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QHypergeometricPFQ
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QHypergeometricPFQ

Updated in 14

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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 .
  • QHypergeometricPFQ automatically threads over lists. »

Examples

open allclose all

Basic Examples  (4)Summary of the most common use cases

Evaluate numerically:

In[19]:=19
✖
https://wolfram.com/xid/0dc3vc01mxvk-ca6r73
Out[19]=19

Plot over a subset of the reals:

In[1]:=1
✖
https://wolfram.com/xid/0dc3vc01mxvk-eew11v
Out[1]=1

Plot over a subset of the complexes:

In[1]:=1
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https://wolfram.com/xid/0dc3vc01mxvk-kiedlx
Out[1]=1

Series expansion at the origin:

In[1]:=1
✖
https://wolfram.com/xid/0dc3vc01mxvk-f65ufv
Out[1]=1

Scope  (21)Survey of the scope of standard use cases

Numerical Evaluation  (6)

Evaluate numerically:

In[1]:=1
✖
https://wolfram.com/xid/0dc3vc01mxvk-l274ju
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0dc3vc01mxvk-cksbl4
Out[2]=2

Evaluate to high precision:

In[1]:=1
✖
https://wolfram.com/xid/0dc3vc01mxvk-b0wt9
Out[1]=1

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

In[2]:=2
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https://wolfram.com/xid/0dc3vc01mxvk-y7k4a
Out[2]=2

Complex number input:

In[1]:=1
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https://wolfram.com/xid/0dc3vc01mxvk-hfml09
Out[1]=1

Evaluate efficiently at high precision:

In[1]:=1
✖
https://wolfram.com/xid/0dc3vc01mxvk-di5gcr
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0dc3vc01mxvk-bq2c6r
Out[2]=2

QHypergeometricPFQ threads elementwise over lists in its fourth argument:

In[1]:=1
✖
https://wolfram.com/xid/0dc3vc01mxvk-bhng5j
Out[1]=1

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

In[2]:=2
✖
https://wolfram.com/xid/0dc3vc01mxvk-fgz5td
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0dc3vc01mxvk-d45z2i
Out[3]=3

Compute the elementwise values of an array using automatic threading:

In[1]:=1
✖
https://wolfram.com/xid/0dc3vc01mxvk-thgd2
Out[1]=1

Or compute the matrix QHypergeometricPFQ function using MatrixFunction:

In[2]:=2
✖
https://wolfram.com/xid/0dc3vc01mxvk-o5jpo
Out[2]=2

Specific Values  (4)

Value at zero:

In[1]:=1
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https://wolfram.com/xid/0dc3vc01mxvk-jevg27
Out[1]=1

For simple parameters, QHypergeometricPFQ evaluates to simpler functions:

In[1]:=1
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https://wolfram.com/xid/0dc3vc01mxvk-ih9u38
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0dc3vc01mxvk-ocqx3
Out[2]=2

Find a value of x for which QHypergeometricPFQ[{1/2},{3/7},5,x]=2:

In[1]:=1
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https://wolfram.com/xid/0dc3vc01mxvk-f2hrld
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0dc3vc01mxvk-i5y61v
Out[2]=2

TraditionalForm formatting:

In[1]:=1
✖
https://wolfram.com/xid/0dc3vc01mxvk-d4bwh3

Visualization  (2)

Plot the QHypergeometricPFQ function:

In[1]:=1
✖
https://wolfram.com/xid/0dc3vc01mxvk-c0x9p4
Out[1]=1

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

In[1]:=1
✖
https://wolfram.com/xid/0dc3vc01mxvk-dbvuei
Out[1]=1

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

In[2]:=2
✖
https://wolfram.com/xid/0dc3vc01mxvk-fn4pm8
Out[2]=2

Function Properties  (7)

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

In[1]:=1
✖
https://wolfram.com/xid/0dc3vc01mxvk-h5x4l2
Out[1]=1

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

In[1]:=1
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https://wolfram.com/xid/0dc3vc01mxvk-mdtl3h
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0dc3vc01mxvk-mn5jws
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/0dc3vc01mxvk-nlz7s
Out[1]=1

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

In[1]:=1
✖
https://wolfram.com/xid/0dc3vc01mxvk-poz8g
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0dc3vc01mxvk-ctca0g
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/0dc3vc01mxvk-hkqec4
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0dc3vc01mxvk-hdm869
Out[2]=2

QHypergeometricPFQ is neither non-negative nor non-positive:

In[1]:=1
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https://wolfram.com/xid/0dc3vc01mxvk-84dui
Out[1]=1

QHypergeometricPFQ is neither convex nor concave:

In[1]:=1
✖
https://wolfram.com/xid/0dc3vc01mxvk-8kku21
Out[1]=1

Series Expansions  (2)

Find the Taylor expansion using Series:

In[1]:=1
✖
https://wolfram.com/xid/0dc3vc01mxvk-ewr1h8
Out[1]=1

Plots of the first three approximations around :

In[3]:=3
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https://wolfram.com/xid/0dc3vc01mxvk-binhar
Out[4]=4

Series expansion with respect to :

In[1]:=1
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https://wolfram.com/xid/0dc3vc01mxvk-jmafrm
Out[1]=1

Applications  (8)Sample problems that can be solved with this function

Two natural -extensions of the exponential function:

