I would like to approximate the following when $n \gg k$. $\sum_{y = k + 1}^n \frac{\sum_{m = 0}^{k - 1} {y - 2 \choose m} (y - 1)}{\sum_{m = 0}^k {y - 1 \choose m}}.$ The formul …

In one of my complexity analysis, I came up with $O(n^k/2^n)$, where $k$ is a fixed number and $n$ is the size of the data. However I rarely see a big-O written as this. Is there a …

I would like to approximate the following sum when $n \rightarrow \infty$ and $n \gg k$, $\sum_{x = k}^n \sum_{y > x}^n \frac{\sum_{m = 0}^{k - 1} {y - 2 \choose m}}{\sum_{m = 0}^ …

Let $g(x):\mathbb{R}_{\geq0}\rightarrow\mathbb{R}$ be real analytic s.t. $g(0)\neq 0$ and $g(x)=O(x^{-2})$ as $x\rightarrow\infty$. I think the following integral should be bounded …

Where can I find a readable textbook or lecture notes on asymptotic expansions ?

Let $M$ be a smooth manifold with Riemannian metric $g$, which is not smooth but only continuous. Question: Is there still an asymptotic expansion of the heat kernel of the form $ …

This is crosspost from math.stackexchange http://math.stackexchange.com/questions/311213/heat-kernel-asymptotics-on-manifold-with-boundary where the question did not yield any answ …

Let us consider a sequence $(p_l)_l$ of polynomials on $[0,1]$ that converge uniformely, as $l\to \infty$, to a function $f$ defined on $[0,1]$. I denote the polynomials $p_l(t) = …

I toss $n$ biased coins and I want to count the number of times you get a H followed by a T or a T followed by a H. I call these switches. So for example if I get HHTTHTHHHT then I …

Let $n,k$ be positive integers such that $3n=2k$ and $N = \lfloor \alpha n\rfloor$ for some constant $0<\alpha<1$. Let $S_{3n}$ denote the permutation group of order $3n$. Co …

My stackexchange post [http://math.stackexchange.com/questions/275830/schrodinger-kernels-on-manifolds] was somewhat unsatisfactory (also because I may not have stated clear enough …

The $n$-th Bell number $B_n$ represents the number of distinct partitions of a set with $n$ distinguished elements. It can be expressed as the infinite sum $B_n = (1/e)\sum_{k=1}^{ …

This is a cross-post with minor edits of the unanswered question http://math.stackexchange.com/questions/269904/pair-of-recurrence-relations. I have a system of equations as follo …

Let $\Bbb{R}^{+}_{0}$ be the set of non-negative real numbers and $\Bbb{R}^{+}$be the set of positive reals. Let us say that a function $f \colon \Bbb{R}^{+}_{0} \to \Bbb{R}^{+}_{0 …

For $\tau > 0$ define $\theta_{\tau}(x) = e^{\tau(x-x^{2})}$. I am curious about the asymptotics of $\widehat{\theta}_{\tau}(\tau)$, that is $\int_{\mathbb{R}} e^{\tau(x - x^{2})} …