# BilateralHypergeometricPFQ

BilateralHypergeometricPFQ[{a1,,ap},{b1,,bq},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 , 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 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:

## 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:

Wolfram Research (2023), BilateralHypergeometricPFQ, Wolfram Language function, https://reference.wolfram.com/language/ref/BilateralHypergeometricPFQ.html.

#### Text

Wolfram Research (2023), BilateralHypergeometricPFQ, Wolfram Language function, https://reference.wolfram.com/language/ref/BilateralHypergeometricPFQ.html.

#### CMS

Wolfram Language. 2023. "BilateralHypergeometricPFQ." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/BilateralHypergeometricPFQ.html.

#### APA

Wolfram Language. (2023). BilateralHypergeometricPFQ. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/BilateralHypergeometricPFQ.html

#### BibTeX

@misc{reference.wolfram_2024_bilateralhypergeometricpfq, author="Wolfram Research", title="{BilateralHypergeometricPFQ}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/BilateralHypergeometricPFQ.html}", note=[Accessed: 18-July-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_bilateralhypergeometricpfq, organization={Wolfram Research}, title={BilateralHypergeometricPFQ}, year={2023}, url={https://reference.wolfram.com/language/ref/BilateralHypergeometricPFQ.html}, note=[Accessed: 18-July-2024 ]}