What is the formula for F(a, a+1/2, 3/2, z^2) and its general form?

[Gypsumfantastic]

How would i go about showing the special case F(1, b, b; x) of the hypergeometic function is the geometric series and also how the geometric series is = 1/ (1 -x)


Cheers,

Dave

 

[dextercioby]

The geometric series ?? I get the series of $e^{x}$.

Daniel.

 

[Gypsumfantastic]

The geometric series ?? I get the series of $e^{x}$.

Daniel.

I checked it on mathworld that one of the special cases off the hypergeometric function is F(1,1,1;x) is 1 / (1-x) and i want to know how to show it one of my questions is also show that F(1, b, b;x) is the sum to infinity of x^n

 

[dextercioby]

Ok, my mistake. The factorial in the denominator simplifies through. So

$$_{2}F_{1}\left(1,b;b;x\right)=\sum_{\nu=0}^{\infty} x^{\nu}$$

which converges for |x|<1 to $\frac{1}{1-x}$

Daniel.

 

[Gypsumfantastic]

Ok, my mistake. The factorial in the denominator simplifies through. So

$$_{2}F_{1}\left(1,b;b;x\right)=\sum_{\nu=0}^{\infty} x^{\nu}$$

which converges for |x|<1 to $\frac{1}{1-x}$

Daniel.

Cheers thanks

 

[VatanparvaR]

Sorry in advance that I'm posting the same thing in two threads.
I really need it !


From Abramowitz's book I got this one


$$F(a, a+\frac{1}{2}, \frac{3}{2}, z^2)=\frac{1}{2}z^{-1}(1-2a)^{-1}[(1+z)^{1-2a}-(1-z)^{1-2a}]$$

Now I need to find


$$F(a, a+\frac{1}{2}, \frac{5}{2}, z^2)$$


$$F(a, a+\frac{1}{2}, \frac{7}{2}, z^2)$$


and, it would be great if I find

$$F(a, a+\frac{1}{2}, n+ \frac{1}{2}, z^2)$$


are there any books, handbooks, or websites that I could find this guy?