All Questions
Tagged with bell-numbers calculus
6
questions
2
votes
1
answer
176
views
Integral representation of Bell Polynomial?
From Wolfram Alpha: https://functions.wolfram.com/IntegerFunctions/BellB/07/01/0001/,
we have an integral representation for Bell numbers as:
$B_n = \frac{2n!}{\pi e} \displaystyle{\int_{0}^{\pi}} e^{...
- 317
2
votes
1
answer
313
views
Extension of the Multivariate Faa di bruno's formula with more than two composite functions
The Faa di bruno formula for one variable (Wikipedia) is
The combinatorial forms in terms of bell polynomials are also included
Similarly, the multivariate formula (Wikipedia) is expressed ...
- 89
2
votes
1
answer
240
views
prove for bell number using induction on n [duplicate]
hi guys I have to prove this equality $$B_n=e^{-1}\sum_{k=0}^{\infty}\frac{k^n}{k!},$$ that is called bell equality only using induction on $n$ . How can i do this? I have tried by substituting the ...
6
votes
3
answers
221
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Compact formula for the $n$th derivative of $f\left(\sqrt{x+1}\right)$?
I'm interested in a general formula for
$$\frac{d^n}{dx^n}\Big[f\left(\sqrt{x+1}\right)\Big].$$
In particular, Fàa di Bruno's formula gives
$$\frac{d^n}{dx^n}\Big[f\left(\sqrt{x+1}\right)\Big]=\sum_{k=...
- 6,692
1
vote
1
answer
55
views
Alternative representation for $B_{n,k}(1!,\dots,(n-k+1)!)$
Let $B_{n,k}$ be the Exponential Bell Polynomial and $\hat {B}_{n,k}$ the Ordinary Bell Polynomial. One has the identity:
$$
B_{n,k}(0!,1!,\dots,(n-k)!)
=
|s(n,k)|,
$$
with $s(n,k)$ the Stirling ...
- 1,135
8
votes
1
answer
763
views
Sum of Bell Polynomials of the Second Kind
A problem of interest that has come up for me recently is solving the following.
$$\frac{d^{n}}{dt^{n}}e^{g(t)}$$
There is a formula for a general $n$-th order derivative of a composition as shown ...
- 2,120