28 votes
Accepted

Is there an explicit expression for Chebyshev polynomials modulo $x^r-1$?

There's a rapid algorithm to compute $T_n(x)$ modulo $(n,x^r-1)$. Note that $$ \pmatrix{T_n(x) \\ T_{n-1}(x)} = \pmatrix { 2x & -1 \\ 1&0} \pmatrix{T_{n-1}(x) \\ T_{n-2}(x)} = \pmatrix { ...
  • 42.7k
20 votes
Accepted

What is the story behind the Chebyshev polynomials?

The Chebyshev polynomials first appeared in his paper Théorie des mécanismes connus sous le nom de parallélogrammes (1854). The remarkable "mechanisms" described in this work can be seen in action ...
16 votes

Why decompose a function with eigenvectors of Laplace operator?

The exponentials used in Fourier series are eigenvalues of shifts, and thus of any operator commuting with shifts, not just Laplacian. Similarly, spherical harmonics carry irreducible representations ...
  • 6,319
15 votes

Polynomials for which $f''$ divides $f$

Recording a CW-answer to take this off the unanswered list. The Gegenbauer polynomials are defined by the differential equation $$(1-x^2) g'' - (2 \alpha+1) x g' +n(n+2 \alpha) g =0.$$ Putting $\alpha ...
13 votes
Accepted

Is this Hermite polynomial identity known?

If we define the generating functions $F(x,t)=\sum_{n=0}^{\infty}H_{2n}(x)\frac{t^n}{n!}$ and $G(x,t)=\sum_{n=0}^{\infty}H_{2n+1}(x)\frac{t^n}{n!}$ then your identity is equivalent to $$F(x,t)F(y,t)F(...
12 votes

Do you recognize this sequence of polynomials?

The recurrence $f_{n+1} = (t-2)f_n - f_{n-1}$ with $f_0=1$ and $f_1=t-1$ suggests that $f_n$ can be expressed in terms of Chebyshev polynomials as: \begin{split} f_n(t) &= T_n(\tfrac{t}2-1) + \...
11 votes
Accepted

Do you recognize this sequence of polynomials?

Let $b_n(t)$ be the Morgan-Voyce polynomial defined by $$\begin{eqnarray}b_0(t) &=& 1 \\ b_1(t) &=& t + 1 \\ b_n(t) &=& (t+2) b_{n-1}(t) - b_{n-2}(t) \end{eqnarray}$$ Then $f_n(...
  • 4,933
11 votes
Accepted

How are Sheffer polynomials related to Lie theory?

First, a general set of Sheffer polynomials is not orthogonal with respect to some weight function; for example, the prototypical sequence $p_n(x) = x^n$, which belongs to both special sub-groups of ...
  • 8,551
10 votes
Accepted

What are the orthogonal polynomials w.r.t. Maxwell distribution

These are the socalled Maxwell polynomials $M_n^{(p)}(x)$, see page 75 and following of Spectral Methods in Chemistry and Physics. The coefficients for $p=2$, which is the case you need, are given in ...
10 votes
Accepted

Any reference for the series expansion of $\Bigr[-\log(1-t)\Bigr]^x$?

I edit my post to answer Carlo Beenakker's remark and also because I would like to add a reference, possibly more accurate than the two below. Theorem 7.1 p.13 of A. Adelberg, A finite difference ...
  • 3,433
9 votes
Accepted

The Stone-Weiestrass convergence for polynomials in different bases

Regarding the density of the span of monomials in the algebra of continuous functions with the uniform norm, there is the Müntz–Szász theorem. One simple version says that a necessary and sufficient ...
8 votes
Accepted

Characteristic polynomial of a simple matrix: Chebyshev?

Yes, the characteristic polynomial is given by $(-1)^m U_{2m}(1/2\lambda ) \lambda^{2m} $. The inverse matrix is given by $$\begin{pmatrix} 2 & -1 & 0 & 0 & \dots \\ -1 & 2 & -...
  • 122k
8 votes
Accepted

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 $2n-1$?

Everything becomes simpler if add some parameters and start the sum at $k=-n$ instead of $k=0$. Note that if $k$ is a negative integer with $-n\le k \le -1$ then $L_n^k(x)$ is a polynomial in $x$ ...
  • 14.4k
7 votes

Computing Gauss Legendre quadrature for large $N$

To add to Fredrik Johansson's answer: A nice history of algorithms for computing Gauss quadrature rules can be found in this SIAM News article by Alex Townsend. Therein, it is stated that the "...
7 votes
Accepted

Orthogonal basis of polynomials?

