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
0 answers
60 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(...
user avatar
  • 101
1 vote
0 answers
59 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 ...
user avatar
1 vote
1 answer
48 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, ...
user avatar
1 vote
0 answers
81 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": $$ \...
user avatar
4 votes
3 answers
196 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 ...
user avatar
  • 83
4 votes
1 answer
232 views

Characteristic polynomial of a simple matrix: Chebyshev?

In my recent MO question, Darij Grinberg mentioned a closely related (structure-wise) determinant, that is, $$\det\left(x_{\min\{i,j\}}\right)_{i,j}^{1,m}=x_1(x_2-x_1)(x_3-x_2)\cdots(x_m-x_{m-1}).$$ ...
user avatar
1 vote
3 answers
175 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)=\...
user avatar
  • 43
3 votes
1 answer
213 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{\...
user avatar
  • 43
6 votes
1 answer
315 views

How are Sheffer polynomials related to Lie theory?

Sheffer polynomials are orthogonal polynomial sets $\{P_n(x)\}$ having 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 ...
user avatar
2 votes
1 answer
68 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 $P_i(z_j)=\delta_{ij}$?
user avatar
  • 3,343
5 votes
1 answer
411 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 ...
user avatar
  • 2,400
1 vote
1 answer
109 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 \...
user avatar
  • 343
0 votes
0 answers
39 views

Orthogonal polynomials of generalised Wilson weight

The Wilson polynomials $W(x;a,b,c,d)$ are orthogonal on the interval $(0,\infty)$ with weight function $$ w(x;a,b,c,d) = \left| \frac{\Gamma(a+ix)\Gamma(b+ix)\Gamma(c+ix)\Gamma(d+ix)}{\Gamma(2ix)} \...
user avatar
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 ...
user avatar
4 votes
2 answers
215 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 ...
user avatar
0 votes
1 answer
119 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 $T_n(x) = \cos(n \cos^{-1}(x))$? ...
user avatar
  • 9
2 votes
0 answers
68 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 ...
user avatar
  • 233
1 vote
0 answers
105 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}, \...
user avatar
  • 621
0 votes
1 answer
151 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}^{...
user avatar
1 vote
0 answers
63 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 ...
user avatar
  • 11
2 votes
0 answers
98 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 ...
user avatar
6 votes
1 answer
241 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 ...
user avatar
  • 215
3 votes
2 answers
157 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 $w\in\mathbb{C}$...
user avatar
2 votes
0 answers
65 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]\}...
user avatar
  • 49
9 votes
1 answer
367 views

Two-term recurrence relation

We consider the following system of recurrence relations for $n \in \mathbb Z$ and $\vert \lambda \vert=1$ with $\lambda \in \mathbb{C}$ $$a_{n+1} = \lambda a_{n-1}+ \lambda^* a_n + \lambda^* n b_n $$ ...
user avatar
  • 369
5 votes
2 answers
187 views

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, ...
user avatar
0 votes
1 answer
111 views

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 ...
user avatar
0 votes
1 answer
244 views

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 $I$ ...
user avatar
1 vote
0 answers
75 views

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 ...
user avatar
  • 11
4 votes
0 answers
236 views

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}) := \, \...
user avatar
2 votes
1 answer
90 views

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 $L^{\alpha}_n$ ...
user avatar
  • 71
10 votes
1 answer
279 views

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 $C_m$...
user avatar
  • 10.3k
3 votes
0 answers
54 views

Are there extensions of Hilb's and Laplace's formulas to Jacobi polynomials with $\alpha,\beta\le-1$?

In Szegő's Orthogonal Polynomials book, he gives two interesting asymptotic formulas for Jacobi polynomials with $\alpha,\beta>-1$. The first (Theorem 8.21.12, page 197 is a generalization of Hilb'...
user avatar
0 votes
1 answer
83 views

Transformation which "opens up" an arc

I am reading Harold Widom's paper "Extremal Polynomials Associated with a System of Curves in the Complex Plane". At the beginning of section 11 he states that: [There is] a simple transformation ...
user avatar
1 vote
0 answers
34 views

Higher-order inner products of an orthonormal basis

Let $\pi$ be a probability measure on some space $\mathcal{X}$, and let $\Phi = \{ \phi_k \}_{k \geqslant 0}$ be some (possibly complex-valued) orthonormal basis for $L^2 ( \pi )$, with $\phi_0 \equiv ...
user avatar
  • 267
2 votes
1 answer
457 views

Integral involving associated Laguerre polynomial and Bessel function

In a quantum mechanics problem I encountered the following integral \begin{equation*} \int_0^\infty t^{\nu+1}J_\nu(\beta t)L_{\mu-\nu}^{2\nu}(t)e^{-t/2}dt\,, \end{equation*} where $L$ denotes the ...
user avatar
  • 23
4 votes
1 answer
447 views

Gegenbauer's addition theorem for Jacobi polynomials

I have the following identity, $$\int_{-1}^{1} \! dz \, j_0\bigl(\sqrt{x^2 + y^2 - 2xy z}\bigr) \, P_n(z) = 2 \, j_n(x) \, j_n(y) \;,$$ where $x, y > 0$, $P_n$ is a Legendre polynomial, and $...
user avatar
1 vote
1 answer
322 views

