48
votes
To prove irrationality, why integrate?
Hermite's approximations to values of $e^x$ are based on good rational function approximations to $e^x$, which nowadays go under the name of Padé approximations (a name that came much later: Padé was ...
- 50.6k
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
$$
\pmatrix{T_n(x) \\ T_{n-1}(x)} = \pmatrix { 2x & -1 \\ 1&0} \pmatrix{T_{n-1}(x) \\ T_{n-2}(x)} = \pmatrix { ...
- 43.7k
23
votes
Accepted
To prove irrationality, why integrate?
Here's an exposition of Niven's proof that makes the connection to orthogonal polynomials explicit. We start with an observation, easily proven by induction, that if $P\in \mathbb{Z}[x]$, then $\int_0^...
- 8,992
15
votes
Polynomials for which $f''$ divides $f$
Recording a CW-answer to take this off the unanswered list. The Gegenbauer polynomials are defined by the differential equation
$$(1-x^2) g'' - (2 \alpha+1) x g' +n(n+2 \alpha) g =0.$$
Putting $\alpha ...
15
votes
Accepted
How are Sheffer polynomials related to Lie theory?
First, a general set of Sheffer polynomials is not orthogonal with respect to some weight function; for example, the prototypical sequence $p_n(x) = x^n$, which belongs to both special sub-groups of ...
- 10.5k
13
votes
Accepted
Is this Hermite polynomial identity known?
If we define the generating functions $F(x,t)=\sum_{n=0}^{\infty}H_{2n}(x)\frac{t^n}{n!}$ and $G(x,t)=\sum_{n=0}^{\infty}H_{2n+1}(x)\frac{t^n}{n!}$ then your identity is equivalent to
$$F(x,t)F(y,t)F(...
- 85.6k
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) + \...
- 34.3k
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(...
- 7,226
9
votes
Sum of squared hypergeometric polynomials
The identity can be derived by differentiating both sides with respect to $u$:
The rhs
\begin{align}\tag{1}\label{eq:1}
R(u)=\frac 1 4-\frac 1 2 \log u
\end{align}
simply gives
\begin{align}\tag{2}\...
- 3,671
9
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 & -...
- 148k
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 $x$ ...
- 17k
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 ...
- 12.1k
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 ...
- 4,034
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’,\...
- 23.7k
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 = ...
- 163
6
votes
What is the story behind the Chebyshev polynomials?
An earlier, but different, approach to the questions posed and solved by next-to-be Tchebyshev Polynomials was given by Augustin-Louis Cauchy in his Cours d'Analyse (1821). See p. 230 and ff.
- 161
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 ...
- 12.1k
6
votes
Accepted
Real zeroes of the determinant of a tridiagonal matrix
For $\epsilon_1=\epsilon_2=-1$ and $\epsilon_3=\epsilon_4=\epsilon_5=1$ you get the counterexample $\operatorname{det}M(t)=t(t - 1)^2(t + 1)^2$.
Another example, with simple real roots, is $\epsilon_1=...
- 22.5k
5
votes
Sum of squared hypergeometric polynomials
I think the proof below should work. Although it looks more complicated than the one of Fred Hucht, it doesn't involve divergent series.
I write
\begin{equation}\tag{1}S(t)=\sum_{m=1}^\infty\frac {t^m}...
- 2,306
5
votes
Generating function for products of complex Hermite polynomials
The OP's question has been addressed By Richard Stanley.
So, we make attempt at Stanley's question on the complex counterpart to
$$\sum_{n\geq0}\frac{H_n(x)H_n(y)}{n!}\left(\frac{u}2\right)^2
=\...
- 43.2k
5
votes
Generating function for products of complex Hermite polynomials
The formula (*) is trivial since it breaks up into a product of a sum over $m$ with a sum over $n$. The same reasoning works for the complex analog as a sum over $m,n,k,l$, using
$$ \sum_{m,n\geq 0}...
- 50.8k
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 ...
- 34.3k
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
$$I:=\int_0^\infty e^{-a\,t} t^{\nu+1} J_\nu(b\,t) L_n^{2v}(t) \,dt =\frac{(2b)^\nu \Gamma(\nu+1/2)}...
- 894
5
votes
Complex Hermite polynomial orthogonality on weighted space
Fix $\xi$ and assume $k>n$ (the other case is similar). Integrate over $x$. You'll get $\int_{\mathbb R+i\xi}H_k(z)P_n(z-i\xi,\xi)e^{-z^2/2-\xi^2}\,dz$ where $P_n$ is some polynomial of 2 variables ...
- 61.9k
5
votes
Do you recognize this sequence of polynomials?
See OEIS A129818 and the unsigned version A085478 as well as A049310, A011973, and A054142.
Relations among the Gegenbauer, Jacobi, Legendre, and the polynomials highlighted in this post can be found ...
- 10.5k
5
votes
Accepted
Riemann-Hilbert approach to Selberg integral
Let me formulate the problem in a slightly more general way: We seek to evaluate the large-$N$ limit of the matrix integral
$$\int e^{-\beta\,{\rm Tr}\,V(X)}|\Delta(X)|^\beta dX\equiv e^{-\beta N^2 F},...
- 188k
5
votes
Accepted
Generating function of the product of Legendre polynomials
Using the recursion relation
$$
P_{n-1} (x) = x P_n (x) - \frac{x^2 -1}{n} \frac{d}{dx} P_n (x) \ ,
$$
you can reduce your expression to a sum of the generating function you quote and a combined ...
- 6,696
5
votes
Accepted
Can the Chebyshev polynomials be constructed from the extremal property?
The argument in the answer you refer to actually shows that (using the notation from that answer) $t_2, \ldots, t_n$ are extrema of $p$ with value $\pm 1$ and thus double roots of $1-p^2$. Hence $(1-x^...
- 24.2k
4
votes
Any reference for the series expansion of $\Bigr[-\log(1-t)\Bigr]^x$?
It should be remarked that the expansion is an immediate consequence of the Lagrange Inversion Formula, and of the definition of the Stirling polynomials via their generating function (as given e.g. ...
- 60.5k
4
votes
Accepted
Sturm Liouville problems for non-classical orthogonal polynomials
A reference in english for Bochner's theorem is section 20.1, p.508, of the book by Mourad E.H.Ismail,
Classical and quantum orthogonal polynomials in one variable, Encyclopedia of Mathematics and ...
- 4,034
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