Hypergeometric Function
   

   

 Math Help -> Geometry and Trigonometry -> Trigonometric 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 F(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

 

 


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