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I am writing up some notes and the following occurred to me and I would like to see if there are a variety of ways to prove it. Just for reference, the identity pops out of equality between constant ...

We say a rational function $F(z)$ is positive if the coefficients of its Maclaurin expansion, in the variable $z$, are non-negative. In this context, let $$F_r(z):=\frac{1 - 2z + z^r - (1 - z)^r}{(1 - ...

Here is an identity for which I outlined two different arguments. I'm collecting further alternative proofs, so QUESTION. can you provide another verification for the problem below? Problem. Prove ...

Let $I$ be a (nonempty) compact interval in $\mathbb R$ and $a_1,b_1,\ldots,a_L,b_L \in \mathbb R$. Let $\varphi$ be a piecewise function with $T \ge 2$ pieces(for example $T=2$ for the choice $\...