Try to prove it isn't necessary and derive a contradiction.
Hello everyone, this is my first post and might be trivial.
After some calculations, i come up with the following equality:
2F1(1,1/n,1+1/n,a c)==2F1(1,1/n,1+1/n,b c) (1)
where, 2F1 is Gauss' hypergeometric function, n,c real positive quantities.
Obviously, the equality holds when a==b (2). My problem is that i want to show that (2) is not only sufficient but also necessary, i.e. (1) holds only when (2) holds.
Any ideas?
TIA
Ok, we define Gauss' hypergeometric function by an infinite sum as follows:
= =
Where (.)n is Pochhammer's symbol. Changing the notation in my example:
2F1(1,1/m,1+1/m,x*d)==2F1(1,1/m,1+1/m,y*d) (1)
where m,d positive real quantities. The equality holds when x==y (2)
I want to show that (2) is a necessary condition.
Assuming that x<>y, (1) can be written in the form:
(frac1)*((x-y)d)+(frac2)*((x^2-y^2)*d^2)+(frac3)*...==0 (3)
It is sufficient to show that (3) cannot hold... Obviously, we can show this if each term is positive (negative), thus the LHS is positive (negative) as an infinite sum of positive (negative) values.
I think that this is true if we assume (additionally) that |x|<=1,|y|<=1. Am I right? Is there an elegant way to present this?
TIA