6
votes
2answers
174 views

How prove this $p(x)>0$ if $p(x)=\sum_{i=0}^{n}\binom{n}{i}a_{i}x^i(1-x)^{n-i}$

let the polynomials $$p(x)=\sum_{i=0}^{n}\binom{n}{i}a_{i}x^i(1-x)^{n-i}$$ and such $$a_{0}+\sum_{a_{i}<0}(1-\dfrac{i}{n})\binom{n}{i}a_{i}>0$$ and ...
0
votes
1answer
596 views

Calculation of Chebyshev coefficients

The Chebyshev polynomials can be defined recursively as: $T_0(x)=1$; $T_1(x)=x$; $T_{n+1}(x)=2xT_n(x) + T_{n-1}(x)$ The coefficients of these polynomails for a function, $\space f(x)$, under ...
4
votes
1answer
111 views

Why the sum of the squares of the roots of the $n$th Hermite polynomial is equal to $n(n-1)/2$?

How to prove that the sum of the squares of the roots of the $n$th Hermite polynomial is $\frac{n(n-1)}{2}$? I tried with Vieta formulas, but it's hard. I appreciate a proof or reference to it. ...
2
votes
2answers
195 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
218 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
113 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
116 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
508 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
495 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
120 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
2answers
150 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
368 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
148 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 ...
11
votes
1answer
502 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 ...
10
votes
1answer
279 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 ...