**2**

votes

**1**answer

158 views

### Chebyshev polynomials factoring uniformly modulo all primes

Consider the Chebyshev polynomial of the first kind $T_n(x)$ and its factorization in $\mathbb F_p$ for a given prime $p$. Most often, this factorization is not uniform (meaning that the irreducible ...

**0**

votes

**0**answers

36 views

### How to find out if a given sequence of orthogonal polynomials belongs to the Askey scheme?

I am studying some classes of orthogonal polynomials and want to find out which of them belong to the Askey scheme. To give a simple example consider the polynomials
$${p_n}(x,r) = \sum\limits_{k = ...

**5**

votes

**1**answer

134 views

### Orthogonal polynomial under linear transformation

Let $M_n(x) = x^n$ be the standard monomials. The binomial formula allows one to expand $M_n(ax+b)$ as a linear combination of $M_k(x)$, for $k \leq n$, giving
$$
M_n(ax+b) = (ax+b)^n = \sum_{k=0}^n ...

**11**

votes

**0**answers

230 views

### Ratio of entries of A and log A where A is a triangular matrix

Consider triangular matrices $A = \left( {a(n,k)} \right)$ of arbitrary order with $a(n,k) = 0$ if $n + k$ is odd and $a(n,n - 2k) = \frac{{n!}}{{k!(n - 2k)!}}\frac{{(m + n - k - 1)!}}{{(m + n - ...

**2**

votes

**1**answer

96 views

### Polynomials orthogonal w.r.t. the logarithmic weight

Recently, I have encountered the family of orthogonal polynomials $p_{n}(x)$ which is orthogonal w.r.t. the function $-\ln(x)$ on $(0,1)$. This means we have
...

**7**

votes

**4**answers

249 views

### Numerical integration of legendre polynomials

I hope that numerical questions are also permitted here.
I want to expand a smooth functions $f \in C^{\infty}$in terms of Legendre polynomials. Thus I need to calculate integrals of the form ...

**4**

votes

**1**answer

79 views

### Legendre Q(n,x) function coefficients in terms of P(n,x) coefficients

Empirically, the Legendre functions of second kind, $Q_n(x)$, appear to be of form
$$
Q_n(x)=\frac{P_n(x)}{2} \cdot\ln(\frac{1+x}{1-x})+p_n(x),
$$
with $P_n(x)$ the Legendre polynomials of first kind ...

**3**

votes

**0**answers

54 views

### Polynomials connected with Gale's condition and cyclic polytopes

I was reading about cyclic polytopes and (I think naturally) became interested in polynomials $f(X) \in \mathbb{R}[X]$ which have a factorization of the form $f(X)=g(X)g(X+1)$ for some $g(X) \in ...

**20**

votes

**1**answer

989 views

### A conjectured formula for Apéry numbers

A conjecture by the late Romanian mathematician Alexandru Lupas.
Posted in sci.math in 2005, but no proof was found.
Physicist Alan Sokal just reminded me of it, saying it was related to something he ...

**4**

votes

**1**answer

128 views

### Differential Operator Simplification

Does anyone know the explicit formulation for the $q_k$'s in, $$(x+D)^n=\sum_{k=0}^n q_k(x)D^k\ \ \ \ ?$$
I know that $e^{-x^2/2+x}$ is a fixed point of $(x+D)$. I also, know that ...

**0**

votes

**0**answers

26 views

### discrete Associated Legendre polynomials summation

I really apreciate if you can answer me:
How I can compute a discrete Associated Legendre polynomials summation, i.e.,
$$
\sum_{j=1}^{N} P_{\ell}^{|m|}(x_j)P_{\ell \,'}^{|m'|}(x_j)
$$
The ...

**8**

votes

**0**answers

159 views

### Littlewood-Richardson coefficients for Jack symmetric functions

Let $\Lambda$ be the algebra of symmetric functions over $\mathbb{Q}(\alpha)$.
We define a scalar product $\langle \cdot,\cdot\rangle_\alpha$ on $\Lambda$ by setting $\langle ...

**2**

votes

**1**answer

233 views

### What does this ODE have to do with the associated Legendre polynomials?

I am currently struggeling with the following differential equation:
$$(t^2-1)f''(t)+tf'(t)(1-8a+8at^2)-4(a+a^2-2at^2+\phi (-a+2at^2))f(t)= 4\lambda f(t),$$
where $a \in \mathbb{R}$ constant, $\phi ...

