Does this distribution have a name? Or what is a stochastic process that could generate it? A discrete distribution with mass function
$$p(x;k) = \frac{k}{(x+k)(x+k-1)},\quad x = 1,2,\ldots$$
arises on page 9 of this paper.
For $k=1$ it is a Yule-Simon distribution with $\rho=1$, but I haven't found any other examples.
Does it have a name? Does it appear in any other contexts? Is there a simple stochastic process that might generate it?
 A: It's a discrete power law.
(This is a description--whose meaning will be made precise below--rather than a technical term.  The phrase "discrete power law" has a slightly different technical meaning, as indicated by @Cardinal in comments to this answer.)
To see this, observe that the partial fraction decomposition can be written
$$p(x;k) = \frac{k}{(x+k)(x+k-1)} = \frac{1}{1 + (x-1)/k} - \frac{1}{1 + x/k}.$$
The CDF telescopes into a closed form:
$$\eqalign{
&\text{CDF}(i) = \sum_{x=1}^i p(x;k) \\
              = &[\frac{1}{1 + 0/k} - \frac{1}{1 + 1/k}] +  [\frac{1}{1 + 1/k} - \frac{1}{1 + 2/k}] + \cdots + [\frac{1}{1 + (i-1)/k} - \frac{1}{1 + i/k}] \\
              = &\frac{1}{1 + 0/k} + [- \frac{1}{1 + 1/k} +  \frac{1}{1 + 1/k}] + [ - \frac{1}{1 + 2/k} + \cdots + \frac{1}{1 + (i-1)/k}] - \frac{1}{1 + i/k} \\
              = &1 + 0 + \cdots + 0 - \frac{1}{1 + i/k} \\
              = &\frac{i}{i+k}.
}$$
(Incidentally, because this is easily inverted, it immediately provides an efficient way to generate random variables from this distribution: simply compute $\lceil \frac{k u}{1 - u} \rceil$ where $u$ is uniformly distributed on $(0,1)$.)
Differentiating this expression with respect to $i$ shows how the CDF can be written as an integral,
$$\text{CDF}(i) = \frac{i}{i+k} = \int_0^i \frac{dt/k}{(1 + t/k)^2} = \sum_{x=1}^i \int_{x-1}^x \frac{dt/k}{(1 + t/k)^2},$$
whence
$$p(x;k) = \int_{x-1}^x \frac{dt/k}{(1 + t/k)^2}.$$
This form of writing it exhibits $k$ as a scale parameter for the family of (continuous) distributions determined by the density
$$f(\xi)d\xi = (1 + \xi)^{-2}\, d\xi$$
and shows how $p(x;k)$ is the discretized version of $f$ (scaled by $k$) obtained by integrating the continuous probability over the interval from $x-1$ to $x$.  That's obviously a power law with exponent $-2$.  This observation gives you an entrance into extensive literature on power laws and how they arise in science, engineering, and statistics, which may suggest many answers to your last two questions.
A: Okay, after a bit more investigation, I found some more details.
It's a special case of a continuous mixture of a geometric distribution with a Beta, so could be called a Beta-geometric distribution. Specifically, if:
$$P \sim \mathrm{Beta}(1,k) $$
and:
$$X|P \sim \mathrm{Geometric}(P)$$
then the marginal distribution of $Y = X+1$ has this distribution. As such, it's a special case of a Beta-Negative binomial distribution.
It has a couple of other interesting properties:


*

*It has an infinite mean

*It describes its own tail distribution: if $X$ has this distribution with parameter $k$, then $X-t | X>t$ has parameter $t+k$.

