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Hypergeometric functions. Gauss hypergeometric differential equation. Confluent hypergeometric differential equation.

Hypergeometric differential equation. The equation

1)        x(1 - x) y" + [c - (a + b + 1)x] y' - aby = 0

Syn. Gauss hypergeometric differential equation

Hypergeometric function. The function given by

or, more concisely,

where a, b and c are fixed parameters and

            (a)_n denotes the product a(a + 1)(a + 2)(a + 3) ....... (a + n -1)                     (n factors)

            (b)_n denotes the product b(b + 1)(b + 2)(b + 3) ....... (a + n -1)                    (n factors)

            (c)_n denotes the product c(c + 1)(c + 2)(c + 3) ....... (a + n -1)                     (n factors)

Example.

            (6)₄ = 6 ·7 · 8 · 9, 

            (-7)₃ = (-7)(-6)(-5)

The series

 is called the hypergeometric series. It converges for -1 < x < 1 provided a, b, c are real and c - (a + b) > -1.

The notation F(a,b;c;x). In the notation F(a,b;c;x) of 3), the arguments listed before the first semicolon refer to numerator parameters (the products (a)_n and (b)_n in this case). The arguments listed between the first and second semicolons refer to denominator parameters (the product (c)_n in this case). The argument after the second semicolon refers to the variable (x in this case). The notation F(a,b;c;x) may also be written as ₂F₁(a,b;c;x) where subscripts before and after the F denote the number of numerator and denominator parameters, respectively. Functions of hypergeometric type with any number of numerator and denominator parameters have been studied for many years. The subscripts on the F are useful when one is making general statements regarding functions as in “Any ₀F₁ is essentially a Bessel function of the first kind” or “The Laguerre polynomial is a terminating ₁F₁. Specifically,

Generalized hypergeometric function. The hypergeometric function 3) above is a special case of the generalized hypergeometric function

where

            (a)_n denotes the product a(a + 1)(a + 2)(a + 3) ....... (a + n -1)                     (n factors)

            (b)_n denotes the product b(b + 1)(b + 2)(b + 3) ....... (a + n -1)                    (n factors)

Special instances

General solution of the hypergeometric equation. If c, a - b and c - a - b are all non-integers, the general solution of the hypergeometric equation valid for |x| < 1 is

14)      y = A F(a, b; c; x) + Bx^{1 - c} F(a - c + 1, b - c + 1; 2 - c; x)

where A and B are arbitrary constants.

Miscellaneous properties

The confluent hypergeometric differential equation. The equation

19)      xy" + (b - x)y' - ay = 0

Syn. Kummer’s equation

The simplest solution is Kummer’s function ₁F₁(a; b; x).

General solution of the confluent hypergeometric equation. The general solution if b is not an integer is

20)      y = A ₁F₁(a; b; x) + B x^{1- b} ₁F₁(a + 1 - b; 2 - b; x)

where A and B are arbitrary constants. If b is an integer there may be a logarithmic solution.

References.

1. James/James. Mathematics Dictionary

2. The International Dictionary of Applied Mathematics. D. Van Nostrand Co.

3. Murray R. Spiegel. Mathematical Handbook of Formulas and Tables. (Schaum)

4. Rainville. Elementary Differential Equations