The -digamma function , also denoted , is defined as
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(1) |
where is the q-gamma function. It is also given by the sum
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(2) |
The -polygamma function (also denoted ) is defined by
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(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
The -polygamma functions are related to the Lambert series
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(6) | |||
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(7) | |||
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(8) |
(Borwein and Borwein 1987, pp. 91 and 95).
An identity connecting -polygamma to elliptic functions is given by
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(9) |
where is the golden ratio and is an Jacobi theta function.
See also
False Logarithmic Series, Polygamma Function
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References
Borwein, J. M. and Borwein, P. B. "Evaluation of Sums of Reciprocals of Fibonacci Sequences." ยง3.7 in Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 91-101, 1987.
Referenced on Wolfram|Alpha
Cite this as:
Weisstein, Eric W. "q-Polygamma Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/q-PolygammaFunction.html