Questions tagged [orthogonal-polynomials]
A familly 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.
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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 ...
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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 $...
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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 ...
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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 ...
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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 ...
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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)....
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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 ...
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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 ...
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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 $$\sum^{\infty}_{k=0}\...
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Closed formula for Hermite polynomials
Hermite polynomials $H_k(x), x \in \mathbb{R}, k \in \mathbb{N}$ are defined by the formula
$$
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 ...
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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 ...
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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 ...
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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 ...
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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{...
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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}(...
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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(...
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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$$
Where $L_{n}^{\alpha}(x)$ is the generalized Laguerre ...
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Relation between the local maxima and the local minima for approximating the generalized Laguerre polynomial
I have already asked my question in the link below:
Minima approximation for Laguerre polynomials
I have suggested to anyone to give me the approximations of the minima for the Laguerre polynomial, ...
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Generating Hermite polynomial with coefficient recurrance relation algorithm
I am writing a math paper for my numerical analysis class about orthogonal Hermite polynomials. I want to implement the algorithm for generating the "probabilist's Hermite polynomials":
$$
\...
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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 ...
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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,
$$\det\left(x_{\min\{i,j\}}\right)_{i,j}^{1,m}=x_1(x_2-x_1)(x_3-x_2)\cdots(x_m-x_{m-1}).$$
...
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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
$$
R=R(z, t)=\...
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Generating function of the product of Legendre polynomials
The generating function of the product of Legendre polynomials for the same $n$ is given by
\begin{aligned}
\sum_{n=0}^{\infty} z^{n} \mathrm{P}_{n}(\cos \alpha) \mathrm{P}_{n}(\cos \beta)&=\frac{\...
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How are Sheffer polynomials related to Lie theory?
Sheffer polynomials $\{P_n(x)\}$ have generating function $P(x,t) = \sum_{n=0}^{\infty}P_n(x)t^n=A(t)e^{xu(t)}$.
This form reminds me of the Lie group–Lie algebra correspondence. Is there any ...
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Two variable polynomials that behave like Lagrange polynomials [closed]
Let us consider different points $z_i=(x_i,y_i)$ in the plane where $i=1,\cdots n$.
Q Do there exist two variable polynomials $P_i(x,y)$ with minimal degree such that $P_i(z_j)=\delta_{ij}$?
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Riemann-Hilbert approach to Selberg integral
I am interested in matrix integrals, and I have seen many mentions to a certain Riemann-Hilbert approach that indicate that this is a very powerful tool to can be used in this area, when coupled with ...
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Recursive formula from given explicit formula for normalized Chebyshev polynomials
The Chebyshev polynomials $(T_k)_{k \in \mathbb{N}_0}$ are defined recursively by
$$
T_0(x)=1 , \ \ T_1(x)=x, \ \ T_{k+1}(x)=2x\,T_k(x)-T_{k-1}(x) \ .
$$
With this one can find the explicit formulas
\...
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Do you recognize this sequence of polynomials?
In teaching my linear algebra students about Gram-Schmidt orthogonalization, I found a curious sequence of polynomials. They are closely related to Legendre polynomials, but they also appear to be ...
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A special class of weighted Motzkin paths
Consider Motzkin paths with the following weight:
All up-steps and the horizontal steps on height $0$ have weight $1$, all down-steps have weight $t$ and the horizontal steps on even heights have ...
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Defining Legendre polynomials in terms of a sinusoidal function for $|x| \leq 1$
Would it be possible to define Legendre polynomials in terms of a sinusoidal function for $|x|\leq 1$ in a similar manner to Chebyshev polynomials being defined as $T_n(x) = \cos(n \cos^{-1}(x))$?
...
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Finite version of Mehlers formula?
This is a crosspost from Math Stack Exchange, please let me know if this is not an appropriate use of crossposting, and I will delete.
Mehler's formula is the following identity for Hermite ...
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A holonomic function and its singularity
The following series where $q_i , h$ are constant parameters. $G(z)$ is a rational function.
$$F(x):=\sum_{d= 1}^\infty \sum_{k=1}^d (-1)^{d-k} \, s_{(k, 1^{d-k})}(\tfrac{q_1}{h}, \tfrac{q_2}{h}, \...
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Laplace transform and Laguerre Polynomials
What is the kernel $K(t)$ of the following Laplace transform equation:
$$\int_{0}^{+\infty}e^{-(x+y)t} K(t) dt= \sum_{n=0}^{\infty}\varphi_{n}^{\alpha}(x)\varphi_{n}^{\alpha}(y),$$
where $\varphi_{n}^{...
