following extremal property:
Theorem: Of all monic degree $n$ polynomials, $\frac1{2^{n-1}} T_n$ has ...
- 63
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2
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Question about the Bessel operator
For $\nu>-1$ denote by $\{\lambda_{k,\nu}\}_{k\in\mathbb{N}}$ the succesive positive zeros of the Bessel function of the first kind $J_{\nu}$. The Bessel operator is given by
\begin{equation*}
L_\...
- 421
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0
answers
115
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Product of d-dimensional Legendre polynomials
Let $P_n:\mathbb{R}\rightarrow\mathbb{R}$ be the $d$-dimensional Legendre polynomials, that is they are orthonormal polynomials w.r.t the probability measure $\mu_d$ on $[-1,1]$ given by $\mu_d= \...
- 155
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99
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Iterated chaos expansion
Using the notations from Normal Approximations with Malliavin Calculus, Chapter 2
random variables $F$ in the probability space generated by an iso-normal Gaussian family $X(h)$,
$$E[X(h)X(g)] = \...
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The $n$-th reproducing kernel of orthogonal polynomial
Let $N$ be a non negative integer. Define the sequence of monic orthogonal polynomials $\{P_n(x)\}_{n}$ with respect to the inner product
$$
\langle f , g\rangle =\sum^{N}_{k=0}{f(k)g(k)\rho(k)}
$$
...
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vote
2
answers
522
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A closed formula for a sum involving hypergeometric function
Let ${ }_1 F_1(a ; c ; z)$ be Kummer's function defined by the function, and all its analytic continuations, represented by the infinite series $\sum_{n=0}^{\infty} \frac{(a)_n}{(c)_n} \frac{z^n}{n !...
- 571
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1
answer
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Orthogonal vectors translation using standard vectors
When $n=2m$, let us consider the following vectors $\mathbf{v}_1,\ldots, \mathbf{v}_n$ in $\mathbb{R}^n$
$$\mathbf{v}_q=(v_{1q},\ldots,v_{n,q})$$
$$v_{p,q}=\sin\Big(\frac{pq}{n+1}\pi\Big)$$
It is ...
- 4,150
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answer
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Laguerre polynomial and series
Let $L^\alpha_n(x)$ be Laguerre polynomials of type $n$.
Consider the sum
$$\sum^\infty_{j=0} \frac{1}{(b-j)}L^{m}_j(x)$$
where $b\not\in\Bbb N,x>0$ and $m\in \Bbb N$.
I have found this series ...
- 571
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0
answers
146
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Closed formula for Laguerre
Let $L^\alpha_n(x)$ be Laguerre polynomials of type $n$.
Assume $0<\beta<1$. Is there a closed formula for this sum
$$\sum^\infty_{j=0} \frac{1}{(b+j)^{1-\beta}}L^{m}_j(x)$$
where $b>0$ and $...
- 571
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votes
1
answer
139
views
Verify directly that $\{p_n\}$ are the orthogonal polynomials
I have no idea about an exercise in the book by Percy Deift.
Let $\mu$ be a given positive Borel measure with bounded or unbounded support on $\mathbb{R}$. If the support is unbounded, it requires ...
- 179
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vote
0
answers
212
views
Algorithm for converting from 2D Legendre basis to 2D Monomial basis
I am dealing with function written in a 2D Legendre polynomial basis and I need to convert it so that it's written in a 2D monomial basis. I've found of an algorithm that allows for change of basis ...
- 111
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0
answers
238
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Proving that quotient of orthogonal polynomials is a Padé approximant of Stieltjes transform
This question is reposted from Math Stack Exchange (you can see the original post here). The motivation for reposting is that I feel like the question isn't getting much attention in MSE - if there is ...
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Real zeroes of the determinant of a tridiagonal matrix
Let $\epsilon_1,\ldots,\epsilon_n$ be a sequence of signs and $M(t)$ be the tridiagonal matrix whose diagonal entries are $\epsilon_1 t,\ldots, \epsilon_n t$ and off-diagonal entries equal to $1$. Is ...
