New answers tagged generating-functions
0
votes
Product of three or more independent sub-Gaussian varibles
As indicated in the comment, the boundedness can be used to prove that the product is sub-Gaussian. Suppose $X_{1}, \ldots, X_{n}$ are bounded such that for $i \in [n]$ there exists a constant such ...
3
votes
Accepted
Ask for a generating function or an explicit expression of a triangle of positive integers
The generating function:
$${\cal C}(x,y) = \sum_{n,k\geq 0} C_{n,k} x^n y^{2k}$$
has the following explicit form:
$${\cal C}(x,y) = \frac{\arctan(y)}{y(1-x(1+y^2))}.$$
For "one more problem",...
5
votes
Accepted
Identities for the generating functions of a sort of convolution powers of the Narayana numbers
First off, it should be added that $C_0(t)=1$, which does not follow from the given formula.
Let's show that
$$h_{2k+1}(x,t) = L_{2k}(1-(1+t)x,-tx^2)tx + L_{2k+2}(1-(1+t)x,-tx^2).$$
First notice that $...
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