QPolyGamma [z,q]
gives the -digamma function TemplateBox[{z, q}, QPolyGamma].
QPolyGamma [n,z,q]
gives the ^(th) derivative of the -digamma function TemplateBox[{n, z, q}, QPolyGamma3].
QPolyGamma
QPolyGamma [z,q]
gives the -digamma function TemplateBox[{z, q}, QPolyGamma].
QPolyGamma [n,z,q]
gives the ^(th) derivative of the -digamma function TemplateBox[{n, z, q}, QPolyGamma3].
Details
- Mathematical function, suitable for both symbolic and numerical manipulation.
- TemplateBox[{z, q}, QPolyGamma]=partial_z TemplateBox[{z, q}, QGamma]/TemplateBox[{z, q}, QGamma]⩵-log(1-q)+log(q)sum_(n=0)^inftyq^(n+z)/(1-q^(n+z)).
- TemplateBox[{n, z, q}, QPolyGamma3]=d^nTemplateBox[{z, q}, QPolyGamma]/d z^n.
- QPolyGamma automatically threads over lists.
Examples
open all close allBasic Examples (6)
Evaluate numerically:
Plot over a subset of the reals:
Plot over a subset of the complexes:
Series expansion at the origin:
Series expansion at Infinity :
Series expansion at a singular point:
Scope (26)
Numerical Evaluation (6)
Evaluate numerically:
Evaluate to high precision:
The precision of the output tracks the precision of the input:
Complex number input:
Evaluate efficiently at high precision:
Compute average-case statistical intervals using Around :
Compute the elementwise values of an array:
Or compute the matrix QPolyGamma function using MatrixFunction :
Specific Values (5)
Evaluate at exact arguments:
Evaluate symbolically:
Some singular points of QPolyGamma :
Values at infinity:
Find a value of x for which QPolyGamma [x,6]=3:
Visualization (3)
Plot the QPolyGamma function:
Plot the QPolyGamma as a function of its second parameter q:
Plot the real part of TemplateBox[{0, z, {1, /, 2}}, QPolyGamma3]:
Plot the imaginary part of TemplateBox[{0, z, {1, /, 2}}, QPolyGamma3]:
Function Properties (7)
The real domain of QPolyGamma :
The complex domain:
QPolyGamma threads elementwise over lists:
TemplateBox[{x, {2, /, 3}}, QPolyGamma] is neither nonincreasing nor nondecreasing:
QPochhammer is not injective:
QPolyGamma is neither non-negative nor non-positive:
QPolyGamma is neither convex nor concave:
TraditionalForm formatting:
Differentiation (3)
First derivative with respect to z:
Higher derivatives with respect to z:
Plot the higher derivatives with respect to z when q=3:
Formula for the ^(th) derivative with respect to z:
Series Expansions (2)
Find the Taylor expansion using Series :
Plots of the first three approximations around :
The Taylor expansion at a generic point:
Applications (3)
Express certain sums in closed form:
In general, all basic -rational sums can be computed using QPolyGamma :
Use DifferenceDelta to verify:
Compute an approximation for a finite sum:
Compute the numerical approximation for increasing values of n:
Compare with the exact results given by Sum :
The Lambert series can be expressed in terms of the -digamma function:
Verify the identity through series expansion:
The Lambert series is related to the generating function for the number of divisors:
Properties & Relations (2)
Differences of QPolyGamma are -rational functions:
Derivatives of QGamma involve QPolyGamma :
Related Guides
Related Links
History
Text
Wolfram Research (2008), QPolyGamma, Wolfram Language function, https://reference.wolfram.com/language/ref/QPolyGamma.html.
CMS
Wolfram Language. 2008. "QPolyGamma." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/QPolyGamma.html.
APA
Wolfram Language. (2008). QPolyGamma. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/QPolyGamma.html
BibTeX
@misc{reference.wolfram_2025_qpolygamma, author="Wolfram Research", title="{QPolyGamma}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/QPolyGamma.html}", note=[Accessed: 04-January-2026]}
BibLaTeX
@online{reference.wolfram_2025_qpolygamma, organization={Wolfram Research}, title={QPolyGamma}, year={2008}, url={https://reference.wolfram.com/language/ref/QPolyGamma.html}, note=[Accessed: 04-January-2026]}