Re: [Reduce-algebra-developers] expansion of gamma function

[Alan B. <ala...@gm...>]

On 10/10/15 15:38, Andrey G. Grozin wrote:
>
> Can't
>
> log(gamma(1+x)) = - euler_gamma*x + 
> sum((-1)^n*zeta(n)/n*x^n,n,2,infinity)
>
> be used by taylor and/or tps, to calculate the series for gamma(1+x) 
> as the exponent of the known series?
>
> By the way, why, after loading specfn and sfgamma, psi(1) is not 
> simpilfied to -euler_gamma?
>
> Andrey
>
In tps at least this should be possible (RTFM!).

Redfront 3.2, built 23-Nov-2014 ...
Reduce (Free PSL version), 23-Nov-2014 ...

1: load_package specfn, sfgamma, tps;

2: r:=pssum(n=2, (-1)^n*zeta(n)/n,x,0,n);

         2                         4 6
       pi    2    zeta(3)   3    pi    4    zeta(5)   5 pi     6      7
r := -----*x  - ---------*x  + -----*x  - ---------*x  + ------*x  + O(x )
       12            3           360           5 5670

3: ps(r-euler_gamma*x,x,0);
                       2 4                         6
                     pi    2    zeta(3)   3    pi    4 zeta(5)   5    
pi     6
  - euler_gamma*x + -----*x  - ---------*x  + -----*x  - ---------*x  + 
------*x
                     12            3           360 5           5670

       7
  + O(x )

4: psexplim 12;

6

5: ws 3;

                       2 4                         6
                     pi    2    zeta(3)   3    pi    4 zeta(5)   5    
pi     6
  - euler_gamma*x + -----*x  - ---------*x  + -----*x  - ---------*x  + 
------*x
                     12            3           360 5           5670

                       8                           10
     zeta(7)   7     pi     8    zeta(9)   9     pi 10    zeta(11)   11
  - ---------*x  + -------*x  - ---------*x  + --------*x - ----------*x
        7           75600           9 935550            11

            12
      691*pi      12      13ps(exp(ws 3
  + ------------*x   + O(x  )
     7662154500


However there seems to be a problem with manipulating this series 
(probably a bug in tps connected with simplification of pssum).

It ought to be possible to do

6: ws^2;
^C***** Interrupt in assoc*

7: ps(exp(ws 3),x,0);

^C
***** Interrupt in mksq

but these seem to hang forever whereas simpler cases are fine; eg

8: ps(sin x,x,0);

      1   3     1    5     1     7      1      9 1       11      13
x - ---*x  + -----*x  - ------*x  + --------*x  - ----------*x   + O(x  )
      6        120        5040        362880        39916800

9: ws^2;

  2    1   4    2    6     1    8      2     10      2 12      13
x  - ---*x  + ----*x  - -----*x  + -------*x   - --------*x   + O(x  )
       3        45        315        14175         467775

10: ps(exp(ws 8),x,0);

          1   2    1   4    1    5     1    6    1    7 31    8     1     9
1 + x + ---*x  - ---*x  - ----*x  - -----*x  + ----*x  + ------*x  + 
------*x
          2        8        15        240        90 5760        5670

      2951     10     1     11      181      12      13
  - ---------*x   - ------*x   + ----------*x   + O(x  )
     3628800         3150         14515200


I will investigate the cause of the problem over the next week or so.

Alan