# QPolyGamma

QPolyGamma[z,q]

gives the -digamma function .

QPolyGamma[n,z,q]

gives the 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(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 :

Plot the imaginary part of :

### Function Properties(7)

The real domain of QPolyGamma:

The complex domain:

is neither nonincreasing nor nondecreasing:

QPochhammer is not injective:

QPolyGamma is neither non-negative nor non-positive:

QPolyGamma is neither convex nor concave:

### 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 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:

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.

#### 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_2024_qpolygamma, author="Wolfram Research", title="{QPolyGamma}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/QPolyGamma.html}", note=[Accessed: 17-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_qpolygamma, organization={Wolfram Research}, title={QPolyGamma}, year={2008}, url={https://reference.wolfram.com/language/ref/QPolyGamma.html}, note=[Accessed: 17-September-2024 ]}