If a sequence of monic polynomials is orthogonal with respect a measure, it satisfies a three-term recurrence \[ p_{n+1}(t) = (t-a_n)p_n(t) - b_n p_{n-1}(t) \] where $b_n>0$. From this it ...
  • 11.8k
7 votes
Accepted

Riemann-Hilbert and orthogonal polynomials

Let $P_{n}(z)=\gamma_{n}z^{n}+\cdots$ be a sequence of orthonormal polynomials with respect to some weight $w$ on $\mathbb{R}$. Given an $n\geq0$, consider the following Riemann-Hilbert problem for ...
  • 3,433
7 votes
Accepted

Is there a bijective proof of an identity enumerating independent sets in cycles?

It seems that I’ve seen this question here before, but I am not sure whether it had a bijective answer. Anyway, here you are. Enumerate the vertices in two copies of $C_m$ as $1,2,\dots,m$ and $1’,2’,\...
6 votes
Accepted

Computing Gauss Legendre quadrature for large $N$

There are asymptotic methods that essentially give you $N$ nodes and weights in $O(N)$ time if the precision is assumed to be fixed (e.g. at double precision). See Nicholas Hale and Alex Townsend, &...
6 votes

Summation of an integral involving Laguerre polynomial and Bessel function

This integral is in the literature (e.g. Bateman manuscript project Vol. 2, Equation 8.9(5)). $ \int_0^\infty x^\mu \: L_n^{(\mu)}(\alpha x^2) \: J_\mu(xy) \: \mathrm{e}^{-\beta x^2} \: \mathrm{d}x = ...
6 votes

Applications of space filling curves

It's used in Google maps to introduce cache locality: When you move a little bit while viewing the map, you want to be moving only a little bit in the memory, which is after all arranged linearly. ...
6 votes

Recurrence of Legendre polynomial roots/ quadrature points

It is a conjecture of Stieltjes, apparently still open, see T.J. Stieltjes, Letter No. 275 of Oct. 2, 1890, in Correspondance d'Hermite et de Stieltjes, vol 2, Gauthier-Villars, Paris, 1905. that ...
  • 3,433
6 votes
Accepted

Are there graphs whose matching polynomials are Legendre?

Any family of orthogonal polynomials can be realized as the characteristic polynomials of a sequence of weighted paths, possibly with loops. If the implicit weight function is symmetric about the ...
  • 11.8k
5 votes
Accepted

How do I Calculate :$\int_{0}^{1}x^{k}\psi(x)dx$ where $k\geq 3$ is an integer?

This integral has been observed by Donal F. Connor in 2010 (you can find the link here, pg. 94). As far as I know, he found a closed form for the odd case, but I believe the even case is somewhere in ...
  • 1,367
5 votes

Applications of space filling curves

In a very different style, N. Katz has studied "space filling curves" over finite fields (https://web.math.princeton.edu/~nmk/spacefill.pdf), and given some applications (e.g., every abelian variety ...
5 votes

Applications of space filling curves

Space filling curves have been used a good deal by Katsuya Eda and his coauthors to compute singular homology groups of Peano continua (which are precisely the continuous images of $[0,1]$). For ...
5 votes

Is there an explicit expression for Chebyshev polynomials modulo $x^r-1$?

The coefficient of $x^j$ in $(T_n(x)\bmod (x^r-1))$ equals the coefficient of $t^{n+r-j-1}$ in $$\frac{(1+t^2)^{r-j}}{2^{r-j}} \frac{((1+t^2)^{r-1}t - 2^{r-1}t^{r-1})}{((1+t^2)^r - 2^rt^r)}.$$ This ...
5 votes
Accepted

Integral involving associated Laguerre polynomial and Bessel function

Let $n=\mu - \nu$ be an integer, as specified in the problem. Then I'll indicate how to prove that $$I:=\int_0^\infty e^{-a\,t} t^{\nu+1} J_\nu(b\,t) L_n^{2v}(t) \,dt =\frac{(2b)^\nu \Gamma(\nu+1/2)}...
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5 votes

Complex Hermite polynomial orthogonality on weighted space

Fix $\xi$ and assume $k>n$ (the other case is similar). Integrate over $x$. You'll get $\int_{\mathbb R+i\xi}H_k(z)P_n(z-i\xi,\xi)e^{-z^2/2-\xi^2}\,dz$ where $P_n$ is some polynomial of 2 variables ...
  • 54.3k
5 votes

Do you recognize this sequence of polynomials?

See OEIS A129818 and the unsigned version A085478 as well as A049310, A011973, and A054142. Relations among the Gegenbauer, Jacobi, Legendre, and the polynomials highlighted in this post can be found ...
  • 8,551
5 votes
Accepted

Riemann-Hilbert approach to Selberg integral

Let me formulate the problem in a slightly more general way: We seek to evaluate the large-$N$ limit of the matrix integral $$\int e^{-\beta\,{\rm Tr}\,V(X)}|\Delta(X)|^\beta dX\equiv e^{-\beta N^2 F},...

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