# Relations between special functions

The diagram below illustrates the relationships between many common special functions.

The **hypergeometric functon _{2}F_{1}** has many special cases. (See these notes on hypergeometric functions for definitions and notation.)

The **Jacobi polynomials** are related to _{2}*F*_{1} by

Jacobi polynomials with α = β are called **Gegenbauer** or **ultraspherical **polynomials and are denoted *C*^{(α)}* _{n}*. The

**Legendre**polynomials

*P*are Gegenbauer polynomials with α = 0. The

_{n}**Chebyshev**polynomials of the

**first kind**

*T*are Gegenbauer polynomials with α = -½. The

_{n}**Chebyshev**polynomials of the

**second kind**

*U*are Gegenbauer polynomials with α = ½.

_{n}The **incomplete beta function** *B _{x}*(

*a*,

*b*) is related to

_{2}

*F*

_{1}(

*x*) by

The **beta function** *B*(*a*,*b*) is defined to be Γ(*a*) Γ(*b*) / Γ(*a*+*b*) and equals *B*_{1}(*a*, *b*).

The **complete elliptic integrals** *K*(*z*) and *E*(*z*) are related to _{2}*F*_{1} by

The complete elliptic integrals are the values of the **incomplete elliptic integrals** *F*(φ, *z*) and *E*(φ, *z*) at φ = φ/2.

The inverse of the integral *F*(φ, *z*) is the **Jacobi elliptic function sn**. The Jacobi functions **sn**, **cn**, and **dn** are intimately related, much like the elementary trigonometric functions.

The **hypergeometric functon _{1}F_{1}** is also known as the

**confluent hypergeometric function**. It is related to the hypergeometric function

_{2}

*F*

_{1}by

The **incomplete gamma function** γ(*a*, *z*) is related to _{1}*F*_{1} by

The limit of γ(*a*, *z*) as *z* goes to infinity is the **gamma function** Γ(*a*).

The derivative of the logarithm of the gamma function Γ(*z*) is the function ψ(*z*).

The **error function** erf(*z*) is related to _{1}*F*_{1} by

The error function erf(*z*) is related to the **Fresnel integrals** *C*(*z*) and *S*(*z*) by

The **Laguerre polynomials** *L*^{α}* _{n}* are related to

_{1}

*F*

_{1}by

The **Hermite polynomials** are related to the Laguerre polynomials by

The **hypergeometric functon _{0}F_{1}** is related to the hypergeometric function

_{1}

*F*

_{1}by

Only **Bessel functions of the first kind** *J*_{ν} are shown on the diagram. Other Bessel functions are related to *J*_{ν} as described here.

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