BilateralHypergeometricPFQ

BilateralHypergeometricPFQ[{a₁,…,a_p},{b₁,…,b_q},z]

is the bilateral hypergeometric function .

Details

The bilateral hypergeometric series has a similar definition for its terms as the generalized hypergeometric series but sums over all integers, thus forming a doubly infinite series.
Mathematical function, suitable for both symbolic and numerical manipulation.
has the series expansion $sum_(k=-infty)^(infty)TemplateBox[{{a, _, 1}, k}, Pochhammer]...TemplateBox[{{a, _, p}, k}, Pochhammer]/TemplateBox[{{b, _, 1}, k}, Pochhammer]...TemplateBox[{{b, _, q}, k}, Pochhammer]z^k$ , where is the Pochhammer symbol.
The bilateral hypergeometric series is convergent if and .
The bilateral hypergeometric function for the case when is calculated using Borel regularization.
None of the parameters can be positive integers and none of the can be negative integers.
BilateralHypergeometricPFQ can be evaluated to arbitrary numerical precision.
For certain special arguments, BilateralHypergeometricPFQ automatically evaluates to exact values.
BilateralHypergeometricPFQ automatically threads over lists.

Examples

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Basic Examples (3)

Evaluate numerically:

Plot the real and complex parts of :

Series expansion at the origin:

Scope (18)

Numerical Evaluation (4)

Evaluate to high precision:

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

Evaluate for complex arguments and parameters:

Evaluate BilateralHypergeometricPFQ efficiently at high precision:

BilateralHypergeometricPFQ threads elementwise over lists in its third argument:

Specific Values (3)

BilateralHypergeometricPFQ automatically evaluates to simpler functions for certain parameters:

BilateralHypergeometricPFQ at :

BilateralHypergeometricPFQ at for the case :

Integration (2)

Integrate BilateralHypergeometricPFQ:

Definite integral of BilateralHypergeometricPFQ:

Differentiation (1)

The first derivative of a specific BilateralHypergeometricPFQ:

The th derivative of this BilateralHypergeometricPFQ:

Series Expansions (3)

Calculate the series expansion of BilateralHypergeometricPFQ at the origin:

Calculate the series expansion of BilateralHypergeometricPFQ at Infinity:

Calculate the series expansion of BilateralHypergeometricPFQ at a generic point:

Visualization (2)

Plot the real and complex parts of :

Plot the real part of :

Plot the imaginary part of :

Function Properties (3)

at converges when :

has parameter-order symmetry:

Traditional formatting:

Applications (1)

Compute doubly infinite sums via BilateralHypergeometricPFQ:

Properties & Relations (2)

BilateralHypergeometricPFQ may be written as a sum of two HypergeometricPFQ:

BilateralHypergeometricPFQ may simplify to elementary functions:

Possible Issues (1)

When , BilateralHypergeometricPFQ uses Borel regularization, which may be time-consuming:

The evaluation is fast for the case :

Neat Examples (1)

BilateralHypergeometricPFQ may autosimplify to simpler special functions:

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