gives the Gegenbauer polynomial TemplateBox[{n, m, x}, GegenbauerC].


gives the renormalized form TemplateBox[{{TemplateBox[{n, m, x}, GegenbauerC], /, m}, m, 0}, Limit2Arg].


  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • Explicit polynomials are given for integer n and for any m.
  • TemplateBox[{n, m, x}, GegenbauerC] satisfies the differential equation .
  • The Gegenbauer polynomials are orthogonal on the interval with weight function , corresponding to integration over a unit hypersphere.
  • For certain special arguments, GegenbauerC automatically evaluates to exact values.
  • GegenbauerC can be evaluated to arbitrary numerical precision.
  • GegenbauerC automatically threads over lists.
  • GegenbauerC[n,0,x] is always zero.
  • GegenbauerC[n,m,z] has a branch cut discontinuity in the complex z plane running from to .
  • GegenbauerC can be used with Interval and CenteredInterval objects. »


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Basic Examples  (7)

Evaluate numerically:

Compute the 10^(th) Gegenbauer polynomial:

Compute the 10^(th) renormalized Gegenbauer polynomial:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Asymptotic expansion at Infinity:

Asymptotic expansion at a singular point:

Scope  (43)

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:

GegenbauerC can be used with Interval and CenteredInterval objects:

Specific Values  (8)

Values of GegenbauerC at fixed points:

Simple cases give exact symbolic results:

GegenbauerC for symbolic n:

Values at zero:

Find the first positive maximum of GegenbauerC[10,x ]:

Compute the associated GegenbauerC[7,x] polynomial:

Compute the associated GegenbauerC[1/2,x] polynomial for half-integer n:

Different GegenbauerC types give different symbolic forms:

Visualization  (4)

Plot the GegenbauerC function for various orders:

Plot the real part of :

Plot the imaginary part of :

Plot as real parts of two parameters vary:

Types 2 and 3 of GegenbauerC function have different branch cut structures:

Function Properties  (14)

Domain of GegenbauerC of integer orders:

The range for GegenbauerC of integer orders:

The range for complex values is the whole plane:

Gegenbauer polynomial of an odd order is odd:

Gegenbauer polynomial of an even order is even:

GegenbauerC threads elementwise over lists:

GegenbauerC has the mirror property :

Gegenbauer polynomials are analytic:

However, the GegenbauerC function is generally not analytic for noninteger parameters:

Nor is it meromorphic:

TemplateBox[{2, x}, GegenbauerC2] is neither non-decreasing nor non-increasing:

TemplateBox[{2, x}, GegenbauerC2] is not injective:

TemplateBox[{2, x}, GegenbauerC2] is not surjective:

TemplateBox[{2, x}, GegenbauerC2] is neither non-negative nor non-positive:

TemplateBox[{n, x}, GegenbauerC2] has singularities or discontinuities when is not an integer and :

TemplateBox[{n, m, x}, GegenbauerC] has additional singularities when is noninteger:

TemplateBox[{2, x}, GegenbauerC2] is convex:

TraditionalForm formatting:

Differentiation  (3)

First derivatives with respect to x:

Higher derivatives with respect to x:

Plot the higher derivatives with respect to x when n=10 and m=1/3:

Formula for the ^(th) derivative with respect to x:

Integration  (3)

Compute the indefinite integral using Integrate:

Verify the anti-derivative:

Definite integral:

More integrals:

Series Expansions  (2)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

Taylor expansion at a generic point:

Function Identities and Simplifications  (4)

GegenbauerC is a special case of JacobiP:

Derivative identity of GegenbauerC:

Generating function of Gegenbauer polynomials:

Recurrence relations:

Generalizations & Extensions  (2)

Apply GegenbauerC to a power series:

GegenbauerC can deal with real-valued intervals:

Applications  (3)

Eigenfunctions of the angular part of the four-dimensional Laplace operator:

Radial part of the hydrogen atom eigenfunction in momentum representation:

In an n-point GaussLobatto quadrature rule, the values of the two extreme nodes are fixed, and the other n-2 nodes are computed from the roots of a certain Gegenbauer polynomial. Compute the nodes and weights of an n-point GaussLobatto quadrature rule:

Use the n-point GaussLobatto quadrature rule to numerically evaluate an integral:

Compare the result of the GaussLobatto quadrature with the result from NIntegrate:

Properties & Relations  (5)

Use FunctionExpand to expand GegenbauerC into other functions:

GegenbauerC can be represented as a DifferenceRoot:

General term in the series expansion of GegenbauerC:

The generating function for GegenbauerC:

Define an inner product on functions using Integrate:

Construct an orthonormal basis using Orthogonalize:

This inner product produces the GegenbauerC polynomials:

Possible Issues  (1)

Cancellations in the polynomial form may lead to inaccurate numerical results:

Evaluate the function directly:

Wolfram Research (1988), GegenbauerC, Wolfram Language function, (updated 2022).


Wolfram Research (1988), GegenbauerC, Wolfram Language function, (updated 2022).


Wolfram Language. 1988. "GegenbauerC." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022.


Wolfram Language. (1988). GegenbauerC. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2024_gegenbauerc, author="Wolfram Research", title="{GegenbauerC}", year="2022", howpublished="\url{}", note=[Accessed: 20-July-2024 ]}


@online{reference.wolfram_2024_gegenbauerc, organization={Wolfram Research}, title={GegenbauerC}, year={2022}, url={}, note=[Accessed: 20-July-2024 ]}