How exactly are the beta and gamma distributions related?

According to Wikipedia, the Beta distribution is related to the gamma distribution by the following relation:

$$\lim_{n\to\infty}n B(k, n) = \Gamma(k, 1)$$

Can you point me to a derivation of this fact? Can it be generalized? For example, is there a similar relation that results in something other than a constant 1 for the Gamma second parameter? What if we have

$$\lim_{n\to\infty,m\to\infty,n=mb}n B(k, m) $$

That is, the two variables go to infinity while maintaining a constant ratio b.

The reason I'm asking is because I'm trying to figure out how to simplify a hieraerchical bayesian model involving the beta distribution.

(This is my first post; sorry for the math notation, the MathJaX syntax was too daunting, but I'll try to learn)

This concerns the relationship between the Gamma and Beta distributions as opposed to the Gamma and Beta functions. Let $X \sim \mbox{Gamma}(\alpha, 1)$ and $Y \sim \mbox{Gamma}(\beta, 1)$ where the paramaterization is such that $\alpha$ is the shape parameter. Then
$$ \frac{X}{X + Y} \sim \mbox{Beta}(\alpha, \beta). $$

To prove this, write the joint pdf $f_{X, Y} (x, y) = \frac{1}{\Gamma(\alpha) \Gamma(\beta)} x^{\alpha - 1} y^{\beta - 1} e^{-(x + y)}$ (on $\mathbb R^2_+$) and make the transformation $U = \frac{X}{X + Y}$ and $V = X + Y$. The Jacobian of the transformation $X = VU, Y = V(1 - U)$ is equal to $V$ so the joint distribution of $U$ and $V$ has pdf $$ \frac{v}{\Gamma(\alpha)\Gamma(\beta)} (vu)^{\alpha - 1} (v (1 - u))^{\beta - 1} e^{-v} = \frac{1}{\Gamma(\alpha)\Gamma(\beta)}v^{\alpha + \beta - 1} e^{-v} u^{\alpha - 1} (1 - u)^{\beta - 1} $$ (on $\mathbb R_+ \times [0, 1]$) and hence $U$ and $V$ are independent (because the pdf factors over $u$ and $v$) with $V \sim \mbox{Gamma}(\alpha + \beta, 1)$ and $U \sim \mbox{Beta}(\alpha, \beta)$ which is apparent from the terms $v^{\alpha + \beta - 1} e^{-v}$ and $u^{\alpha - 1}(1 - u)^{\beta - 1}$ respectively.

Fix some $k$ and, for every $n$, let $X_n$ denote a random variable with beta distribution $\mathrm B(k,n)$ and $Y_n=nX_n$. Then, for every $s\geqslant0$, $\mathrm E(Y_n^s)=n^s\mathrm E(X_n^s)$ and one knows the value of $\mathrm E(X_n^s)$, hence $$ \mathrm E(Y_n^s)=n^s\frac{\mathrm B(k+s,n)}{\mathrm B(k,n)}=n^s\frac{\Gamma(k+s)\Gamma(k+n)}{\Gamma(k+s+n)\Gamma(k)}\longrightarrow\frac{\Gamma(k+s)}{\Gamma(k)}. $$ This is $\mathrm E(Z^s)$ for any random variable $Z$ with gamma distribution $\Gamma(k)$ hence $Y_n\to Z$ in distribution.

Let $X'_n$ denote a random variable with beta distribution $\mathrm B(k,n/b)$, and $Y'_n=nX'_n$. Then, $X'_n$ is distributed like $X_{n/b}$ hence $Y'_n$ is distributed like $bY_{n/b}$ and $Y'_n\to bZ$ in distribution.

There is another way to view the relationship between the gamma distribution and the beta distribution through the Dirichlet distribution. This post (https://math.stackexchange.com/q/190695) talks about exactly how they are related without the Dirichlet distribution, but here is a slightly broader view:

Let $Z_1, Z_2, \ldots, Z_n$ be independent random variables such that $$ Z_i \sim \text{Gamma}(\alpha_i,1), \quad i=1,\ldots, n$$ where $\alpha_i \geq 0$ are the shape parameters. If $Y_j$ is defined as follows, $$Y_j = Z_j/\sum_{i=1}^n Z_i, \quad j=1, \ldots, n$$ then $(Y_1, \ldots, Y_n)$ is Dirichlet distributed with parameter $(\alpha_1, \ldots, \alpha_n)$. When $n=2$, the Dirichlet distribution reduces to the Beta distribution, denoted by $\text{Beta}(\alpha_1, \alpha_2)$.

Here is a link to Ferguson's paper that mentions the above relationship: http://www.cis.upenn.edu/~taskar/courses/cis700-sp08/papers/ferguson.pdf