**3**

votes

**0**answers

87 views

### Orthogonal basis for the multilinear polynomials with zero “trace”

We say that a multilinear polynomial $P(x_1,\ldots,x_n)$ in $n$ commuting variables over $\mathbb{R}$ has zero trace if
$$ \frac{d}{dt} P(t,\ldots,t) = 0. $$
Equivalently,
$$ \left(\sum_{i=1}^n ...

**1**

vote

**1**answer

174 views

### The Largest Root of Associated Laguerre Polynomial

The Laguerre polynomial $L_n(x)$ is the solution to the Laguerre differential equation
\begin{equation*}
x\,y'' + (1 - x)\,y' + n\,y = 0.
\end{equation*}
The associated Laguerre polynomial ...

**0**

votes

**0**answers

44 views

### Explicit constructions of fundamental systems for spherical harmonics

For definitions and notations on the theory of spherical harmonics I refer to www.cis.upenn.edu/~cis610/sharmonics.pdf
Let $n,k\geq 0$, and let $S^n$ be the unit sphere on $\mathbb{C}^{n+1}$. Let ...

**2**

votes

**3**answers

197 views

### Multivariate Hermite Polynomials

Let $h_0, h_1, \dots$ be the classical univariate Hermite polynomials, renormalized to have constant norm. Is
$$x\mapsto\prod_{j=1}^n h_{l_j}(x_j), \quad l_j\in \mathbb N$$
a complete orthogonal ...

**1**

vote

**1**answer

147 views

### The asymptotic behavior of hypergeometric function around -1

Recently, in studing some specific orthogonal polynomials on unit circle, I was lead to study the asymptotic behavior of the following hypergeometric function at the neighberhood of $-1$:
$$ ...

**1**

vote

**1**answer

157 views

### Accurate bounds for derivatives of Legendre polynomials

Let $P_n(x)$ denote the $n$th Legendre polynomial. What bounds can one give for $d_{n,m}(x) = |\frac{d^m}{dt^m}P_n(t)|_{t=x}$ assuming that $|x| \le 1$? Clearly
$$d_{n,m}(x) \le d_{n,m}(1) = ...

**3**

votes

**0**answers

140 views

### Generalization of Frobenius formula involving Macdonald polynomials

Given a vector $\vec k=(k_1,k_2,\cdots)$ with $k_i$ are non-negative integers, the Newton polynomial $p_{\vec k}(x)$ is defined as
\begin{equation}
p_{\vec k}(x)=\prod_{j=1}^n p_j^{k_j}(x)~,
...

**5**

votes

**1**answer

388 views

### Integral of a product of Laguerre polynomials

In order to estimate the non linear term in a particular PDE, I have to decompose $L_k^\alpha(x)^3\cdot x^{-\delta}$ (with $0<\delta<\alpha+1$) into a basis consisting of Laguerre polynomials ...

**2**

votes

**0**answers

127 views

### A generalization of Macdonald functions?

I am interested in finding a set of functions $f(z_1,\cdots ,z_k;q,\,t)$, conjecturally polynomials, which depend on two parameters $(q,t)$ and an integer $k$, and are orthogonal under the following ...

**4**

votes

**2**answers

319 views

### On $a^4+nb^4 = c^4+nd^4$ and Chebyshev polynomials

In a 1995 paper, Choudhry gave a table of solutions to the quartic Diophantine equation,
$a^4+nb^4 = c^4+nd^4\tag{1}$
for $n\leq101$. Seiji Tomita recently extended this to $n<1000$ and solved ...

**5**

votes

**1**answer

287 views

### Lower bound of integral involving Laguerre polynomials

I want to lower bound the expected value of the square root of a randomly chosen eigenvalue of a Wishart matrix.
To get the bound I want I need a lower bound on
$$T_n = ...

**1**

vote

**0**answers

62 views

### Jack symmetric functions and their inner products

I have some questions regarding Jack polynomials. I use the notation of of I.G. Macdonald's book "Symmetric Functions and Hall polynomials".
Let $\Lambda$ be the ring of symmetric functions over ...

