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BilateralHypergeometricPFQ[{a1,…,ap},{b1,…,bq},z]

is the bilateral hypergeometric function .

Details
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Examples  
Basic Examples  
Scope  
Numerical Evaluation  
Specific Values  
Integration  
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Differentiation  
Series Expansions  
Visualization  
Function Properties  
Applications  
Properties & Relations  
Possible Issues  
Neat Examples  
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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 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.

Examples

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Basic Examples  (3)

Evaluate numerically:

Wolfram Language code: BilateralHypergeometricPFQ[{1 / 2, 3 / 4}, {1 / 4, 1 / 3}, 5.4]
Wolfram Language code: BilateralHypergeometricPFQ[{1 / 2, 3 / 4}, {1 / 4, -1 / 3}, 5.4]

Plot the real and complex parts of :

Wolfram Language code: ReImPlot[BilateralHypergeometricPFQ[{1 / 2, 3 / 4}, {1 / 4, 1 / 3}, x], {x, 0, 0.9}]

Series expansion at the origin:

Wolfram Language code: Series[ BilateralHypergeometricPFQ[{a1, a2, a3}, {b1, b2}, z], {z, 0, 2}]

Scope  (18)

Numerical Evaluation  (4)

Evaluate to high precision:

Wolfram Language code: N[BilateralHypergeometricPFQ[{1 / 2, 3 / 4}, {1 / 4, 1 / 3}, 5], 30]

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

Wolfram Language code: BilateralHypergeometricPFQ[{1 / 2, 3 / 4}, {1 / 4, 1 / 3}, 5`20]

Evaluate for complex arguments and parameters:

Wolfram Language code: BilateralHypergeometricPFQ[{1 / 2I, 3 / 4I}, {1 / 4I, 1 / 3I}, 5`20I]

Evaluate BilateralHypergeometricPFQ efficiently at high precision:

Wolfram Language code: BilateralHypergeometricPFQ[{1 / 2I, 3 / 4I}, {1 / 4I, 1 / 3I}, 5`100]//Timing
Wolfram Language code: BilateralHypergeometricPFQ[{1 / 2I, 3 / 4I}, {1 / 4I, 1 / 3I}, 5`2000];//Timing

BilateralHypergeometricPFQ threads elementwise over lists in its third argument:

Wolfram Language code: BilateralHypergeometricPFQ[{1 / 2, 3 / 4}, {1 / 4, 1 / 3}, {0.1, 0.3, 0.5}]

Specific Values  (3)

BilateralHypergeometricPFQ automatically evaluates to simpler functions for certain parameters:

Wolfram Language code: BilateralHypergeometricPFQ[{1 / 2}, {1}, z]
Wolfram Language code: BilateralHypergeometricPFQ[{1 / 2, 3 / 2}, {1, 5 / 2}, z]

BilateralHypergeometricPFQ at :

Wolfram Language code: BilateralHypergeometricPFQ[{1 / 2, 1 / 3}, {1 / 4, 1 / 5, 1 / 6}, 1.]

BilateralHypergeometricPFQ at for the case :

Wolfram Language code: BilateralHypergeometricPFQ[{a1, a2}, {b1, b2}, 0]

Integration  (2)

Integrate BilateralHypergeometricPFQ:

Wolfram Language code: Integrate[BilateralHypergeometricPFQ[{1 / 2, 3 / 4}, {1 / 4, 1 / 3}, z], z]

Definite integral of BilateralHypergeometricPFQ:

Wolfram Language code: Integrate[BilateralHypergeometricPFQ[{0.5, 0.75}, {0.25, 0.3}, z], {z, 1 / 3, 1 / 2}]

Differentiation  (1)

The first derivative of a specific BilateralHypergeometricPFQ:

Wolfram Language code: D[BilateralHypergeometricPFQ[{1 / 2, 3 / 4}, {1 / 4, 1 / 3}, z], z]

The th derivative of this BilateralHypergeometricPFQ:

Wolfram Language code: D[BilateralHypergeometricPFQ[{1 / 2, 3 / 4}, {1 / 4, 1 / 3}, z], {z, n}]

Series Expansions  (3)

Calculate the series expansion of BilateralHypergeometricPFQ at the origin:

Wolfram Language code: Series[BilateralHypergeometricPFQ[{1 / 2, 3 / 4}, {1 / 4, 1 / 3}, z], {z, 0, 1}]

Calculate the series expansion of BilateralHypergeometricPFQ at Infinity:

Wolfram Language code: Series[BilateralHypergeometricPFQ[{1 / 2}, {1 / 3}, z], {z, ∞, 1}]

Calculate the series expansion of BilateralHypergeometricPFQ at a generic point:

Wolfram Language code: Series[BilateralHypergeometricPFQ[{1 / 2, 3 / 4}, {1 / 4, 1 / 3}, z], {z, z0, 1}]

