All Questions
3 questions
8
votes
2
answers
484
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Inductive definition of Bernstein polynomials
For $n\in \mathbb{N}$ let $B_n$ be the linear operator taking a function $f$ on the unit interval $I=[0,1]$ to its $n$-th Bernstein polynomial $B_nf$,
$$ B_nf(x):=\sum_{k=0}^n\binom{n}{k} f\Big(\...
7
votes
5
answers
682
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Bound on sum of complex summands involving binomial coefficients
I am trying to find the asymptotic behavior of the sum:
$$ \sum^n_{i=0} \begin{pmatrix} 2n \\ i \end{pmatrix} x^i y^{2n-i}$$
as $n\rightarrow\infty$. Here $x$, $y$ are complex numbers and I have $|x|...
1
vote
1
answer
918
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Pros and cons of probability model for permutations
I am studying probability model of random permetuation
Let $b(n; k)$ denote the number of permutations of {1,...,n} with precisely k
inversions ($inv(\pi)$). The analytic approach was considered by L....