A special function 
corresponding to a polygamma function 
with 
, given by
 | (1) |
An alternative function
 | (2) |
is sometimes called the trigamma function, where
 | (3) |
Sums and differences of 
for small integers 
and 
can be expressed in terms of 
, Catalan's constant 
, and Clausen functions. For example,
 |  |  | (4) |
 |  |  | (5) |
 |  |  | (6) |
 |  |  | (7) |
 |  |  | (8) |
 |  |  | (9) |
 |  |  | (10) |
 |  |  | (11) |
 |  |  | (12) |
 |  |  | (13) |
In general,
 | (14) |
(Lewin 1991).
) and Trigamma (
) Functions."