# 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(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 :

Plot the imaginary part of :

### Function Properties(3)

The real domain of QPolyGamma:

The complex domain:

### 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(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: