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[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.

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$g(x)$ doesn't look like a geometric series, and it doesn't look like a Taylor expansion, so I'm stuck at the beginning. –  Gerry Myerson Nov 9 '10 at 3:08
    
Well, it is meant that we take each term of g(x) as formal powerseries and construct the formal powerseries g(x) as alternating sums over the coefficients at equal powers of x . g(x) = (1 - x + x^2 - x^3 + ... ) -1/2(1 -1/2 x + 1/4 x^2 - 1/8 x^3 + ...) +1/6(1 -1/3 x + 1/9 x^2 - 1/27 x^3 + ...) -1/24 ( ...) + (...) –  Gottfried Helms Nov 9 '10 at 5:50
    
I've put my notes together in go.helms-net.de/math/musings/ReducingGamma%28draft%29.pdf It is still a draft but meant to illustrate the exploration which led me to my question. Also while I'm working this out I'm getting the impression that this all might be a bit trivial for MO and better be stacked elsewhere. Sorry, if that is the case (it was my first question here) –  Gottfried Helms Nov 10 '10 at 23:47
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1 Answer

up vote 0 down vote accepted

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.

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