q-Polygamma Function

The -polygamma functions are the derivatives of the -digamma function , also denoted . For real with , the latter is defined by

(1)

where is the q-gamma function and is away from its poles. For , it is also given by the sum

(2)

For , the corresponding convention is

(3)

(Qi 2015).

For a nonnegative integer , the -polygamma function is defined by

(4)

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 the following, with real in the first identity:

			(5)
			(6)

The -polygamma functions are related to the Lambert series

			(7)
			(8)
			(9)

for (Borwein and Borwein 1987, pp. 91 and 95).

An identity connecting -polygamma to elliptic functions is given by

(10)

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