Integration on sphere $\mathbb{S}^{d-1}$ for $d$ large -- Change of variables

I'm trying to integrate a function over two vectors which lie on the surface of the unit sphere in D dimensions. The function depends only on the difference between the two vectors, and their dot ...
user avatar
1 vote
1 answer
134 views

mollifier satisfying moment conditions

I wish to find a mollifier $\psi\in C_0^{d+1}(-1,1)$ such that $$ \int_{-1}^1 x^k \psi(x)dx = \begin{cases} 1, & k=0;\\ 0, & k=1,\dots,d. \end{cases} $$ This paper (https://home.cscamm....
user avatar
  • 580
2 votes
0 answers
54 views

Convergence of gPC expansions for random variables in the total variation distance

Suppose that a random variable $Y$ can be written as $Y=g(Z)$, where $g$ is a function and $Z$ is a random variable. When $Z$ is a continuous random variable with finite absolute moments, we consider ...
user avatar
1 vote
0 answers
68 views

Recurrence involving families of orthogonal polynomials

Let $ \forall n \in N, n\geq 1$ $$ R_n(x)=(-1)^n n! \displaystyle \frac{(x-1)...(x-n)}{(x(x+1)..(x+n))^2}$$ thus by decomposition in simple element it's easy to see that $$ (1): \quad R_n(x)= \...
user avatar
  • 417
0 votes
0 answers
74 views

About a family of orthogonal polynoms satisfying a recurrence relation

let $P_0(x)=0$;$P_1(x)=1$ Let $\forall n $ integer $ \geq 2$, $\forall x$ real, $$P_n(x)=\displaystyle \sum_{k=0}^{n-1} C_{n+k}^n (-x)^k \alpha_{n,k}$$ and where $\forall k$ such that $0 \leq k \...
user avatar
  • 417
0 votes
1 answer
178 views

Upper bound over $[0,1] $ for strange family of polynomials

Let $n$ integer $\geq 2,$ $x$ real, and $$P_n(x)=\displaystyle \sum_{k=0}^{n-1} C_{n+k}^n (-x)^k \alpha_{n,k}$$ and where $\forall k$ such that $0 \leq k \leq n-1 $ $$ \alpha_{n,k}= \displaystyle \...
user avatar
  • 417
8 votes
2 answers
344 views

Riemann-Hilbert and orthogonal polynomials

Sorry for perhaps naive questions, I am not at all a specialist in the subject but I need it for my research. I know that there are close relations between Riemann-Hilbert problems and orthogonal ...
user avatar
  • 8,928
3 votes
1 answer
382 views

Evaluating an integral with Jacobi and Legendre polynomials

The following integral came up in one of my studies: $$\int_{-1}^1 (1-x)^\alpha (1+x)^\beta P_n^{(\alpha,\beta)}(x)\,P_{n+j}(x)\,dx$$ where $P_n^{(\alpha,\beta)}(x)$ is a Jacobi polynomial and $P_m(...
user avatar
  • 31
1 vote
0 answers
73 views

An asymptotic behavior of a sequence of special polynomials

For $n\to\infty$, I would like to know the asymptotic behavior of the polynomials defined in terms of the Gauss hypergeometric series: $$ p_{n}(z):={}_{2}F_{1}(-n,-nz+\alpha;1;\beta), $$ where $\alpha,...
user avatar
  • 2,086
4 votes
1 answer
758 views

Is this Hermite polynomial identity known?

In some physics related problem, I found out the curious identity $$\sum\limits_{n_1+n_2+n_3=n}\frac{n!}{n_1!\,n_2!\,n_3!}\,H_{2n_1}(x)\,H_{2n_2}(y)\,H_{2n_3}(z)=\frac{H_{2n+1}(r)}{2r},$$ where $H_n(x)...
user avatar
4 votes
2 answers
410 views

Orthogonal Polynomials and Sturm Liouville operators

Classical Orthogonal polynomials (e.g., Hermite, Legendre) are eigenfunctions of Sturm Liouville operators. For example, define $L[u]=u''-xu'$, then the $n$-th order Hermite polynomial satisfies $...
user avatar
  • 3,200
7 votes
0 answers
233 views

Determinants associated to orthogonal polynomials

Let $${p_n}(x) = \sum\limits_{j = 0}^{n } {{{( - 1)}^{n - j}}p(n,j){x^j}} $$ be orthogonal polynomials satisfying $${p_n}(x) = (x - {s_{n - 1}}){p_{n - 1}}(x) - {t_{n - 2}}{p_{n - 2}}(x)$$ with ...
user avatar
4 votes
1 answer
794 views

Orthogonal basis of polynomials?

Let us define the basis of polynomials given by: $$ \begin{array}\ P_0=1, \\ P_1=x, \\ P_2=x(x-1), \\ P_3=x(x-1)(x-2), \\ P_4=x(x-1)(x-2)(x-3), \ldots\\ \end{array} $$ I would like to know if this ...
user avatar
  • 303