Tagged Questions
0
votes
0answers
47 views
Zero's of the General septic equation in nonhypergeo?
Can the zero's of the general septic equation be expressed in special functions apart from the hypergeometric ones ?
We know that the zero's for the general quintic can be expressed by other ...
0
votes
0answers
45 views
if system is localized
Consider function $g=e^{-|x|}, x \in R$.
Let $\psi_n$ be a Hermite function (see http://en.wikipedia.org/wiki/Hermite_polynomials for definition). Consider system $\Psi=\{\psi_n\}, n \in N$.
Let ...
2
votes
2answers
144 views
Polynomials in Fourier trigonometric series
I'm successively integrating $x^{n} \cos{k x}$ for increasing values of positive integer n. I'm finding:
$\frac{\sin{kx}}{k}$,
$\frac{\cos{kx}}{k^2}+\frac{x\sin{kx}}{k}$,
$\frac{2 x ...
2
votes
0answers
208 views
$L_2$-norm representation of the function
Let
$$
f^{\alpha}_+(x)=\frac{1}{\Gamma(\alpha+1)}\sum_{k\ge 0}(-1)^k{\alpha+1 \choose k}(x-k)^{\alpha}_+,
$$
where $\alpha > -\frac 12$(see for reference ...
5
votes
0answers
83 views
relationship between solution of quintic in terms of $_{4}F_{3}$ hypergeometric function and theta functions
There is one approach (Bring radical/method of differential resolvents) to the general solution to the quintic that gives the solution for a particular root $v\in\{v_{1},v_{2},v_{3},v_{4},v_{5}\}$ in ...
3
votes
0answers
81 views
The polynomial where only the terms in the multinomial series where each variable's exponent is $>0$ are kept?
I'm wondering if there's a special polynomial with a name out there with $x_1,x_2,\ldots,x_k$ as variables that's defined like this:
$$
\sum_{\substack{i_1>0,i_2>0, \ldots,i_k>0 \\ i_1 ...
2
votes
2answers
312 views
The relationship between Legendre Polynomials and monomial basis polynomials
I am currently doing filter designs and stumbled across this mathematical problem which I cannot understand. I was hoping for some insight from experts around this field to help me with this.
...
13
votes
4answers
447 views
Intriguing polynomials coming from a combinatorial physics problem
For real $0<q<1$, integer $n >0 $ and integer $k\ge 0$, define
$$[k, n]_q \equiv -\sum_{m=1}^{n} q^{m(k+1)} (q^{-n}; q)_m = -\sum_{m=1}^{n} q^{m(k+1)} \prod_{l=0}^{m-1} (1-q^{l-n})$$
...
5
votes
1answer
110 views
orthonormal polynomials
Here is the question:
Suppose $P_0, P_1, P_2, \dots$ are polynomials orthonormal with respect to the inner product $$(f,g)=\int_a^b f(x)g(x)W(x)dx,$$ where $W(x) > 0$ is a weight
function and ...
8
votes
1answer
119 views
Is there a neat way to show $\int_{-1}^1 \frac{ U_n(z) U_n(z)}{\sqrt{1-z^2}} \mathrm{d} z = \pi (n+1)$
In answering a question on math.SE, I attempted to find integral of Fejér kernel by using
$$
K_n(t) = \frac{1}{n} U_{n-1}^2\left( \cos \frac{t}{2} \right)
$$
where $U_n(z)$ stands for the ...
2
votes
2answers
252 views
Representing affine transform of Legendre polynomials
I have a function defined as a set of weighted Legendre polynomials:
$f(x)=\alpha_0 P_0(x) + \alpha_1 P_1(x) + \alpha_2 P_2(x) +\ldots$.
I have another function similarly defined with Legendre basis ...
1
vote
0answers
122 views
Orthogonal polynomial interpolation of a function
I want to write down an arbitrary function $f$ as an (infinite) sum of orthogonal polynomials, e.g. for $f(x) = e^{\sin(x)}$, $f(x) = \sum{a_n T_n}$, where $a_n$ are the coefficients and $T_n$ are the ...
10
votes
0answers
305 views
Hermite's solution of the general quintic in terms of theta functions
Can someone point me at or produce a translation or modern exposition of Hermite's solution of the general quintic in terms of theta functions? (the "before" and "after" steps are on the mathworld ...
9
votes
1answer
245 views
density of roots of a family of polynomials: $(1-x^2)^{v+n}$
My research has brought me to the following, very general problem.
Given a fixed, but arbitrary, natural number, $\displaystyle v$, consider the following family of polynomials: The $\displaystyle ...
