The digamma function , also denoted , is defined as
(1)

where is the qgamma function. It is also given by the sum
(2)

The polygamma function (also denoted ) is defined by
(3)

It is implemented in the Wolfram Language as QPolyGamma[n, z, q], with the digamma function implemented as the special case QPolyGamma[z, q].
Certain classes of sums can be expressed in closed form using the polygamma function, including
(4)
 
(5)

The polygamma functions are related to the Lambert series
(6)
 
(7)
 
(8)

(Borwein and Borwein 1987, pp. 91 and 95).
An identity connecting polygamma to elliptic functions is given by
(9)

where is the golden ratio and is an Jacobi theta function.