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 .
- For , the basic hypergeometric series is defined for .
- QHypergeometricPFQ automatically threads over lists. »
Examples
open allclose allBasic Examples (4)
Scope (21)
Numerical Evaluation (6)
The precision of the output tracks the precision of the 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:
Compute the elementwise values of an array using automatic threading:
Or compute the matrix QHypergeometricPFQ function using MatrixFunction:
Specific Values (4)
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)
Function Properties (7)
has no singularities or discontinuities:
is neither nonincreasing nor nondecreasing:
QHypergeometricPFQ is neither non-negative nor non-positive:
QHypergeometricPFQ is neither convex nor concave:
Series Expansions (2)
Find the Taylor expansion using Series:
Applications (8)
Two natural -extensions of the exponential function:
A -analog of the Legendre polynomial:
Recover the Legendre polynomial as :
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 Stieltjes–Wigert 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 :
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:
Text
Wolfram Research (2008), QHypergeometricPFQ, Wolfram Language function, https://reference.wolfram.com/language/ref/QHypergeometricPFQ.html (updated 2024).
CMS
Wolfram Language. 2008. "QHypergeometricPFQ." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2024. 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