Question: About a solution of a ODE

HI all,

After long manipulations, I was able to write the first solution y1 in this form (following the method of Frobenius):

> y1c := 1+sum((product((k-1)^2+1, k = 1 .. n))*x^n/factorial(n)^2, n = 1 .. 5);

                                                                        2
               GAMMA(1 - I) GAMMA(1 + I) x   GAMMA(2 - I) GAMMA(2 + I) x 
    y1c := 1 + --------------------------- + ----------------------------
                   GAMMA(-I) GAMMA(I)            4 GAMMA(-I) GAMMA(I)    

                                    3                              4
         GAMMA(3 - I) GAMMA(3 + I) x    GAMMA(4 - I) GAMMA(4 + I) x 
       + ---------------------------- + ----------------------------
            36 GAMMA(-I) GAMMA(I)          576 GAMMA(-I) GAMMA(I)   

                                    5
         GAMMA(5 - I) GAMMA(5 + I) x 
       + ----------------------------
           14400 GAMMA(-I) GAMMA(I)  

As you can see,  Maple made a change (I will investigate later why).  Maple introduce GAMMA(n+I)* GAMMA(n-I).  So it's simply the number with his conjugate.  The product of those two function is a real number which I confirm my self by doing

> evalf(GAMMA(5-I)*GAMMA(5+I)/(14400*GAMMA(-I)*GAMMA(I)));

                             0.1180555555 + 0. I

is there a way to take only the real part fo the product of the function GAMMA for each power of x^n so to be able to plot it in the real plane.

By the way, if I put n=infinity. I recieve that function

> y1g := 1+sum(GAMMA(n-I)*GAMMA(n+I)*x^n/(GAMMA(-I)*GAMMA(I)*factorial(n)^2), n = 1 .. infinity);

            y1g := 1 + x hypergeom([1, 1 + I, 1 - I], [2, 2], x)

that does not plot on the real plane. I am a bit confuse.  Maby  i have to do some assuming......

Any help will be greatly appreciate.

Mario