QPolyGamma

QPolyGamma[z,q]

gives the -digamma function .

QPolyGamma[n,z,q]

gives the ^(th) derivative of the -digamma function .

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • .
  • .
  • QPolyGamma automatically threads over lists.

Examples

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Basic 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  (20)

Numerical Evaluation  (4)

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:

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, {x, +, {i,  , y}}, {1, /, 2}}, QPolyGamma3]:

Plot the imaginary part of TemplateBox[{0, {x, +, {i,  , y}}, {1, /, 2}}, QPolyGamma3]:

Function Properties  (3)

The real domain of QPolyGamma:

The complex domain:

QPolyGamma threads elementwise over lists:

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  (1)

Express certain sums in closed form:

In general, all basic -rational sums can be computed using QPolyGamma:

Use DifferenceDelta to verify:

Properties & Relations  (2)

Differences of QPolyGamma are -rational functions:

Derivatives of QGamma involve QPolyGamma:

Wolfram Research (2008), QPolyGamma, Wolfram Language function, https://reference.wolfram.com/language/ref/QPolyGamma.html.

Text

Wolfram Research (2008), QPolyGamma, Wolfram Language function, https://reference.wolfram.com/language/ref/QPolyGamma.html.

BibTeX

@misc{reference.wolfram_2021_qpolygamma, author="Wolfram Research", title="{QPolyGamma}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/QPolyGamma.html}", note=[Accessed: 25-October-2021 ]}

BibLaTeX

@online{reference.wolfram_2021_qpolygamma, organization={Wolfram Research}, title={QPolyGamma}, year={2008}, url={https://reference.wolfram.com/language/ref/QPolyGamma.html}, note=[Accessed: 25-October-2021 ]}

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