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Results tagged with asymptotics
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user 48831
Asymptotic behavior of functions, asymptotic series and related topics
21
votes
Accepted
Asymptotic expansion of $\sum\limits_{n=1}^{\infty} \frac{x^{2n+1}}{n!{\sqrt{n}} }$
I sketch the arguments for $C(x)$, the arguments for $L(x)$ are essentially the same.
The specific form of the sum suggests probabilistic arguments.
Let $X_x$ be a $\mathrm{Poiss}(x^2)$-distributed …
- 3,255
3
votes
A balls and urns model for a hashing problem
General solution:
assume there are $n$ passwords, $k$ hashes and $x_i$ passwords hashing to $i$. The expected time (drawing without replacement) for the drawing of the first password
with hash $i$ is …
- 3,255
2
votes
Accepted
Monotonicity of the sum of coefficients of a family of generating functions
Your conjecture is true. First, observe that
\begin{align*}
K_{w+1}(z)&=(1+z)\,K_w(z) & \mbox{ if } w \mbox{ is odd, }\\
K_{w+1}(z)&=(1+z)\,K_w(z) +\frac{1}{2}{w\choose \frac{w}{2}}z^{w/2}(1-z) & …
- 3,255