Using this identity to express the hypergeometric function as a Gegenbauer function, and this identity which gives the value of the Gegenbauer function evaluated at 1 1 $1$, the hypergeometric function in question may then be expressed as a ratio of gamma functions whose arguments are each positive integers. These can be reorganized into binomial terms for a compact final expression:

$\begin{array}{rl}& \frac{1}{2}{\int }_{\mathrm{\Omega }}|u(t,x){|}^{2}\phantom{\rule{thinmathspace}{0ex}}dx+\nu {\int }_{0}^{t}{\int }_{\mathrm{\Omega }}|\mathrm ...

also note it says for n ≥ 2 n ≥ 2 $n\ge 2$ so your base case needs to be n = 2 n = 2 $n=2$

${B}_{0}+{B}_{1}x+\frac{{B}_{2}{x}^{2}}{2}+\sum _{n=3}^{\mathrm{\infty }}\frac{{B}_{n}{x}^{n}}{n!}=\frac{x}{{e}^{x}-1}$ and I believe this is not much help. I want to ...

This is being repurposed in an effort to cut down on duplicates, see here: Coping with abstract duplicate questions. and here: List of abstract duplicates.

Picard's little theorem says that If there exist two complex numbers $a,b$ such that $f: \Bbb {C} \to \Bbb {C}\setminus \ {a,b\}$ is holomorphic then $f$ is constant ...

$\\text{What is the ratio of legs in a right triangle with angles of 15, 75, and 90?}$ I know the ratio of legs in a $30, 60, 90$ triangle, which is the lengths $1$, $\\sqrt{3}$, and $2$ respectively...

Before I begin, I did a search through math.stackexchange and came across two previous attempts to get people to solve probability problems involving bingo. Neither produced a response. So what m...

Another example is afforded by Q[x] Q [x] $\mathbb{Q}[x]$, the ring of rational polynomials. In this ring, polynomials are associates, if one is a rational multiple of the other-for example, 9x2 − 18x + 9 9 x 2 − 18 x + 9 $9{x}^{2}-18x+9$ and 2x2 − 4x + 2 2 x 2 − 4 x + 2 $2{x}^{2}-4x+2$ are associates.

I am having a hard time understanding what is the main difference between a polyhedron and a polytope. Could anyone explain me what is the difference between these two structures?