In[1]:=1
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https://wolfram.com/xid/0dc3vc01mxvk-f3st20
In[2]:=2
✖
https://wolfram.com/xid/0dc3vc01mxvk-b4j5q1
In[3]:=3
✖
https://wolfram.com/xid/0dc3vc01mxvk-bh8cc2
Out[3]=3

The -binomial theorem:

In[1]:=1
✖
https://wolfram.com/xid/0dc3vc01mxvk-fvih2c
Out[1]=1

The -Gauss sum:

In[1]:=1
✖
https://wolfram.com/xid/0dc3vc01mxvk-9b5oi
Out[1]=1

Rogers–Ramanujan identities:

In[1]:=1
✖
https://wolfram.com/xid/0dc3vc01mxvk-hzi8lj
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0dc3vc01mxvk-frshd9
Out[2]=2

A -analog of the Legendre polynomial:

In[1]:=1
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https://wolfram.com/xid/0dc3vc01mxvk-b15hva
In[2]:=2
✖
https://wolfram.com/xid/0dc3vc01mxvk-gw9mfs
Out[2]=2

Recover the Legendre polynomial as :

In[3]:=3
✖
https://wolfram.com/xid/0dc3vc01mxvk-gs464z
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0dc3vc01mxvk-bvresa
Out[4]=4

Euler's -logarithm of base :

In[1]:=1
✖
https://wolfram.com/xid/0dc3vc01mxvk-fygjyv

Compare with the usual logarithm for base :

In[2]:=2
✖
https://wolfram.com/xid/0dc3vc01mxvk-hc4uit
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/0dc3vc01mxvk-pfwuco

Verify the identity through series expansion:

In[2]:=2
✖
https://wolfram.com/xid/0dc3vc01mxvk-lwn5z
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/0dc3vc01mxvk-epdwvy
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0dc3vc01mxvk-dypze5
Out[4]=4

Define the Stieltjes–Wigert polynomials:

In[1]:=1
✖
https://wolfram.com/xid/0dc3vc01mxvk-ihrhzf

Generate the first few polynomials:

In[2]:=2
✖
https://wolfram.com/xid/0dc3vc01mxvk-c878jr
Out[2]=2

Verify an alternative expression for the first few polynomials:

In[3]:=3
✖
https://wolfram.com/xid/0dc3vc01mxvk-bh6buc
Out[3]=3

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

In[4]:=4
✖
https://wolfram.com/xid/0dc3vc01mxvk-i9x7xr
Out[4]=4

Verify the generating function relation for the first few polynomials:

In[5]:=5
✖
https://wolfram.com/xid/0dc3vc01mxvk-dn6vbb
Out[5]=5

Properties & Relations  (3)Properties of the function, and connections to other functions

QHypergeometricPFQ is not closed under differentiation with respect to :

In[1]:=1
✖
https://wolfram.com/xid/0dc3vc01mxvk-c8xo9j
Out[1]=1

It is closed under -difference:

In[2]:=2
✖
https://wolfram.com/xid/0dc3vc01mxvk-bo3rgf
In[3]:=3
✖
https://wolfram.com/xid/0dc3vc01mxvk-ew7z4c
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0dc3vc01mxvk-cnre8b
Out[4]=4

Series expansions:

In[1]:=1
✖
https://wolfram.com/xid/0dc3vc01mxvk-bcggwg
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0dc3vc01mxvk-jy81an
Out[2]=2

-series are building blocks of other -factorial functions:

In[1]:=1
✖
https://wolfram.com/xid/0dc3vc01mxvk-fqpimv
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0dc3vc01mxvk-fhb7mj
Out[2]=2

Possible Issues  (1)Common pitfalls and unexpected behavior

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:

In[1]:=1
✖
https://wolfram.com/xid/0dc3vc01mxvk-bchq4y
Out[1]=1
Wolfram Research (2008), QHypergeometricPFQ, Wolfram Language function, https://reference.wolfram.com/language/ref/QHypergeometricPFQ.html (updated 2024).
✖
Wolfram Research (2008), QHypergeometricPFQ, Wolfram Language function, https://reference.wolfram.com/language/ref/QHypergeometricPFQ.html (updated 2024).

Text

Wolfram Research (2008), QHypergeometricPFQ, Wolfram Language function, https://reference.wolfram.com/language/ref/QHypergeometricPFQ.html (updated 2024).

✖
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.

✖
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

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

BibTeX

@misc{reference.wolfram_2025_qhypergeometricpfq, author="Wolfram Research", title="{QHypergeometricPFQ}", year="2024", howpublished="\url{https://reference.wolfram.com/language/ref/QHypergeometricPFQ.html}", note=[Accessed: 26-March-2025 ]}

✖
@misc{reference.wolfram_2025_qhypergeometricpfq, author="Wolfram Research", title="{QHypergeometricPFQ}", year="2024", howpublished="\url{https://reference.wolfram.com/language/ref/QHypergeometricPFQ.html}", note=[Accessed: 26-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_qhypergeometricpfq, organization={Wolfram Research}, title={QHypergeometricPFQ}, year={2024}, url={https://reference.wolfram.com/language/ref/QHypergeometricPFQ.html}, note=[Accessed: 26-March-2025 ]}

✖
@online{reference.wolfram_2025_qhypergeometricpfq, organization={Wolfram Research}, title={QHypergeometricPFQ}, year={2024}, url={https://reference.wolfram.com/language/ref/QHypergeometricPFQ.html}, note=[Accessed: 26-March-2025 ]}