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
1 vote
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Polynomial filter on tri-diagonal 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 ...
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2 votes
0 answers
70 views
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 ...
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4 votes
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272 views
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 $...
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0 votes
2 answers
200 views
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
5 votes
0 answers
99 views
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
0 votes
0 answers
82 views
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 ...
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3 votes
0 answers
198 views
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 $$_1F_1(a;c;z) = \sum_{n=0}^{\infty} \frac{(a)_n}{(c)_n n!} z^...
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3 votes
0 answers
149 views
Linearization coefficients for Jacobi polynomials
In general, for families of polynomials $\{ Q_n\}, \{ R_n\},\{S_n\}$, there exist linearization coefficients such that one may write the product $Q_m R_n = \sum_k c_{m,n}^k S_k$. Let $P^{\alpha,\beta}...
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3 votes
0 answers
156 views
Deeper reason for why classical orthogonal polynomials have simple generating functions?
Is there a known reason why all classical families of orthogonal polynomials have simple generating functions? I was wondering whether one could get an explanation using the connection with Sturm-...
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0 votes
1 answer
202 views
How to efficiently replace the Paterson-Stockmeyer algorithm and reduce non-scalar multiplications
Paterson-Stockmeyer algorithm If we need to compute a high-degree polynomial expression, such as: $$ P(y) = \sum_{k=0}^{B} a_k y^k $$ the Paterson-Stockmeyer algorithm can process the powers in ...
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1 vote
0 answers
98 views
Asymptotically small submatrices of random matrices
Consider an ensemble of $N \times N$ random Hermitian matrices distributed according to some unitarily invariant measure $$P(M) \mathrm{d}M = \frac{1}{Z_{N}} e^{-\mathrm{tr}[ Q(M)]}\mathrm{d}M,$$ for ...
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1 vote
1 answer
194 views
Closed formula / asymptotics for a generating function involving Gegenbauer / ultraspherical polynomials
Are there asymptotics, or even a closed form, for the following series $$ \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
6 votes
1 answer
341 views
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 ...
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3 votes
2 answers
436 views
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_\...
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1 vote
0 answers
110 views
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= \...
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2 votes
0 answers
98 views
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 $X(h)$, $$E[X(h)X(g)] = \...
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2 votes
0 answers
104 views
The $n$-th reproducing kernel of orthogonal polynomial
Let $N$ be a non negative integer. Define the sequence of monic orthogonal polynomials $\{P_n(x)\}_{n}$ with respect to the inner product $$ \langle f , g\rangle =\sum^{N}_{k=0}{f(k)g(k)\rho(k)} $$ ...
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1 vote
2 answers
492 views
A closed formula for a sum involving hypergeometric function
Let ${ }_1 F_1(a ; c ; z)$ be Kummer's function defined by the function, and all its analytic continuations, represented by the infinite series $\sum_{n=0}^{\infty} \frac{(a)_n}{(c)_n} \frac{z^n}{n !...
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1 vote
1 answer
179 views
Orthogonal vectors translation using standard vectors
When $n=2m$, let us consider the following vectors $\mathbf{v}_1,\ldots, \mathbf{v}_n$ in $\mathbb{R}^n$ $$\mathbf{v}_q=(v_{1q},\ldots,v_{n,q})$$ $$v_{p,q}=\sin\Big(\frac{pq}{n+1}\pi\Big)$$ It is ...
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2 votes
1 answer
286 views
Laguerre polynomial and series
Let $L^\alpha_n(x)$ be Laguerre polynomials of type $n$. Consider the sum $$\sum^\infty_{j=0} \frac{1}{(b-j)}L^{m}_j(x)$$ where $b\not\in\Bbb N,x>0$ and $m\in \Bbb N$. I have found this series ...
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0 votes
0 answers
145 views
Closed formula for Laguerre
Let $L^\alpha_n(x)$ be Laguerre polynomials of type $n$. Assume $0<\beta<1$. Is there a closed formula for this sum $$\sum^\infty_{j=0} \frac{1}{(b+j)^{1-\beta}}L^{m}_j(x)$$ where $b>0$ and $...
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0 votes
1 answer
138 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 ...
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1 vote
0 answers
206 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 ...
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1 vote
0 answers
237 views
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 ...
- 51
6 votes
1 answer
294 views
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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0 votes
1 answer
151 views
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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0 votes
0 answers
156 views
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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0 votes
0 answers
134 views
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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3 votes
0 answers
88 views
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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1 vote
2 answers
630 views
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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69 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 ...
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6 votes
0 answers
212 views
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 votes
0 answers
175 views
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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1 vote
1 answer
454 views
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
210 views
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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2 votes
0 answers
104 views
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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0 votes
0 answers
102 views
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" ...
2 votes
0 answers
144 views
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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3 votes
0 answers
138 views
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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1 vote
0 answers
104 views
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 votes
1 answer
172 views
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
214 views
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": $$ \...
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4 votes
3 answers
529 views
Infinite sum of Laguerre polynomials: is $\sum_{k=0}^{\infty}\frac{n!}{(k+n)!}x^kL_n^k(x)^2=e^x+P(x)$, with $P$ a polynomial of degree $2n-1$?
I have the following expression: $$ \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!} $$ is the usual associated Laguerre ...
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5 votes
1 answer
447 views
Characteristic polynomial of a simple matrix: Chebyshev?
In my recent MO question, Darij Grinberg mentioned a closely related (structure-wise) determinant, that is, $$\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}).$$ ...
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1 vote
3 answers
410 views
Generating function of the square of Jacobi polynomial
The generating function of the Jacobi polynomials is given by $$ \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)=\...
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3 votes
1 answer
682 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
100 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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5 votes
1 answer
599 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 ...
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1 vote
1 answer
242 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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