[Background:]
Looking at the powerseries for the gamma-function 


$ \Gamma(1+x) = 1 + a\_1 x + a\_2 x^2 - a\_3 \* x^3 + ... $


then we can arrive at a decomposition


$ \Gamma(1+x) = r(x) + g(x) $


where g(x) is constructed by the sum of the (taylor-expansions of) geometric series



```
          1      1      1   1     1   1    
  g(x) = ---  - --- +  ---*--- - ---*--- + ...   
         1+x    2+x    3+x  2!   4+x  3!

                                               [edit:corrected a sign-error]

```

and from this the powerseries for r(x) 


$ r(x) = \Gamma(1+x) - g(x) $ 


[end background]


That function r(x) begins with 


$ r(x) = 1/e + 0.21938 \* x + 0.09784 \*x^2 + \ldots $ 


The function has then some nice properties. By heuristics and inspection of its powerseries
it seems for instance, that


* apparently it is entire, has no zero except that $ lim\_{x-> \infty} r(-x) = 0 $
* $ r(0) = 1/e $ where $ e = \exp(1) $
* $ r(k) = r(k-1)\*k + 1/e $


Just today I found, that in fact this is the incomplete gamma-
function as defined/implemented in mathematica as "gamma(1+x,1)".
But this may not be of concern here, because I want to understand how to think the other way round:


* Question 1:


Assume we had only the functional relation and the initial value


$ r(x) = r(x-1)\*x +1/e $   

$ r(0) = 1/e $


what else would we need to make r(x) unique and arrive at the solution


$ r(x) = \Gamma(1+x)-g(x) $


?
* Question 2: 


Is there any way to generalize that construction scheme to
get some function f(x) where the functional equation depends
on a constant parameter c =/= 0


$ f(x) = f(x-1)\*x + c $


For instance let $ c=1/2 $ . What would a -for instance convex - function $ f(x)$ look like?


[update]: Question 2 seems to be easy - at the integer x f(x) is simply a scaling of r(x) by c and e: $ f(x)= r(x)\*c\*e $
so I'd assume the same can be assumed for fractional x.



What are conditions to make f(x) defined by f(x)=f(x-1)*x + 1/e unique(for instance convex)?
I'm answering my own question... :-) 


I found that book of Emile Artin on the "Theory of the Gamma-function" and a very nice text of Philip Davis(1959) on "Leonard Euler's Integral" which both dealt very explanative with the uniqueness-problem and the specific topic of convexity.


That answers my question 1) - I'll just have to get some more practice with this.


