# Tagged Questions

**0**

votes

**0**answers

31 views

### Is there inverse FFT algorithm for Fourier transform of a integer-valued random variable?

In many applications, it is possible to derive an explicit expression for the
fourier transform of a random variable $X$
$$\varphi (\theta ) = \sum\limits_{n = 0}^\infty {{p_n}} {e^{in\theta }}$$
...

**2**

votes

**2**answers

156 views

### Generating function of factorable binary words

A word $w$ on the alphabet $A := \{0, 1\}$ is factorable if
\begin{equation}
w = u^k \mbox{ where } u \in A^* \mbox{ and } k \geq 2.
\end{equation}
Let $L$ be the language of the set of ...

**1**

vote

**0**answers

75 views

### Generating series of free PROs

Let
\begin{equation}
G := \biguplus_{p \geq 0} \: \biguplus_{q \geq 0} G(p, q)
\end{equation}
be a bigraded set of generators and $\mathcal{F}(G)$ be the free PRO generated by $G$ (see [1] for a net ...

**8**

votes

**4**answers

372 views

### Partitions-Sum of divisors identity

A few years ago I first read about the marvelous Euler identity:
$\sum_{n\in\mathbb{N}}p(n)z^n=\prod_{k\geq1}\frac{1}{1-z^k}$,
where $p(n)$ is the number of partitions of $n$ ($p(0)=1$ by ...

**3**

votes

**1**answer

244 views

### The relationship between Stirling number of the second kind and the polylogarithm

It is shown here on Mathworld's page on Stirling number of the second kind that
$$
\sum_{k=1}^n S(n,k) (k-1)! z^k = (-1)^n \text{Li}_{1-n}(1+\frac{1}{z})
$$
where $S(n,k)$ is Stirling number of the ...

**35**

votes

**10**answers

5k views

### The functional equation $f(f(x))=x+f(x)^2$

I'd like to gather information and references on the following functional equation for power series $$f(f(x))=x+f(x)^2,$$$$f(x)=\sum_{k=1}^\infty c_k x^k$$
(so $c_0=0$ is imposed).
First things that ...