is the bilateral hypergeometric function .


  • 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 TemplateBox[{a, k}, Pochhammer] 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.


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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:

Wolfram Research (2023), BilateralHypergeometricPFQ, Wolfram Language function,


Wolfram Research (2023), BilateralHypergeometricPFQ, Wolfram Language function,


Wolfram Language. 2023. "BilateralHypergeometricPFQ." Wolfram Language & System Documentation Center. Wolfram Research.


Wolfram Language. (2023). BilateralHypergeometricPFQ. Wolfram Language & System Documentation Center. Retrieved from


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


@online{reference.wolfram_2024_bilateralhypergeometricpfq, organization={Wolfram Research}, title={BilateralHypergeometricPFQ}, year={2023}, url={}, note=[Accessed: 18-July-2024 ]}