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 ...

- 155k

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 ...

Community wiki

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(...

- 82.7k

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) + \...

- 27.5k

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 ...

- 155k

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 ...

- 3,569

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 "...

- 1,213

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’,\...

- 19.8k

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, &...

- 2,020

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 = ...

- 153

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. ...

Community wiki

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 ...

Community wiki

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 ...

Community wiki

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 ...

- 27.5k

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)}...

- 884

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},...

- 155k

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