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Bounds on coefficients $c_i$ of Chebyshev expansion $f(x) = \sum_{k=0}^{n} c_kT_k(x) : [-1,1] \mapsto [-1,1]$
Let $n$ be a given positive integer and let $f(x) = \sum_{k=0}^{n} c_kT_k(x)$, where $c_i \in \mathbb{R}$, $0 \leq i \leq n$. If
$|f(x)| \leq 1$, for $|x| \leq 1$, is it possible to get the maximum ...
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Deduce Sheffer's classification of orthogonal polynomials of A-type 0
Theorem 1.9 in Daniel Galiffa and Tanya Riston's paper, An elementary approach to characterizing Sheffer A-type 0 orthogonal polynomial sequences, 2015, presents without proof Isador Sheffer's ...
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Are the “generalized Catalan numbers” of Dumitrescu–Mulase the "moments" of some "multivariate Wigner semicircle distribution"?
The classical Catalan numbers
$$ C_n = \frac{1}{n+1} \binom{2n}{n}, $$
well-known for their numerous combinatorial interpretations (the second volume of Stanley's Enumerative Combinatorics famously ...
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Complex Hermite polynomial orthogonality on weighted space
Consider the "probabilist's" Hermite polynomials given by $$H_n(x)=(-1)^ne^{\frac{x^2}{2}}\partial_x^ne^{-\frac{x^2}{2}}.$$
These polynomials trivially extend to functions of $w\in\mathbb{C}$...
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Do you know of orthogonal-polynomial families with complex measure on the square? I'm just looking for family names to read up on
I'm looking for the name(s) of a family or families of polynomials whose normalization and orthogonality are defined by integrals (inner product) over the complex square $\{u+iv\, |\, u, v \in [-1,1]\}...
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Two-term recurrence relation
We consider the following system of recurrence relations for $n \in \mathbb Z$ and $\vert \lambda \vert=1$ with $\lambda \in \mathbb{C}$
$$a_{n+1} = \lambda a_{n-1}+ \lambda^* a_n + \lambda^* n b_n $$
...
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Proving that the primitives of the Laguerre functions are uniformly bounded
Let $(L_k)_{k\geq 0}$ be the Laguerre polynomials. These polynmials are orthogonal with respect to the inner product: $$\langle f,g\rangle = \int_0^\infty f(x)g(x)\mathrm e^{-x}\,\mathrm dx.$$
Hence, ...
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Expansion of white noise into infinite series using orthogonal polynomials
Having a white random process $s(t)$, is it possible to write $$s(t)=\sum_{i=0}^\infty\alpha_i\phi_i(t)$$ where the $\alpha_i$ are random variables and the $\phi_i$ orthogonal polynomials (Jacobi ...
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321
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Gaussian quadrature, with no exact result over polynomial, but on inverse functions
Generally, a Gaussian quadrature of degree $n$ over an interval $I$ is defined so that it integrates exactly polynomials up to degree $2n - 1$. The main tool are the orthogonal polynomials.
When $I$ ...
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97
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Recursive formula for integral of Chebyshev-type integral
Define
$$
I_{m,n}(x,y,r) = \int_a^b T_m(x + r \sin(\gamma)) T_n(y-r \cos(\gamma)) d\gamma
$$
where $T_m(x)$ are the Chebyshev polynomials of the first kind, and $a$ and $b$ are constants. Assume that ...
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Infinite tridiagonal matrices and a special class of totally positive sequences
Let $\Bbb{y} = \big(y_1, y_2, y_3, \dots \big)$ be an infinite sequence of positive real numbers such that following $\Bbb{N} \times \Bbb{N}$ tridiagonal matrix
\begin{equation}
T(\Bbb{y}) := \,
\...
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Inequality for generalized Laguerre polynomials
Please. Does anybody know a proof of this inequality
$$\Big|\frac{n!\Gamma(\alpha+1)}{\Gamma(n+\alpha+1)} L^{\alpha}_n(x)\Big|\leq e^{\frac{x}{2}}$$ where $\alpha\geq0$ and $x\geq0$ and
$L^{\alpha}_n$ ...
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Is there a bijective proof of an identity enumerating independent sets in cycles?
Let $C_m$ be the cycle with $m$ vertices, defined so that $C_1$ has a self-loop on its unique vertex. Let $p_m$ be the generating function enumerating the number of ways to choose $k$ vertices in $C_m$...
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