All Questions

While trying to understand a paper of Cayley, he left something unexplained, I managed to show that it is equivalent to the following formula, which I got stuck at: $$k \cdot (f^k)^{(k-1)} = \sum_{j=0}...

I am currently trying to build the derivatives of $$f(x) = \frac{1}{e^x+e^{-x}}.$$ It is fairly straightforward to obtain $$ \frac{d^n f}{dx^n} = \frac{P_n(e^x)}{e^{(n-1)\cdot x} (e^x+e^{-x})^{n+1}}, $...

I am interested in the following integral related to the Chebyshev polynomials $$I_{n,m}:= \int_0^\pi \left(\frac {\sin nx}{\sin x}\right)^{m} dx,$$ where $n,m\in \mathbb{Z}^+.$ It is easy to see ...

An asymptotic expansion I stumbled upon has real numbers $c^\alpha_{ij}$ as coefficients, where $i, j \in \mathbb{N}_0$ are non-negative integers and $\alpha \in \mathbb{N}_0^n$ is a multi-index. They ...