The -digamma function , also denoted , is defined as
(1)
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where is the q-gamma function. It is also given by the sum
(2)
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The -polygamma function (also denoted ) is defined by
(3)
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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)
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(5)
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The -polygamma functions are related to the Lambert series
(6)
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(7)
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(8)
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(Borwein and Borwein 1987, pp. 91 and 95).
An identity connecting -polygamma to elliptic functions is given by
(9)
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where is the golden ratio and is an Jacobi theta function.