Corner integrals of $\exp$

Question

I feel that there is a good chance to apply certain integrals of $\ \exp(-(\sum_{k=1}^n x_k))\ $ over corners (see below) to the analytic number theory. I have obtained two formulas to start with but am asking about their joint generalization. If all this is well known then let me be in the known too (and I apologize).

A real sequence $\ \mathbf A:=(A_1\ \ldots\ A_n)\in(0;\infty)^n\ $ defines a corner

$$ \Delta(\mathbf A)\ :=\ \{ (x_1\ \ldots\ x_n)\in[0;\infty)^n\ : \ \sum_{k=1}^n\frac{x_k}{A_k}\ \le\ 1 \} $$

The special $n$-dimensional case is $\ \mathbf S\ :=\ (S\ \dots\ S),\ $ where $\ A_1=\ldots=A_n=S>0.\ $ I'll write $\ \Delta(n;\ S):=\Delta(\mathbf S)\ $ to show the dimension $n$ explicitly. Then,

THEOREM 1

$$ \int_{\Delta(n;\ S)} \exp\left(-\sum_{k=1}^n x_k\right)\cdot dx_1\ldots dx_n\,\ = \,\ 1 -\ \sum_{k=0}^{n-1} \frac{S^k}{k!} \cdot e^{-S} $$

Now let's consider the general but only 2-dimensional case, where $\ \mathbf A\ :=\ (A\ B),\ $ so that we can write $\ \Delta(A\ B):= \Delta(\mathbf A)\ $. Then,

THEOREM 2

$$ \int_{\Delta(A\ B)} \exp(-(x+y))\cdot dxdy\,\ = \,\ 1\ +\ \frac B{A-B}\cdot e^{-A}\ +\ \frac A{B-A}\cdot e^{-B} $$

See the comment below about the case $\ A=B>0$.

Q U E S T I O N

I'd welcome the n-dimensional formula for the integral over the n-dimensional corner, especially in the spirit of Theorem 2 above.

A comment: Singularity (removable;)

Theorem 2 doesn't look kosher when $\ A=B>0.\ $ But it is a $2$-dimensional generalization of Theorem 1. To see it, let $\ D:=B-A.\ $ Then

$$ \int_{\Delta(A\ B)} \exp(-(x+y))\cdot dxdy\,\ = \,\ 1\ - \left(1+\frac{1-e^{-D}}D\cdot A\right)\cdot e^{-A} $$

Thus, for $\ D\rightarrow 0,\ $ the above expression approaches the 2-dimensional case of Theorem 1.

Answer 1

By a substitution $x_k = A_k y_k$, $$\begin{aligned} I & := \idotsint\limits_{\;\;\Delta(\mathbf{A})} \exp\left(-\sum_{k=1}^n x_k\right) dx_1 \ldots dx_n \\ & = \idotsint\limits_{\;\;\Delta(\mathbf{1})} \exp\left(-\sum_{k=1}^n A_k y_k\right) A_1 \ldots A_n dy_1 \ldots dy_n \\ & = \idotsint\limits_{\;\;\Delta(\mathbf{1})} \prod_{k = 1}^n \bigl(A_k \exp(-A_k y_k)\bigr) dy_1 \ldots dy_n . \end{aligned}$$ It follows that $I$ is the CDF of the sum of independent exponential random variables with parameters $A_1, \ldots, A_n$, evaluated at $1$. According to this solution of Problem 12 in Chapter I of Feller's book, if all $\lambda_k$ are distinct, the density function of the distribution of this sum is given by $$ f_{\mathbf{A}}(s) = \sum_{j = 1}^n \left(\prod_{k \in \{1, \ldots, n\} \setminus \{j\}} \frac{A_k}{A_k - A_j}\right) A_j e^{-A_j s} $$ for $s > 0$. (This also follows directly from the partial fraction expansion of the Laplace transform of $f_{\mathbf{A}}$). We conclude that $$\begin{aligned} I & = \int_0^1 f_{\mathbf{A}}(s) ds = \sum_{j = 1}^n \left(\prod_{k \in \{1, \ldots, n\} \setminus \{j\}} \frac{A_k}{A_k - A_j}\right) (1 - e^{-A_j}) . \end{aligned}$$