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 sum_(k=0)^infty(TemplateBox[{{a, _, 1}, q, k}, QPochhammer]... TemplateBox[{{a, _, r}, q, k}, QPochhammer])/(TemplateBox[{{b, _, 1}, q, k}, QPochhammer]... TemplateBox[{{b, _, s}, q, k}, QPochhammer])((-1)^kq^(k (k-1)/2))^(1+s-r)(z^k)/(TemplateBox[{q, q, k}, QPochhammer]).
  • 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:

TraditionalForm formatting:

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)

TemplateBox[{{{1, /, 2}}, {{2, /, 3}}, {3, /, 4}, x, 1, 1}, QHypergeometricPFQSeq] is an analytic function of x:

TemplateBox[{{{1, /, 2}}, {{2, /, 3}}, {3, /, 4}, x, 1, 1}, QHypergeometricPFQSeq] has no singularities or discontinuities:

TemplateBox[{{2}, {{2, /, 3}}, {3, /, 4}, x, 1, 1}, QHypergeometricPFQSeq] is neither nonincreasing nor nondecreasing:

TemplateBox[{{2}, {{2, /, 3}}, {3, /, 4}, x, 1, 1}, QHypergeometricPFQSeq] is not injective:

TemplateBox[{{{1, /, 2}}, {{2, /, 3}}, {3, /, 4}, x, 1, 1}, QHypergeometricPFQSeq] 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:

Wolfram Research (2008), QHypergeometricPFQ, Wolfram Language function, https://reference.wolfram.com/language/ref/QHypergeometricPFQ.html.

Text

Wolfram Research (2008), QHypergeometricPFQ, Wolfram Language function, https://reference.wolfram.com/language/ref/QHypergeometricPFQ.html.

CMS

Wolfram Language. 2008. "QHypergeometricPFQ." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/QHypergeometricPFQ.html.

APA

Wolfram Language. (2008). QHypergeometricPFQ. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/QHypergeometricPFQ.html

BibTeX

@misc{reference.wolfram_2023_qhypergeometricpfq, author="Wolfram Research", title="{QHypergeometricPFQ}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/QHypergeometricPFQ.html}", note=[Accessed: 22-September-2023 ]}

BibLaTeX

@online{reference.wolfram_2023_qhypergeometricpfq, organization={Wolfram Research}, title={QHypergeometricPFQ}, year={2008}, url={https://reference.wolfram.com/language/ref/QHypergeometricPFQ.html}, note=[Accessed: 22-September-2023 ]}