**3**

votes

**0**answers

51 views

### Fourier coefficients of positive polynomials

Let $p(x) \geq 0$ be a positive polynomial on the hypersphere ($x \in S^{n-1}$) satisfying $\int_{S^{n-1}} p(x) = 1$. Writing $p(x) = \sum_{j=0}^s p_j(x)$ where $p_j(x) = \sum_m p_{jm} s_{jm}(x)$ with ...

**0**

votes

**0**answers

35 views

### Bounds on the smallest eigenvalue of a Hankel matrix

Let $H=H_n$ be a positive definite Hankel matrix of size $n$ with $\lambda_n$ is it's smallest eigenvalue.
What bounds are known on $\lambda_n$ in terms of the entries on $H$.
I can see some results ...

**3**

votes

**0**answers

83 views

### A “nice” Orthogonal Basis for Translation Invariant Symmetric Polynomials

It is going to be a rather long question, so I will first state it and then try to explain and motivate it.
Take $\Lambda_n $ as the graded ring of symmetric polynomials of a field $F$ in $n$ ...

**1**

vote

**0**answers

92 views

### Evaluation of Macdonald Polynomials at $x_1=x_2=…=x_n = h$

This question is related to my previous question (here).
Let $P_\lambda$(q,t) be the Macdonald polynomials with partition $\lambda$. Let $\Lambda$ denote the ring of symmetric functions over the ...

**11**

votes

**1**answer

300 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 ...

**0**

votes

**2**answers

177 views

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

How do I Calculate, if possible, in terms of well-known constants the integral :
$\int_{0}^{1}x^{k}\psi(x)dx$ , where $k\geq 3$ is an integer ?
note: $\psi(x)$ is digamma function.
Any help would ...

**24**

votes

**4**answers

2k views

### How does this relationship between the Catalan numbers and SU(2) generalize?

This is a question, or really more like a cloud of questions, I wanted to ask awhile ago based on this SBS post and this post I wrote inspired by it, except that Math Overflow didn't exist then.
As ...

**0**

votes

**0**answers

35 views

### Growth of average first derivative of orthogonal polynomials

Let $L_k(t)$ be the Legendre polynomials normalized so that
$$\int_{-1}^1 L_k(t)^2\,\frac{1}{2}\,dt = 1.$$
With a few identities (http://en.wikipedia.org/wiki/Legendre_polynomials), one can show that
...

**9**

votes

**2**answers

217 views

### Computing Gauss Legendre Quadrature for Large N

I've been scanning across the web, and haven't found a good method to compute the Gauss Legendre abscisas and weights $\{ x_j, w^j \} _j$ for large N. My question is how to do it, and why should it ...

**5**

votes

**1**answer

505 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 ...

**4**

votes

**1**answer

149 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

**0**answers

70 views

### q-Hermite polynomials

It is well known that the q-Hermite polynomials defined by $$H_n(\theta; q)= \sum\limits_{k=0}^n \frac{(q;q)_n}{(q;q)_k(q;q)_{n-k}}e^{i(n-2k)\theta}$$
are orthogonal in $\theta \in [0, \pi]$ with ...

**1**

vote

**1**answer

51 views

### Generating function for products of laguerre polynomials?

In a quantum physics context, I would like to evaluate $S=\sum_{n=0}^\infty z^n\cos(\pi L_n(x))$ for $z<1$. I found generating functions for squares of Laguerre polynomials but not for any higher ...

**0**

votes

**1**answer

115 views

### Equality cannot hold unless $x \in \{-1,1\}$ and/or Wronskian is not zero [closed]

By playing around with assoc. Legendre polynomials, I arrived at
$$((l+1)+m) (P_l^m(x))^2+((l+1)-m)(P_{l+1}^m(x))^2 = 2(l+1)x P_l^m(x)P_{l+1}^m(x).$$
Now, I want to show that we don't have equality ...

**8**

votes

**0**answers

168 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 ...

**1**

vote

**1**answer

214 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 ...

**2**

votes

**1**answer

180 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 ...

**1**

vote

**4**answers

1k views

### Functions orthogonal to x^n

I would like to ask if there are any set of functions $u_n(x)$ which is orthogonal to $x^n$?
i.e.:
$\int_0^1 x^n u_m(x) dx = \delta_{n,m}$
Edit: For clarification, this question asked for all ...

**4**

votes

**3**answers

2k views

### Finding a recursion for a sum of Legendre polynomials

The polynomial
$a_n(x):=P_n(x)-\frac{n-1}{n}P_{n-2}(x)$
where $P_n(x)$ is a Legendre polynomial came up while I was investigating methods for estimating the error in Gaussian quadrature.
I am ...

**0**

votes

**0**answers

47 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 = ...

**11**

votes

**0**answers

237 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 - ...

**7**

votes

**4**answers

470 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 ...

**2**

votes

**1**answer

123 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
...

**4**

votes

**1**answer

100 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

55 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

1k 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

138 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 ...

**1**

vote

**1**answer

190 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 $-1$:
$$ ...

**2**

votes

**1**answer

238 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 ...

**2**

votes

**1**answer

679 views

### Closed form for an orthogonal polynomial integral?

The following integral came up in one of my applications:
$\int_{-1}^1P_n(x)T_j(x)T_k(x)\mathrm{d}x$
where $P_n(x)$ is a Legendre polynomial, $T_k(x)$ is a Chebyshev polynomial, and $j$, $k$, and ...

**10**

votes

**3**answers

599 views

### MacWilliams Identity for Asymptotic Weight Spectrum of a Code

Introduction
Let $C$ be a code of block length $n$ having $A_i^C$ words of Hamming weight $i$, for $i\in [n]$, where $[n]:=\{0,\ldots,n\}$. Then, the sequence $\{ A_i^C \}_{i \in [n]} $ is called the ...

**2**

votes

**3**answers

299 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

189 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

154 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)~,
...

**6**

votes

**2**answers

1k 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
...

**2**

votes

**0**answers

147 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

**3**answers

1k views

### Convergence of orthogonal polynomial expansions

"Everyone" knows that for a general $f\in L^2[0,1]$, the Fourier series of $f$ converges to $f$ in the $L^2$ norm but not necessarily in most other senses one might be interested in; but if $f$ is ...

**5**

votes

**1**answer

314 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 = ...

**4**

votes

**2**answers

332 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 ...

**1**

vote

**0**answers

77 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

564 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 $a_k$ ...

**2**

votes

**1**answer

626 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} - ...

**1**

vote

**0**answers

141 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 ...

**7**

votes

**2**answers

895 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} ...

**1**

vote

**1**answer

346 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

95 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), ...

**5**

votes

**1**answer

1k 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 ...

**2**

votes

**0**answers

88 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

557 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}
...