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:

Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

Or compute average-case statistical intervals using Around :

Compute the elementwise values of an array:

Or compute the matrix GegenbauerC function using MatrixFunction :

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:

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:

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:

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:

Compute the indefinite integral using Integrate :

Verify the anti-derivative:

Definite integral:

More integrals:

Find the Taylor expansion using Series :

Plots of the first three approximations around :

Taylor expansion at a generic point:

GegenbauerC is a special case of JacobiP :

Derivative identity of GegenbauerC :

Generating function of Gegenbauer polynomials:

Recurrence relations: