From: israel@math.ubc.ca (Robert Israel)
Subject: Re: Special values of the digamma function
Date: 18 Jul 2001 06:58:08 GMT
Newsgroups: sci.math.research

In article <Pine.SOL.3.91.1010718064047.16261A-100000@ksc9.th.com>,
Warut Roonguthai  <warut@ksc9.th.com> wrote:

>So, for any integer n, Psi(n/4) and Psi(n/6) can always be represented in 
>simple closed form.  Are there any other special values of the digamma 
>function with this property?

As a matter of fact, Psi(r) can be represented in "closed form" 
for every rational r. 

Consider r = a/b with a and b integers, 0 < a < b.
Noting that Psi(1) = -gamma, we have 
Psi(r) = -gamma + sum_{n=0}^infinity (1/(n+1) - 1/(n+a/b))
       = -gamma + b sum_{j=1}^infinity g(j)/j
 
where g(j) = 1 if j = 0 mod b, 
            -1 if j = a mod b, 
             0 otherwise
We can write g(j) = sum_w c(w) w^j where the sum is over the b'th
roots of 1, and c(w) = 1/b sum_{k=0}^{b-1} g(k) w^(-k) = 
(1 - w^(-a))/b.  Note that c(1) = 0.  Then 
sum_{j=1}^infinity g(j)/j = sum_w c(w) sum_{j=1}^infinity w^j/j
= - 1/b sum_w (1 - w^(-a)) Log(1-w) 
where here the sum is over all b'th roots of 1 except 1, and 
Log is the principal branch of the natural logarithm.  Thus

Psi(r) = -gamma - sum_w (1 - w^(-a)) Log(1-w)

Robert Israel                                israel@math.ubc.ca
Department of Mathematics        http://www.math.ubc.ca/~israel 
University of British Columbia            
Vancouver, BC, Canada V6T 1Z2
==============================================================================

From: Raymond Manzoni <raymman@club-internet.fr>
Subject: Re: Special values of the digamma function
Date: Wed, 18 Jul 2001 15:22:37 +0200
Newsgroups: sci.math.research

Warut Roonguthai wrote:
> 
(snip)
> Psi(1/6) = -gamma-sqrt(3)/2*Pi-2*log(2)-3/2*log(3).
> 
> So, for any integer n, Psi(n/4) and Psi(n/6) can always be represented in
> simple closed form.  Are there any other special values of the digamma
> function with this property?


  Yes, any rational! (Knuth's Art of Computer Programming (1981) p.94) :

Psi(p/q):=-gamma-log(2*q)-PI/2*cotan(PI*p/q)+2*sum(cos(2*PI*p*k/q)*log(sin(PI*k/q)),k=1..(ceil(q/2)-1); 
(that is 0<k<q/2)
  (0<p<q is imposed but other cases are straightforward)

  This gives (for example)
  Psi(1/3):= -gamma-3*ln(3)/2-PI*3^(1/2)/6
  Psi(1/4):= -gamma-3*log(2)-PI/2
  ...

  There's no much hope for Gamma itself but 
  there's a general formula too for all derivatives of Psi function
(well it requires Clausen functions!).

  If n>0 integer and 0<p<q then :

  Psi(p/q,n):= -1/2*PT(p/q,n)-(-q)^n*n!*SCl(p/q,n)

  if PT(x,n) is the n th derivative of PI*cotan(PI*x) taken at x=p/q :
    PT(x,n)= PI^(n+1)*(d/dx)^n(1/tan(x))

  and SCl (for sum of Clausen for example) is :

  SCl(p/q,n):=sum(cos(2*PI*k*p/q)*Cl(2*PI*k/q,n+1),k=0..q-1) (n even)
  and
  SCl(p/q,n):=sum(sin(2*PI*k*p/q)*Cl(2*PI*k/q,n+1),k=0..q-1) (n odd)

  This may be found too in Kolbig's preprint at CERN :
http://preprints.cern.ch/cgi-bin/setlink?base=preprint&categ=cern&id=CN-96-005


  Nice discovering,

  Raymond