**4**

votes

**1**answer

473 views

### Why are all these families of polynomials finally log-concave?

This started when I was examining certain families of unimodal polynomials, i.e. $\sum_{k=0}^n a_kx^k$ where $a_0\le a_1\le\cdots \le a_k\ge\cdots \ge a_n$.
(Notation: in the following, the $a_k$ ...

**1**

vote

**0**answers

139 views

### Polynomials satisfying a three-term recurrence

Let ${p_n}(x) = x{p_{n - 1}}(x) - {a_{n - 2}}{p_{n - 2}}(x)$ for some numbers ${a_n}$ with initial values ${p_{ - 1}}(x) = 0$ and ${p_0}(x) = 1.$
By Favard’s theorem about orthogonal polynomials ...

**2**

votes

**0**answers

91 views

### Perturbation analysis for three term recurrences

Jacobi polynomials, denoted by $J^{(\alpha,\beta)}_n$, on $[-1,1]$ satisfy a three term recurrence
$$ J_{n+1}^{(\alpha,\beta)}(x) = (A_n+B_nx)J^{(\alpha,\beta)}_n + C_nJ_{n-1}^{(\alpha,\beta)}(x), ...

**1**

vote

**1**answer

328 views

### Proof of generalized Cauchy formula

I would like to know if there is a proof for the identity used in the superconformal index of 4d ${\cal N}=2$ gauge theory. In the paper by Rastelli el al, it was discovered that Eq. (10) is equal to ...

**2**

votes

**0**answers

82 views

### Lower asymptotic bounds for the derivative of Laguerre polynomials

Let $ L_{d}^{(1)}(x)$ denote the generalized Laguerre polynomial of degree $d$ and order $\alpha=1$. Clearly, since all the roots $r_1,\dots,r_d$ of $L_{d}^{(1)}$ are simple, there exists a strictly ...

**3**

votes

**1**answer

523 views

### Pochhammer symbol of a differential, and hypergeometric polynomials

I have a minor result which I'm sure has come up somewhere before but I can't seem to find it.
Consider a confluent hypergeometric function of the form
$$\newcommand{\ff}{{}_1F_1}
...

**2**

votes

**1**answer

592 views

### Three term recurrence relation.

For given $n,\ell\in\mathbb N_0$, I am interested in studying the following recursion relation for some $\mu\in\mathbb R$:
$$\sqrt{-1} \tfrac{j(\ell-j+1)(n-j+1)(n+j+1)}{2(2j-1)(2j+1)} a_{j-1} - ...

**3**

votes

**1**answer

408 views

### asymptotic behaviour of a sum

I'd like to know the asymptotic behaviour as $N\to\infty$ of the following sum
$$ Z_N(x) := 2^{-N/2} \sum_{k=0}^{N/2} \frac{N!}{k! (N-2k)!} (N-1)^{-k} (\sqrt{2} x)^{N-2k} $$
in order to compute ...

**2**

votes

**0**answers

148 views

### Growth of the recurrence coefficients of orthogonal polynomials

Consider the sequence of measures $$d\mu_N(x)=e^{-NV(x)}dx$$ on the real axis, where $V$ is continuous and satisfies the growth assumption $$\lim_{|x|\rightarrow\infty}(V(x)-2\log|x|)=+\infty.$$
...

**2**

votes

**0**answers

279 views

### About a Christoffel-Darboux-type sum

Hi!
I've been using the Christoffel-Darboux identity for the Hermite polynomials,
$$\sum_{k=0}^n\frac{H_k(x)H_k(y)}{2^k k!}=\frac{1}{2^n n!}\frac{H_{n+1}(x)H_n(y)-H_n(x)H_{n+1}(y)}{x-y},$$
for some ...

**13**

votes

**1**answer

521 views

### Can you name these orthogonal polynomials?

I have a collection of orthogonal polynomials in infinitely commuting variables $x_1, x_2, x_3, \ldots$. I think they must be well known (perhaps Schur or Hermite polynomials or some variant ...

**1**

vote

**0**answers

148 views

### Delta function representation as an integral of Pfaffians over SO(2m)

Define $\mathcal S$ as the set of all $2^m$ skew-symmetric $2m \times 2m$ matrices with the form $\oplus_{j=1}^m\begin{pmatrix} 0&\pm 1 \\ \mp 1&0\end{pmatrix}$.
Let $S_i, S_j \in ...