Visualization  (2)

Plot the real and complex parts of :

Wolfram Language code: ReImPlot[BilateralHypergeometricPFQ[{1 / 2, 3 / 4}, {1 / 4, 1 / 3}, z], {z, -1, 0.9}]

Plot the real part of :

Wolfram Language code: ComplexContourPlot[Re[BilateralHypergeometricPFQ[{1 / 2, 3 / 4}, {1 / 4, 1 / 3}, z]], {z, -1 / 2 - 1 / 2I, 1 / 2 + 1 / 2I}]

Plot the imaginary part of :

Wolfram Language code: ComplexContourPlot[Im[BilateralHypergeometricPFQ[{1 / 2, 3 / 4}, {1 / 4, 1 / 3}, z]], {z, -1 / 2 - 1 / 2I, 1 / 2 + 1 / 2I}]

Function Properties  (3)

at converges when :

Wolfram Language code: BilateralHypergeometricPFQ[{1 / 2, 3 / 2}, {9 / 4, 13 / 4}, 1.]

has parameter-order symmetry:

Wolfram Language code: BilateralHypergeometricPFQ[{a1, a2, a3}, {b1, b2}, z] == BilateralHypergeometricPFQ[{a3, a2, a1}, {b2, b1}, z]

Traditional formatting:

Wolfram Language code: TraditionalForm[BilateralHypergeometricPFQ[{a1, a2, a3}, {b1, b2}, z]]

Applications  (1)

Compute doubly infinite sums via BilateralHypergeometricPFQ:

Wolfram Language code: Sum[(Pochhammer[a, k]/Pochhammer[b, k])z^k, {k, -∞, ∞}]
Wolfram Language code: Sum[(Pochhammer[1 / 2, n]/Pochhammer[1, n])z^n, {n, -∞, ∞}]//AbsoluteTiming
Wolfram Language code: Sum[(Pochhammer[1 / 3, n]/Pochhammer[1, n])z^n, {n, -∞, ∞}]//AbsoluteTiming
Wolfram Language code: Sum[(Pochhammer[a1, n]Pochhammer[a2, n]/Pochhammer[b1, n]Pochhammer[b2, n])z^n, {n, -∞, ∞}]//AbsoluteTiming

Properties & Relations  (2)

BilateralHypergeometricPFQ may be written as a sum of two HypergeometricPFQ:

Wolfram Language code: BilateralHypergeometricPFQ[{1 / 2, 3 / 4}, {1 / 4, 1 / 3}, 3 / 10`20]
Wolfram Language code: HypergeometricPFQ[{1 / 2, 3 / 4, 1}, {1 / 4, 1 / 3}, 3 / 10`10] + HypergeometricPFQ[{1, 3 / 4, 2 / 3}, {1 / 2, 1 / 4}, 10 / 3] - 1

BilateralHypergeometricPFQ may simplify to elementary functions:

Wolfram Language code: BilateralHypergeometricPFQ[{1 / 2}, {1}, z]

Possible Issues  (1)

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

Wolfram Language code: BilateralHypergeometricPFQ[{1 / 2, 3 / 4}, {1 / 4, 1 / 3, 1 / 5}, 0.3]//AbsoluteTiming

The evaluation is fast for the case :

Wolfram Language code: BilateralHypergeometricPFQ[{1 / 2, 3 / 4}, {1 / 4, 1 / 3}, 0.3]//AbsoluteTiming

Neat Examples  (1)

BilateralHypergeometricPFQ may autosimplify to simpler special functions:

Wolfram Language code: BilateralHypergeometricPFQlist = Inactivate[...];
Wolfram Language code: Grid[...]//TraditionalForm
Wolfram Research (2024), BilateralHypergeometricPFQ, Wolfram Language function, https://reference.wolfram.com/language/ref/BilateralHypergeometricPFQ.html.

Text

Wolfram Research (2024), BilateralHypergeometricPFQ, Wolfram Language function, https://reference.wolfram.com/language/ref/BilateralHypergeometricPFQ.html.

CMS

Wolfram Language. 2024. "BilateralHypergeometricPFQ." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/BilateralHypergeometricPFQ.html.

APA

Wolfram Language. (2024). BilateralHypergeometricPFQ. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/BilateralHypergeometricPFQ.html

BibTeX

@misc{reference.wolfram_2026_bilateralhypergeometricpfq, author="Wolfram Research", title="{BilateralHypergeometricPFQ}", year="2024", howpublished="\url{https://reference.wolfram.com/language/ref/BilateralHypergeometricPFQ.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_bilateralhypergeometricpfq, organization={Wolfram Research}, title={BilateralHypergeometricPFQ}, year={2024}, url={https://reference.wolfram.com/language/ref/BilateralHypergeometricPFQ.html}, note=[Accessed: 12-June-2026]}