Book:Murray R. Spiegel/Mathematical Handbook of Formulas and Tables/Chapter 31
Murray R. Spiegel: Mathematical Handbook of Formulas and Tables: Chapter 31
Published $\text {1968}$
$31 \quad$ Hypergeometric Functions
Hypergeometric Differential Equation
$31.1$: Hypergeometric Differential Equation
A hypergeometric differential equation is a second order ODE of the form:
- $x \paren {1 - x} \dfrac {\d^2 y} {\d x^2} + \paren {c - \paren {a + b + 1} x} \dfrac {\d y} {\d x} - a b y = 0$
Hypergeometric Functions
- A solution of $31.1$ is given by
$31.2$: Gaussian Hypergeometric Function
The Gaussian hypergeometric function is an instance of a generalized hypergeometric function, given for $\size z < 1$ by:
| \(\ds \map F {a, b; c; z}\) | \(:=\) | \(\ds \sum_{n \mathop = 0}^\infty \dfrac {a^{\overline n} b^{\overline n} } {c^{\overline n} } \dfrac {z^n} {n!}\) | where $x^{\overline n}$ denotes the $n$th rising factorial power of $z$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds 1 + \dfrac {a b} {1! \, c} z + \dfrac {a \paren {a + 1} b \paren {b + 1} } {2! \, c \paren {c + 1} } z^2 + \dfrac {a \paren {a + 1} \paren {a + 2} b \paren {b + 1} \paren {b + 2} } {3! \, c \paren {c + 1} \paren {c + 2} } z^3 + \cdots\) |
- If $a$, $b$, $c$ are real, then the series converges for $-1 < x < 1$ provided that $c - \paren {a + b} > 1$.
Special Cases
$31.3$: Power of One plus x in terms of Gaussian Hypergeometric Function
- $\map F {-p, 1; 1; -x} = \paren {1 + x}^p$
$31.4$: Logarithm of One plus x in terms of Gaussian Hypergeometric Function
- $\map \ln {1 + x} = x \map F {1, 1; 2; -x}$
$31.5$: Exponential Function as Limit of Gaussian Hypergeometric Function
Exponential Function as Limit of Gaussian Hypergeometric Function
$31.6$: Cosine Function in terms of Gaussian Hypergeometric Function Cosine Function in terms of Gaussian Hypergeometric Function
$31.7$: Secant Function in terms of Gaussian Hypergeometric Function Secant Function in terms of Gaussian Hypergeometric Function
$31.8$: Arcsine Function in terms of Gaussian Hypergeometric Function
- $\arcsin x = x \map F {\dfrac 1 2, \dfrac 1 2; \dfrac 3 2; x^2}$
$31.9$: Arctangent Function in terms of Gaussian Hypergeometric Function
- $\arctan x = x \map F {\dfrac 1 2, 1; \dfrac 3 2; -x^2}$
$31.10$: Reciprocal of One minus x in terms of Gaussian Hypergeometric Function
- $\dfrac 1 {1 - x} = \map F {1, p; p; x}$
$31.11$: Legendre Polynomial in terms of Gaussian Hypergeometric Function
Legendre Polynomial in terms of Gaussian Hypergeometric Function
$31.12$: Chebyshev Polynomial of First Kind in terms of Gaussian Hypergeometric Function Chebyshev Polynomial of First Kind in terms of Gaussian Hypergeometric Function
General Solution of the Hypergeometric Equation
$31.13$: General Solution of the Hypergeometric Equation General Solution of Hypergeometric Equation
Miscellaneous Properties
$31.14$: Gauss's Hypergeometric Theorem
- $\map F {a, b; c; 1} = \dfrac {\map \Gamma c \map \Gamma {c - a - b} } {\map \Gamma {c - a} \map \Gamma {c - b} }$
$31.15$: Derivative of Gaussian Hypergeometric Function
- $\map {\dfrac \d {\d x} } {\map F {a, b; c; x} } = \dfrac {a b} c \map F {a + 1, b + 1; c + 1; x} $
$31.16$: Euler's Integral Representation of Hypergeometric Function
- $\ds \map F {a, b; c; x} = \dfrac {\map \Gamma c } {\map \Gamma b \map \Gamma {c - b} } \int_0^1 t^{b - 1} \paren {1 - t}^{c - b - 1} \paren {1 - x t}^{- a} \rd t$
$31.17$: Euler's Transformation
- $\ds \map F {a, b; c; x} = \paren {1 - x}^{c - a - b} \map F {c - a, c - b; c; x}$