- 1,351
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151
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Relatively explicit orthogonal systems on the sphere that are not spherical harmonics
I am looking for references studying orthonormal systems of functions $\{h_n\}_{n\geq0}$ on a sphere $S^d$ ($d=2$ or $d\geq2$) with respect to weights that are not uniform (unlike spherical harmonics)....
- 313
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Equilibrium position of $ n $ free charges as polynomials roots
I asked the same question on here but received no answer.
The classic problem of the electrostatic equilibrium positions of a linear system of $ n $ free unit charges between two fixed charges is well ...
- 557
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Applications of Jack polynomials
I developed four libraries (Julia, R, Python, Haskell) for the computation of Jack polynomials. I developed them for fun because I found this was programmatically interesting. But now I'd like them to ...
- 2,590
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answers
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Closed formula for $\sum\limits^\infty_{k=0}\frac1{(k+a)(k+b)} L^1_k(x)L^1_k(y) $
Let $ L^{\alpha}_{n}(x)=\sum^{n}_{k=0} \binom{n+\alpha}{n-k}\big(-1\big)^{k}\frac{x^{k}}{k!},\alpha>-1$ be Laguerre polynomials of type $ n$.
Is there a closed formula for $$\sum^{\infty}_{k=0}\...
- 571
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vote
2
answers
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Closed formula for Hermite polynomials
Hermite polynomials $H_k(x), x \in \mathbb{R}, k \in \mathbb{N}$ are defined by the formula
$$
H_k(x)=(-1)^k e^{x^2} \frac{d^k}{d x^k}\left(e^{-x^2}\right) .
$$
Each $H_k(x)$ is a polynomial of exact ...
- 571
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votes
2
answers
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To prove irrationality, why integrate?
I have been reading David Angell's lovely book, Irrationality and Transcendence in Number Theory, which has given me some fresh insights even with some of the easier proofs. But the book reminds me of ...
- 88.5k
6
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3-term recurrence relation including integral or differential operator for polynomials
Sequences of polynomials with a 3-term recurrence relations are well known for orthogonal polynomials. Do recurrence relations using differential or integral operators also appear in some theories?
I ...
3
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0
answers
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Chebyshev-like polynomials [closed]
In some approximation problems I'm working on, the errors turned out to be polynomials of various degrees whose graphs on the interval $[-1,1]$ look like this:
As you can see, these things look a bit ...
- 659
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vote
1
answer
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Orthogonal polynomials w.r.t. an arbitrary measure
Consider a random scalar variable $X$ with arbitrary measure.
I'm after a basis of polynomial functions $\{p_k\}_{k=0}^\infty$ which are orthonormal with respect to $X$ in the sense that
\begin{...
- 463
1
vote
1
answer
211
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Orthogonal functions on circle with constraints
I have a curious question I stumbled upon that may be interesting to some.
Consider real-valued continuous functions on the circle $f_1(x),f_2(x),f_3(x)$ (so they are periodic in $x \mapsto x+2\pi$).
...
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measure corresponding to certain orthogonal polynomials
Given a scalar $\lambda\in (0,1)$, consider a sequence of monic polynomials $\{p_n(x)\}_{n\geq 0}$ over real variables satisfying the following recurrence relations:
$xp_n(x)=p_{n+1}(x)+\lambda p_{n}(...
- 185
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Asymptotic behavior of the square Generalized Laguerre polynomial
The asymptotic begavior of the Generalized Laguerre polynomial is given in the Book " Formulas and theorems in the special functions of mathematical physics. Berlin: Springer-Verlag; 1966" ...
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Explicit error bounds for orthogonal polynomials with exponential weights
Let $\rho > -1$, and define the weight function $W_{\rho}(x) = |x|^{\rho} \exp(-2|x|)$. Associated with this weight is the sequence of orthogonal polynomials $\{ p_{n}(x) \}_{0}^{\infty}$, where $...
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Tight bounds on the iterates of the Clenshaw algorithm for Chebyshev polynomials
I'm trying to bound the iterates of the Clenshaw algorithm when applied to the Chebyshev series, related to a question I'm running into related to the stability of this algorithm. Recall that, for $p(...