**2**

votes

**0**answers

121 views

### Orthogonality of Pfaffian polynomials in $SO(2m)$

I've been struggling here to invert some integral equation involving Pfaffians, and it would be very nice if you could shed some light on the problem.
Let's go to it. Let $V=\{-1,1\}^{m}$ and $S = ...

**3**

votes

**1**answer

422 views

### When can a family of polynomials get a weight function to be made orthogonal?

Let $\lbrace P_n(z)\rbrace_{n\in\mathbb N_0}$ be a family of polynomials defined by a generating function $g(t,z)=\sum\limits_{n=0}^\infty P_n(z)t^n$ or by a contour integral $P_n(z)=\frac1{2\pi ...

**7**

votes

**2**answers

738 views

### Multi-dimensional moment problem

Let $\mu$ be a measure on $\def\r{\mathbb{R}}\r^n$, $1\le n \le \infty$. Given a (finite) multi-index $\bar{i} = (i_1, i_2, \ldots)$, one can define the moment
$$ m_{\bar i} = \int x_i^{i_1} ...

**0**

votes

**1**answer

322 views

### Upper bounds on generalized Laguerre polynomials

I evaluated an integral and obtained an expression with a Laguerre polynomial. I'd like something more explicit and useable.
Are there any known simple (e.g. exponential) upper bounds on the ...

**3**

votes

**0**answers

500 views

### Integral involving Laguerre polynomial

In an engineering application, I've been trying to calculate the following integral involving Laguerre polynomials:
$\int_{-\infty}^\infty dx L_n(x^2+\beta^2) e^{-x^2/2+i x \alpha}$,
where ...

**5**

votes

**1**answer

999 views

### Two-variable generating functions for Laguerre polynomials

Where can I find generating functions for orthogonal polynomials in two variables?
Lebedev's book (Special Functions and their Applications, Dover, 1972) gives a closed form for
$$
\sum_{n=0}^\infty ...

**4**

votes

**4**answers

1k views

### Orthogonal polynomials/functions on the interval [0,1] but with same weight as Gegenbauer polynomials

I am looking for an othogonal basis of functions over the interval $[0,1]$ with weight function $(1-x^2)^{\alpha-1/2}$. Gegenbauer polynomials are frustratingly close to what I need, but they are ...

**5**

votes

**1**answer

401 views

### Numerically finding a Mercer expansion for a given covariance kernel

Let $c(r)$ be a nice, continuous function with compact support. For example, $c(r) = \tfrac 1 5 (1-r)^{11} \big( 5 + 55r + 239 r^2 + 429 r^3 \big)$ for $r \in [0,1]$, and $c(r) = 0$ otherwise.
On ...

**5**

votes

**1**answer

686 views

### Integral identity for Legendre polynomials

How does one prove the following integral identity, where $P_n(x)$ is the $n$th Legendre polynomial?
$$
\int_0^1 P_n(2t^2-1) dt = \frac{(-1)^n}{2n+1}
$$
Notes & Background
A variant of this ...

**0**

votes

**3**answers

1k views

### Inverting a power series? … Cornish Fisher

Hello
In the derivation of the cornish fisher expansion, the following equation is obtained:
$$
\sum_{n=2}^{\infty} b_n H_{n-1}(x_\alpha) = \sum_{j=1}^{\infty}\frac{(x_\alpha - ...

**4**

votes

**2**answers

436 views

### Polynomials from a Recurrence Relation

Hi guys,
I have recently started looking at polynomials $q_n$ generated by initial choices $q_0=1$, $q_1=x$ with, for $n\geq 0$, some recurrence formula
$$q_{n+2}=xq_{n+1}+c_n q_n$$
where $c_n$ is ...

**14**

votes

**2**answers

977 views

### Minimal polynomial with a given maximum in the unit interval

Find the lowest degree polynomial that satisfies the following constraints:
i) $F(0)=0$
ii) $F(1)=0$
iii)The maximum of $F$ on the interval $(0,1)$ occurs at point $c$
iv) $F(x)$ is positive ...

**6**

votes

**2**answers

921 views

### Spread polynomials

Norman Wildberger's "rational trigonometry" has been viewed by some mathematicians as a clever new take on an ancient topic. Wildberger's "spread polynomials" $S_n$ are characterized by the identity
...