# QHypergeometricPFQ

QHypergeometricPFQ[{a1,,ar},{b1,,bs},q,z]

gives the basic hypergeometric series .

# Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• has series expansion .
• For , the basic hypergeometric series is defined for .

# Examples

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## Basic Examples(4)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

## Scope(19)

### 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(4)

Value at zero:

For simple parameters, QHypergeometricPFQ evaluates to simpler functions:

Find a value of x for which QHypergeometricPFQ[{1/2},{3/7},5,x]=2:

### Visualization(2)

Plot the QHypergeometricPFQ function:

Plot the real part of 1ϕ2(1/2;1/2,1/3;3/5,z):

Plot the imaginary part of 1ϕ2(1/2;1/2,1/3;3/5,z):

### Function Properties(7) is an analytic function of x: has no singularities or discontinuities: is neither nonincreasing nor nondecreasing: is not injective: is not surjective:

QHypergeometricPFQ is neither non-negative nor non-positive:

QHypergeometricPFQ is neither convex nor concave:

### Series Expansions(2)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

Series expansion with respect to :

## Applications(4)

Two natural -extensions of the exponential function:

The -binomial theorem:

A -analog of the Legendre polynomial:

Recover the Legendre polynomial as :

Euler's -logarithm of base :

Compare with the usual logarithm for base :

## Properties & Relations(3)

QHypergeometricPFQ is not closed under differentiation with respect to :

It is closed under -difference:

Series expansions: -series are building blocks of other -factorial functions: