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.

Filter by
Sorted by
Tagged with
3 votes
2 answers
210 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_\...
Tony419's user avatar
  • 401
1 vote
0 answers
60 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= \...
giladude's user avatar
  • 155
0 votes
0 answers
55 views

The discrete orthogonal polynomials

I want a document or something that explains the following proposition: The discrete orthogonal polynomials are the polynomial solutions of the given diference equation: $$ \sigma(x)\Delta\nabla P_n(...
Karim's user avatar
  • 21
2 votes
0 answers
60 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)] = \...
Julian's user avatar
  • 613
2 votes
0 answers
62 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 $$ \langle f , g\rangle =\sum^{N}_{k=0}{f(k)g(k)\rho(k)} $$ ...
Karim's user avatar
  • 21
1 vote
2 answers
343 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 !...
zoran  Vicovic's user avatar
0 votes
0 answers
34 views

polynomials defined on a non-uniform and uniform lattice

After reading about the definition of orthogonal dual Hahn polynomials on wiki, you will find the link below. I didn't understand the following sentence : "In mathematics, the dual Hahn ...
Made's user avatar
  • 115
1 vote
1 answer
117 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 ...
ABB's user avatar
  • 3,898
2 votes
1 answer
187 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 ...
zoran  Vicovic's user avatar
0 votes
0 answers
87 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 $...
zoran  Vicovic's user avatar
0 votes
1 answer
99 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 ...
MathRoc's user avatar
  • 159
1 vote
0 answers
108 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 ...
David G.'s user avatar
  • 111
1 vote
0 answers
200 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 ...
Rodrigo's user avatar
  • 51
6 votes
1 answer
223 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 ...
Julien Marché's user avatar
0 votes
1 answer
95 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)....
epsilone's user avatar
  • 313
0 votes
0 answers
118 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 ...
user967210's user avatar
0 votes
0 answers
90 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 ...
Stéphane Laurent's user avatar
3 votes
0 answers
60 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}\...
zoran  Vicovic's user avatar
1 vote
2 answers
294 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 ...
zoran  Vicovic's user avatar
65 votes
2 answers
5k views

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 ...
Timothy Chow's user avatar
6 votes
0 answers
170 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 ...
Christian Sattlecker's user avatar
3 votes
0 answers
146 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 ...
bubba's user avatar
  • 629
0 votes
1 answer
201 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{...
dotdashdashdash's user avatar
1 vote
1 answer
167 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 $x \mapsto x+2\pi$). ...
nervxxx's user avatar
  • 207
2 votes
0 answers
66 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}(...
Manish Kumar's user avatar
0 votes
0 answers
55 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" ...
Assinisa Hamidata's user avatar
2 votes
0 answers
83 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 $...
anon1802's user avatar
  • 121
3 votes
0 answers
103 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(...
Ewin's user avatar
  • 101
1 vote
0 answers
71 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 ...
Assinisa Hamidata's user avatar
2 votes
1 answer
103 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, ...
Assinisa Hamidata's user avatar
1 vote
0 answers
131 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": $$ \...
russloewe's user avatar
4 votes
3 answers
384 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 ...
L. Proz's user avatar
  • 83
4 votes
1 answer
366 views

Characteristic polynomial of a simple matrix: Chebyshev?

In my recent MO question, Darij Grinberg mentioned a closely related (structure-wise) determinant, that is, $$\det\left(x_{\min\{i,j\}}\right)_{i,j}^{1,m}=x_1(x_2-x_1)(x_3-x_2)\cdots(x_m-x_{m-1}).$$ ...
T. Amdeberhan's user avatar
1 vote
3 answers
269 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)=\...
Kane's user avatar
  • 43
3 votes
1 answer
450 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{\...
Kane's user avatar
  • 43
12 votes
1 answer
812 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 ...
Andrius Kulikauskas's user avatar
2 votes
1 answer
74 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 $P_i(z_j)=\delta_{ij}$?
ABB's user avatar
  • 3,898
5 votes
1 answer
503 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 ...
Marcel's user avatar
  • 2,510
1 vote
1 answer
139 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 \...
Tardis's user avatar
  • 1,077
15 votes
3 answers
2k views

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 ...
David Richter's user avatar
4 votes
2 answers
254 views

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 ...
Johann Cigler's user avatar
0 votes
1 answer
158 views

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 $T_n(x) = \cos(n \cos^{-1}(x))$? ...
MHF's user avatar
  • 9
2 votes
0 answers
84 views

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 ...
fewfew4's user avatar
  • 233
1 vote
0 answers
136 views

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}, \...
GGT's user avatar
  • 685
0 votes
1 answer
261 views

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}^{...
Adam Hammam's user avatar
1 vote
0 answers
175 views

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 ...
NickVO's user avatar
  • 11
2 votes
0 answers
114 views

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 ...
Andrius Kulikauskas's user avatar
6 votes
1 answer
342 views

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 ...
mlbaker's user avatar
  • 215
3 votes
2 answers
279 views

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 $w\in\mathbb{C}$...
Yonah Borns-Weil's user avatar
2 votes
0 answers
100 views

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]\}...
J. M.'s user avatar
  • 49

1
2 3 4 5