# QBinomial

QBinomial[n,m,q]

gives the -binomial coefficient .

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

# Examples

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## Basic Examples(6)

Exact evaluation with numbers:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Series expansion at Infinity:

Use FunctionExpand to obtain Gaussian polynomials:

## Scope(19)

### Numerical Evaluation(4)

Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Complex number inputs:

Evaluate efficiently at high precision:

### Specific Values(5)

QBinomial for symbolic parameters:

Value at zero:

Find the minimum of QBinomial[3,2,q]:

### Visualization(2)

Plot the QBinomial function for various parameters:

Plot the real part of :

Plot the imaginary part of :

### Function Properties(4) has both singularities and discontinuities for and for : is neither non-negative nor non-positive:

QBinomial is neither convex nor concave:

### Differentiation(2)

The first derivative with respect to n when m=1/2:

The first derivative with respect to m when n=1 and q=2:

Higher derivatives with respect to m:

Plot the higher derivatives with respect to m when n=3 and q=2:

### Series Expansions(2)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

The Taylor expansion at a generic point:

## Generalizations & Extensions(1)

QBinomial can be applied to a power series:

## Applications(4)

Explicit combinatorial construction of QBinomial: -binomial is a generating function for the sequence in a grid-shading problem:

Compare to explicit counting:

Elements in the -Pascal triangle satisfy two recurrence relations:

The number of subspaces in the -dimensional vector space over with prime-power :

Total number of subspaces in three-dimensional vector space over :

Check using recurrence equation for Galois numbers:

## Properties & Relations(2)

Use FunctionExpand and FullSimplify to manipulate expressions containing QBinomial:

Build series expansions: