Let $\{a_n\}_{n\geq 0}$ be a sequence of integers. We're interested in specific limits of the formal series $$f(q) = \sum_{n=0}^{\infty}a_n q^n.$$ Let $\zeta$ be a root of unity. Say $\zeta^k=1$. Formally, when $q = \zeta e^\hbar$, $$f(\zeta e^\hbar) = \sum_{n=0}^{\infty}a_n(\zeta e^\hbar)^n = ...

In this paper, Shanks uses the following formula: $$ \sum_{s=0}^{n-1}q^{s(2n+1)} \times \left( \prod_{k=s+1}^{n} \dfrac{1-q^{2k}}{1-q^{2k-1}}\right) = \sum_{s=1}^{2n} q^{\frac{s(s-1)}{2}}$$ to get a one line computation of the Gauss sum: $$\sum_{k=0}^{m-1} e^{\frac{2ik^2 \pi}{m}}$$ in the odd cas...

Let $X_t= X_0 + \int_0^t \mu(s,X_s)ds + \int_0^t \sigma(s,X_s)dW_s$ where $\mu$ and $\sigma$ are $C^1$ functions satisfying the usual growth restriction and $W_t$ is a $d$-dimensional Brownian motion. Let $f\in C^2(\mathbb{R}^d;\mathbb{R})$ and let $D_t$ denote the Malliavin derivative at time $...

I included this footnote in a paper in which I mentioned that the number of partitions of the empty set is 1 (every member of any partition is a non-empty set, and of course every member of the empty set is a non-empty set): "Perhaps as a result of studying set theory, I was surprised when I ...

In the litterature on D-modules, there are many definitions of regularity of holonomic D-modules. (1) Bernstein first defines regularity on a curve then says a holonomic D-module is regular if its restriction to any curve is regular (2) Mebkhout defines the irregularity complexes of a complex ...