I'm trying to simplify the following sum:
$P(a)=\sum_{n=0}^{\infty}\dfrac{x^{n-a}}{(n-a)!}$
I'm tempted to turn this into some sort of exponential function because it closely ressembles the Maclaurin series $e^x=\sum_{n=0}^{\infty}\dfrac{x^n}{n!}$. Below is my attempt at this:
let $N=n-a$ so index now starts at $N=a$
$P(a)=\sum_{N=a}^{\infty}\dfrac{x^N}{N!}$
So now I have succeeded in making a function that is of the same form. Unfortunately, I don't really know what to do about the sum starting at N=a. Rewriting, I get:
$P(a)=\sum_{N=0}^{\infty}\dfrac{x^N}{N!}-\sum_{N=0}^a\dfrac{x^N}{N!}$
Now I definitely can write the first term as an exponential yaaay! But I'm still left with a finite sum that I have to subtract. So I guess what I'm really asking is what do I do about the second term?
Edit: It seems my first equation is incorrect. Here is the context surrounding my question:
Note: The variables were relabeled in my original question and constants factors were pulled out to make things a little easier on the eye. I'm currently on part b). Subbing the given $P(n|N)$ and $P(N)$ expressions into P(n), I find that a negative factorial is inevitable but maybe I'm missing something.
The partial sum (lower or upper) of the series for $e^x$ is related to the Regularized Incomplete Gamma function $$ Q\left( {n,z} \right)\quad \left| {\;0 \le n \in Z} \right.\;\; = {{\Gamma \left( {n,z} \right)} \over {\Gamma \left( n \right)}}\quad = e^{\,\, - z} \sum\limits_{0\, \le \,k\, \le \,n - 1} {{{z^{\,k} } \over {k!\,}}} $$
Coming to your problem, we have $$ P(n\;\left| N \right.) = \left( \matrix{ N \cr n \cr} \right) \eta ^{\,n} \left( {1 - \eta } \right)^{\,N - n} $$ which is clearly the probability of $n$ successes in $N$ trials.
We assume $N$ to follow a Poisson distribution with average $\overline N $ $$ P(N) = e^{\, - \,\overline N } {{\overline N ^{\,N} } \over {N!}} $$ Now we have $$ P(n\;\left| N \right.) = {{P\left( {n \wedge N} \right)} \over {P(N)}} $$ and as rightly hinted in the text, for $P(n)$ we will have $$ \eqalign{ & P(n) = \sum\limits_{N = 0}^\infty {P\left( {n \wedge N} \right)} = \sum\limits_{N = 0}^\infty {P(n\;\left| N \right.)P(N)} = \cr & = \sum\limits_{N = 0}^\infty {\left( \matrix{ N \cr n \cr} \right)\eta ^{\,n} \left( {1 - \eta } \right)^{\,N - n} e^{\, - \,\overline N } {{\overline N ^{\,N} } \over {N!}}} = \cr & = \eta ^{\,n} \left( {1 - \eta } \right)^{\, - n} e^{\, - \,\overline N } \sum\limits_{N = 0}^\infty {\left( \matrix{ N \cr n \cr} \right) {{\left( {\overline N \left( {1 - \eta } \right)} \right)^{\,N} } \over {N!}}} = \cr & = \eta ^{\,n} \left( {1 - \eta } \right)^{\, - n} e^{\, - \,\overline N } \sum\limits_{N = n}^\infty {\left( \matrix{ N \cr n \cr} \right) {{\left( {\overline N \left( {1 - \eta } \right)} \right)^{\,N} } \over {N!}}} = \cr & = {{\eta ^{\,n} \left( {1 - \eta } \right)^{\, - n} e^{\, - \,\overline N } } \over {n!}} \sum\limits_{N = n}^\infty {{{\left( {\overline N \left( {1 - \eta } \right)} \right)^{\,N} } \over {\left( {N - n} \right)!}}} = \cr & = {{\eta ^{\,n} \left( {1 - \eta } \right)^{\, - n} e^{\, - \,\overline N } } \over {n!}} \sum\limits_{N - n = 0}^\infty {{{\left( {\overline N \left( {1 - \eta } \right)} \right)^{\,N - n + n} } \over {\left( {N - n} \right)!}}} = \cr & = {{\eta ^{\,n} \left( {1 - \eta } \right)^{\, - n} \left( {\overline N \left( {1 - \eta } \right)} \right)^{\,n} } \over {n!}}e^{\, - \,\overline N } \sum\limits_{N - n = 0}^\infty {{{\left( {\overline N \left( {1 - \eta } \right)} \right)^{\,N - n} } \over {\left( {N - n} \right)!}}} = \cr & = {{\left( {\eta \overline N } \right)^{\,n} } \over {n!}}e^{\, - \,\overline N } \sum\limits_{k = 0}^\infty {{{\left( {\overline N \left( {1 - \eta } \right)} \right)^k } \over {k!}}} = \cr & = {{\left( {\eta \overline N } \right)^{\,n} } \over {n!}}e^{\, - \,\overline N } e^{\,\overline N \left( {1 - \eta } \right)} = \cr & = {{\left( {\eta \overline N } \right)^{\,n} } \over {n!}} e^{\, - \left( {\eta \,\overline N } \right)} \cr} $$
That is, the key passage you got astray is at $$ \eqalign{ & \sum\limits_{N = n}^\infty {{{\left( {\overline N \left( {1 - \eta } \right)} \right)^{\,N} } \over {\left( {N - n} \right)!}}} \; \Rightarrow \sum\limits_{N - n = 0}^\infty {{{\left( {\overline N \left( {1 - \eta } \right)} \right)^{\,N - n + n} } \over {\left( {N - n} \right)!}}} \Rightarrow \cr & \Rightarrow \left( {\overline N \left( {1 - \eta } \right)} \right)^{\,n} \sum\limits_{k = 0}^\infty {{{\left( {\overline N \left( {1 - \eta } \right)} \right)^{\,k} } \over {k!}}} \cr} $$ But it is good in any case that you learned about the occurring of $Q$ !