# HypergeometricPFQ

HypergeometricPFQ[{a1,,ap},{b1,,bq},z]

is the generalized hypergeometric function .

# Examples

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## Basic Examples(5)

Evaluate numerically:

Plot over a subset of the reals:

Evaluate symbolically:

Series expansion at the origin:

Series expansion at Infinity:

## Scope(33)

### Numerical Evaluation(5)

Evaluate to high precision:

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

Evaluate for complex arguments and parameters:

Evaluate HypergeometricPFQ efficiently at high precision:

HypergeometricPFQ threads elementwise over lists in its third argument:

HypergeometricPFQ threads elementwise over sparse and structured arrays in its third argument:

HypergeometricPFQ can be used with Interval and CenteredInterval objects:

### Specific Values(4)

For simple parameters, HypergeometricPFQ evaluates to simpler functions:

HypergeometricPFQ evaluates to a polynomial if any of the parameters ak is a non-positive integer:

Value at the origin:

Find a value of satisfying the equation :

### Visualization(2)

Plot the HypergeometricPFQ function:

Plot the real part of :

Plot the imaginary part of :

### Function Properties(9)

Domain of HypergeometricPFQ:

Permutation symmetry:

HypergeometricPFQ is an analytic function of z for specific values:

HypergeometricPFQ is neither non-decreasing nor non-increasing for specific values:

HypergeometricPFQ[{1,1,1},{3,3,3},z] is injective:

HypergeometricPFQ[{1,1,1},{3,3,3},z] is not surjective:

HypergeometricPFQ is neither non-negative nor non-positive:

HypergeometricPFQ[{1,1,2},{3,3},z] has both singularity and discontinuity for z1 and at zero:

HypergeometricPFQ is neither convex nor concave:

### Differentiation(2)

First derivative:

Higher derivatives:

Plot higher derivatives for some values parameters:

### Integration(3)

Indefinite integral of HypergeometricPFQ:

Definite integral of HypergeometricPFQ:

Integral with a power function:

### Series Expansions(4)

Taylor expansion for HypergeometricPFQ:

Plot the first three approximations for around :

General term in the series expansion of HypergeometricPFQ:

Expand HypergeometricPFQ of type into a series at the branch point :

Expand HypergeometricPFQ into a series around :

### Function Representations(4)

Primary definition:

HypergeometricPFQ can be represented as a DifferentialRoot:

HypergeometricPFQ can be represented in terms of MeijerG:

## Applications(7)

Solve a differential equation of hypergeometric type:

Solve a third-order singular ODE in terms of the HypergeometricPFQ and MeijerG functions:

Verify that the components of the general solution for an ODE are linearly independent:

A formula for solutions to the trinomial equation :

First root of the quintic :

Check the solution:

Effective confining potential in random matrix theory for a Gaussian density of states:

Its series expansion at infinity reveals logarithmic growth:

An expression for the surface tension of an electrolyte solution as a function of concentration y:

OnsagerSamaras limiting law for very low concentrations:

Fractional derivative of Sin:

Derivative of order of Sin:

Plot a smooth transition between the derivative and integral of Sin:

Define the Gram polynomial in terms of HypergeometricPFQ:

Verify a discrete orthogonality relation satisfied by the Gram polynomials:

Use the Gram polynomial to compute the SavitzkyGolay smoothing coefficients:

Compare with the result of SavitzkyGolayMatrix:

## Properties & Relations(3)

Integrate frequently returns results containing HypergeometricPFQ:

Sum may return results containing HypergeometricPFQ:

Use FunctionExpand to transform HypergeometricPFQ into less general functions:

## Possible Issues(2)

Machine-precision input may be insufficient to get a correct answer:

With exact input, the answer is correct:

Common symbolic parameters in HypergeometricPFQ generically cancel:

However, when there is a negative integer among common elements, HypergeometricPFQ is interpreted as a polynomial:

## Neat Examples(1)

The period of an anharmonic oscillator with Hamiltonian :

Period for quartic anharmonicity:

Limit of pure quartic potential:

Wolfram Research (1996), HypergeometricPFQ, Wolfram Language function, https://reference.wolfram.com/language/ref/HypergeometricPFQ.html (updated 2022).

#### Text

Wolfram Research (1996), HypergeometricPFQ, Wolfram Language function, https://reference.wolfram.com/language/ref/HypergeometricPFQ.html (updated 2022).

#### CMS

Wolfram Language. 1996. "HypergeometricPFQ." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/HypergeometricPFQ.html.

#### APA

Wolfram Language. (1996). HypergeometricPFQ. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/HypergeometricPFQ.html

#### BibTeX

@misc{reference.wolfram_2024_hypergeometricpfq, author="Wolfram Research", title="{HypergeometricPFQ}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/HypergeometricPFQ.html}", note=[Accessed: 17-June-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_hypergeometricpfq, organization={Wolfram Research}, title={HypergeometricPFQ}, year={2022}, url={https://reference.wolfram.com/language/ref/HypergeometricPFQ.html}, note=[Accessed: 17-June-2024 ]}