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 Math Help > Trigonometry > Trig Equivalences > Hypergeometric Function

# Hypergeometric Function

 The simplest form of the Hypergeometric Function is given by the general form of its power series expansion,
 F(a,x) = 1 + 1 x + 1 x² + 1 x³ + ... . a 1! a(a+1) 2! a(a+1)(a+2) 3!
By selecting a and x carefully, we can coax the Hypergeometric Function to mimic sinh or cosh.

### Proof that sinh(x) = x F(3/2, x²/4)

 F(3/2,x²/4) = 1 + 1 x2/4 + 1 x4/16 + 1 x6/64 + ... . 3/2 1! (3/2)(5/2) 2! (3/2)(5/2)(7/2) 3!
 F(3/2,x²/4) = 1 + 1 x2/2 + 1 x4/4 + 1 x6/8 + ... . 3 1! (3)(5) 2! (3)(5)(7) 3!
 F(3/2,x²/4) = 1 + 1 x2 + 1 x4 + 1 x6 + ... . 3 2 (3)(5) (2)(4) (3)(5)(7) (2)(4)(6)
 F(3/2,x²/4) = 1 + x2 + x4 + x6 + ... . 3! 5! 7!
 x F(3/2,x²/4) = x + x3 + x5 + x7 + ... = sinh(x). 3! 5! 7!
Similarly, it can be shown that

cosh(x) = F(1/2, x²/4)

### General Form of the Hypergeometric Function

From Mathworld, we see that the general form is given by

pFq(a1,a2,...,ap;b1,b2,...,bq;x), in which successive terms of the series have the ratio

 ck+1 = P(k) = (k+a1)(k+a2)...(k+ap) x c Q(k) (k+b1)(k+b2)...(k+bq)(k+1)
The successive terms of F(a,x) -- the simplest form, above -- have the ratio

x/((a)(1)), x/((a+1)(2)), x/((a+2)(3)), ...

So F(a,x) can be written 0F1(;a;x), and is called a "Confluent Hypergeometric Limit Function"

Other forms include 2F1(a,b;c;x), sometimes called The Hypergeometric Function, or Gauss's Hypergeometric Function.

1F1(a;b;x), called the "Confluent Hypergeometric Function of the First Kind", and sometimes written
M(a,b,x) or Φ(a,b,x), and also known as Kummer's function.

### Recurrence Relations

 Using the general form of the Hypergeometric function, recurrence relations are given by 2F1(a,b;c;z) = (1-z)-a 2F1(a,c-b;c;z/(z-1))
 2F1(a,b;c;z) = (1-z)-b 2F1(c-a,b;c;z/(z-1))
 2F1(a,b;c;z) = (1-z)c-a-b 2F1(c-a,c-b;c;z)
 0F1(;a;x) = F(a,x) has the recurrence relation,
 F(a-1,x) = F(a,x) + x F(a+1,x) (a-1)(a)
 Which is used by the Nrich website to develop a continued fraction for tanh(x)
 1F1(a;b;x) = F(a,b,x) has the recurrence relations,
 F(a,b,x) = F(a+1,b+1,x) + (a-b) x F(a+1,b+2,x) (b)(b+1)
 F(a,b,x) = F(a,b+1,x) + a x F(a+1,b+2,x) (b)(b+1)
Which can be used to develop a continued fraction for ex = F(1,1,x), using a technique similar to the one from Nrich, referenced above.

### Internet references

The general form of the Hypergeometric Function pFq(a1,a2,...,ap;b1,b2,...,bq;x) is described in Mathworld.  In this page, p=q=1, and
F(a,x) = 1F1(1;a;x)

The representation of sinh and cosh in terms of the simplified Hypergeometric Function is described in nrich.maths.org.

### Related Pages in this website

Trigonometric Equivalences

Hyperbolic Functions

Euler and Exponents -- Euler's formula, and xy

The Quadratic Formula -- another proof by working backwards

The webmaster and author of this Math Help site is Graeme McRae.