QHypergeometricPFQ

QHypergeometricPFQ[{a1,,ar},{b1,,bs},q,z]

gives the basic hypergeometric series .

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • has the 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  (20)

Numerical Evaluation  (5)

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:

QHypergeometricPFQ threads elementwise over lists in its fourth argument:

QHypergeometricPFQ threads elementwise over sparse and structured arrays in its fourth argument:

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 TemplateBox[{{1, /, 2}, {{1, /, 4}, ,, {1, /, 3}}, {3, /, 5}, z, 1, 2}, QHypergeometricPFQSeq]:

Plot the imaginary part of TemplateBox[{{1, /, 2}, {{1, /, 4}, ,, {1, /, 3}}, {3, /, 5}, z, 1, 2}, QHypergeometricPFQSeq]:

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  (8)

Two natural -extensions of the exponential function:

The -binomial theorem:

The -Gauss sum:

RogersRamanujan identities:

A -analog of the Legendre polynomial:

Recover the Legendre polynomial as :

Euler's -logarithm of base :

Compare with the usual logarithm for base :

The Lambert series can be expressed in terms of the basic hypergeometric series:

Verify the identity through series expansion:

The Lambert series is related to the generating function for the number of divisors:

Define the StieltjesWigert polynomials:

Generate the first few polynomials:

Verify an alternative expression for the first few polynomials:

Verify the three-term recurrence relation for the first few polynomials:

Verify the generating function relation for the first few polynomials:

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:

Possible Issues  (1)

Some older references omit the factor in the defining series for the basic hypergeometric function. To express these in terms of QHypergeometricPFQ, add zero parameters until the condition is satisfied. For example, a function according to the old definition can be expressed in terms of as currently defined:

Wolfram Research (2008), QHypergeometricPFQ, Wolfram Language function, https://reference.wolfram.com/language/ref/QHypergeometricPFQ.html (updated 2023).

Text

Wolfram Research (2008), QHypergeometricPFQ, Wolfram Language function, https://reference.wolfram.com/language/ref/QHypergeometricPFQ.html (updated 2023).

CMS

Wolfram Language. 2008. "QHypergeometricPFQ." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. 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="2023", howpublished="\url{https://reference.wolfram.com/language/ref/QHypergeometricPFQ.html}", note=[Accessed: 13-April-2024 ]}

BibLaTeX

@online{reference.wolfram_2023_qhypergeometricpfq, organization={Wolfram Research}, title={QHypergeometricPFQ}, year={2023}, url={https://reference.wolfram.com/language/ref/QHypergeometricPFQ.html}, note=[Accessed: 13-April-2024 ]}