All Questions
Tagged with chebyshev-polynomials generating-functions
8
questions
3
votes
3
answers
310
views
Closed form for infinite sum involving Chebyshev polynomials
There exists a generating function for the Chebyshev polynomials in the following form:
$$\sum\limits_{n=1}^{\infty}T_{n}(x) \frac{t^n}{n} = \ln\left( \frac{1}{\sqrt{ 1 - 2tx + t^2 }}\right)$$
...
- 243
0
votes
1
answer
165
views
Reference for the exponential generating function for the Chebyshev polynomials of the second kind?
Does anyone have a reference for the exponential generating function for the Chebyshev polynomials of the second kind?
$$
\sum_{n=0}^{\infty}\frac{t^n U_n(x)}{n!}= etc
$$
I know what it is from the ...
0
votes
1
answer
43
views
Can we define the generating function for all $x$ and all $t$
The Chebyshev polynomials of second kind are defined for any $x \in \Bbb R$ (or
even $x \in \Bbb C$), e.g. via the recurrence relation
$$
U_0(x) = 1 \\
U_1(x) = 2x \\
U_{n+1}(x) = 2x U_n(x) - U_{...
- 3,854
2
votes
3
answers
1k
views
Derive the Rodrigues' formula for Chebyshev Polynomials
Rodrigues' formula for Chebyshev Polynomials is stated as $$T_n(x)=(-1)^n2^n\frac{n!}{(2n)!}\sqrt{1-x^2}\frac{d^n}{dx^n}(1-x^2)^{n-1/2}$$
I understand how the Rodrigues formula for all other special ...
- 7,674
0
votes
1
answer
170
views
Show that first order Chebyshev polynomials are in fact polynomials
Given the 1st order Chebyshev polynomials
$$
G(x,t) = \sum_{n=0}^{+\infty} T_n(x) t^n = \frac{1-tx}{1-2xt+t^2}
$$
I'm wondering how can I show that $T_n(x)$ are polynomials ?
- 607
2
votes
1
answer
185
views
Relationship between Poisson's integral formula and the generating function of Chebyshev polynomials
On the disk $\{z:|z|<R\}$, Poisson's integral formula is $$u(r,\theta)=\frac1{2\pi}\int_0^{2\pi}\frac{(R^2-r^2)f(\phi)}{R^2-2Rr\cos(\theta-\phi)+r^2}\,d\phi$$ which solves the Dirichlet problem. ...
- 27.1k
1
vote
2
answers
3k
views
Derive recurrence relation for Chebyshev polynomials from generating function
Hej,
I have a question about the following problem:
Derive a recurrence formula for $m \ge 0$ given the generating function formula
$$
\frac{1}{1-2xt+t^{2}}=\sum_{m=0}^\infty U_m(x)t^{m}.
$$
What I ...
- 107
8
votes
1
answer
702
views
How to get from Chebyshev to Ihara?
I have competing answers on my question about "Returning Paths on Cubic Graphs Without Backtracking". Assuming Chris is right the following should work. Up to one thing:
The number of returning paths ...
- 18.6k