- 101
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0
answers
104
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Geometric series involving the Laguerre polynomials
Let put $\alpha=5$ and $x=3$. Consider the following set given by
$$M=\lbrace \; n \in N, \; \; 0 < |L_{n}^{5}(3)| < 1 \; \rbrace$$
Where $L_{n}^{\alpha}(x)$ is the generalized Laguerre ...
3
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1
answer
176
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Relation between the local maxima and the local minima for approximating the generalized Laguerre polynomial
I have already asked my question in the link below:
Minima approximation for Laguerre polynomials
I have suggested to anyone to give me the approximations of the minima for the Laguerre polynomial, ...
1
vote
0
answers
224
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Generating Hermite polynomial with coefficient recurrance relation algorithm
I am writing a math paper for my numerical analysis class about orthogonal Hermite polynomials. I want to implement the algorithm for generating the "probabilist's Hermite polynomials":
$$
\...
- 11
4
votes
3
answers
542
views
Infinite sum of Laguerre polynomials: is $\sum_{k=0}^{\infty}\frac{n!}{(k+n)!}x^kL_n^k(x)^2=e^x+P(x)$, with $P$ a polynomial of degree $2n-1$?
I have the following expression:
$$
\sum_{k=0}^{\infty}\frac{n!}{(k+n)!}x^k(L_n^k(x))^2,
$$
where
$$
L_n^k(x)=\sum_{j=0}^n(-1)^j\binom{n+k}{n-j}\frac{x^j}{j!}
$$
is the usual associated Laguerre ...
- 83
5
votes
1
answer
453
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Characteristic polynomial of a simple matrix: Chebyshev?
In my recent MO question, Darij Grinberg mentioned a closely related (structure-wise) determinant, that is,
$$\det\left(x_{\min\{i,j\}}\right)_{i,j}^{1,m}=x_1(x_2-x_1)(x_3-x_2)\cdots(x_m-x_{m-1}).$$
...
- 43.9k
1
vote
3
answers
418
views
Generating function of the square of Jacobi polynomial
The generating function of the Jacobi polynomials is given by
$$
\sum_{n=0}^{\infty} P_{n}^{(\alpha, \beta)}(z) t^{n}=2^{\alpha+\beta} R^{-1}(1-t+R)^{-\alpha}(1+t+R)^{-\beta}
$$
where
$$
R=R(z, t)=\...
- 43
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votes
1
answer
703
views
Generating function of the product of Legendre polynomials
The generating function of the product of Legendre polynomials for the same $n$ is given by
\begin{aligned}
\sum_{n=0}^{\infty} z^{n} \mathrm{P}_{n}(\cos \alpha) \mathrm{P}_{n}(\cos \beta)&=\frac{\...
- 43
12
votes
1
answer
1k
views
How are Sheffer polynomials related to Lie theory?
Sheffer polynomials $\{P_n(x)\}$ have generating function $P(x,t) = \sum_{n=0}^{\infty}P_n(x)t^n=A(t)e^{xu(t)}$.
This form reminds me of the Lie group–Lie algebra correspondence. Is there any ...
2
votes
1
answer
101
views
Two variable polynomials that behave like Lagrange polynomials [closed]
Let us consider different points $z_i=(x_i,y_i)$ in the plane where $i=1,\cdots n$.
Q Do there exist two variable polynomials $P_i(x,y)$ with minimal degree such that $P_i(z_j)=\delta_{ij}$?
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votes
1
answer
603
views
Riemann-Hilbert approach to Selberg integral
I am interested in matrix integrals, and I have seen many mentions to a certain Riemann-Hilbert approach that indicate that this is a very powerful tool to can be used in this area, when coupled with ...
- 2,572
1
vote
1
answer
248
views
Recursive formula from given explicit formula for normalized Chebyshev polynomials
The Chebyshev polynomials $(T_k)_{k \in \mathbb{N}_0}$ are defined recursively by
$$
T_0(x)=1 , \ \ T_1(x)=x, \ \ T_{k+1}(x)=2x\,T_k(x)-T_{k-1}(x) \ .
$$
With this one can find the explicit formulas
\...
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