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Hypergeometric function 2 F 1
Tapas Mazumdar (2018A8B40427P)
October 2019
1 Introduction
The hypergeometric function is the solution of the differential equation
x(1 − x)y′′^ + [c − (a + b + 1)x]y′^ − aby = 0 (1)
where a, b and c are constants.
2 Derivations
Dividing (1) throughout by the coefficient of y′′^ and comparing it with standard linear double differential equation y′′^ + P (x)y′^ + Q(x)y = 0
we get
P (x) =
c − (a + b + 1)x x(1 − x) ;^ Q(x) =^
−ab x(1 − x)
It is easy to verify that x = 0, 1 are regular singular points
- Solution around x = 0: The indicial equation obtained is
m(m − 1) + cm = 0 =⇒ m = 0, 1 − c
- Case I: c is not zero or negative integer
In this case there is definite existence of solution corresponding to m = 0. This solution is
y = 1 +
∑^ ∞
n=
(a)n(b)n n!(c)n
xn
where (q)n is the rising Pochammer symbol given by
(q)n :=
1 n = 0 q(q + 1)(q + 2) · · · (q + n − 1) n > 0
Definition 1. The solution to the hypergeometric equation is called the hypergeometric series, denoted by F (a, b, c, x) and given by
F (a, b, c, x) = 1 +
∑^ ∞
n=
(a)n(b)n n!(c)n
xn
The series converges absolutely for |x| < 1.
- Case II: c is not positive integer
In this case there is definite existence of solution corresponding to m = 1 − c. The solution is
y = x^1 −cF (a − c + 1, b − c + 1, 2 − c, x)
- Case III: c is not an integer
The two solutions discussed above become linearly independent in this case and hence the general solu- tion is y = c 1 F (a, b, c, x) + c 2 x^1 −cF (a − c + 1, b − c + 1, 2 − c, x)
- Solution around x = 1: Transformation of t = 1 − x is desirable since it converts (1) to
t(1 − t)y′′^ + [(a + b − c + 1) − (a + b + 1)t]y′^ − aby = 0
It should be noted that this resembles (1) with the only changes being that x is replaced by 1 − t and c is replaced by a + b − c + 1. The general solution when c − a − b is not an integer is
y = c 1 F (a, b, a + b − c + 1, 1 − x) + c 2 (1 − x)c−a−bF (c − a, c − b, c − a − b + 1, 1 − x)
3 Conversion to Hypergeometric form
Any differential equation of the given form
(x − A)(x − B)y′′^ + (C + Dx)y′^ + Ey = 0
can be converted to (1) via substitution
t =
x − A B − A then x = A =⇒ t = 0 and x = B =⇒ t = 1.
4 Properties
- Derivatives:
dn dxn^
F (a, b, c, x) =
(a)n(b)n (c)n
F (a + n, b + n, c + n, x)
- F (a, b, c, x) = F (b, a, c, x)
- Special cases:
- (1 + x)p^ = F (−p, b, b, −x)
- ln(1 + x) = xF (1, 1 , 2 , −x)
- arcsin(x) = xF
2 ,^
2 ,^
2 , x
2
2 ,^1 ,^
2 ,^ −x
2
F
a, b, a, x b
[
a^ lim→∞ F
a, a,
−x^2 4 a^2
)]
a, a,
−x^2 4 a^2
