# Relations between special functions

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

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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_{n} are Gegenbauer polynomials with α = 0.
The **Chebyshev** polynomials of the **first kind** T_{n} are Gegenbauer polynomials with α = -½.
The **Chebyshev** polynomials of the **second kind** U_{n} are Gegenbauer polynomials with α = ½.

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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