Question: BesselK(p,x) when p is an integer

Hi all,

I know that if p is not an integer, the solution of the modified Bessel equation is

> BesselK(p, x) = (1/2)*Pi*(BesselI(-p, x)-BesselI(p, x))/sin(p*Pi);

                             Pi (BesselI(-p, x) - BesselI(p, x))
             BesselK(p, x) = -----------------------------------
                                         2 sin(p Pi)            

But if p is an integer

> series(BesselK(1, x), x, 10);

 (-1)   /  1         1         1         1\  
x     + |- - ln(2) + - ln(x) + - gamma - -| x
        \  2         2         2         4/  

     /  1          1          5    1       \  3
   + |- -- ln(2) + -- ln(x) - -- + -- gamma| x 
     \  16         16         64   16      /   

     /   5      1           1           1       \  5
   + |- ---- + --- gamma - --- ln(2) + --- ln(x)| x 
     \  1152   384         384         384      /   

     /    47       1             1             1        \  7
   + |- ------ + ----- gamma - ----- ln(2) + ----- ln(x)| x 
     \  442368   18432         18432         18432      /   

     /     1               1              131         1         \  9    / 10\
   + |- ------- ln(2) + ------- ln(x) - -------- + ------- gamma| x  + O\x  /
     \  1474560         1474560         88473600   1474560      /            

where gamma is Euler constant

Does anyone has the Sum form for obtainig this series?

A Maple worksheet would be appreciate if the computatiion is long.

Thanks in advance

Mario