Find all real valued functions G(x) on integers is such that:

(i) G(1) = 5/2, and:
(ii) G(0) is not zero, and:
(iii) G(A)*G(B) = G(A+B)+G(A-B), for all A and B

Using (ii) and (iii) with A=B=0 gives G(0)=2.  Then building up successive values gives G(2)=17/4 and G(3)=65/8, which is enough to conjecture that G(x)=2^x + 2^-x.  Substituting in (iii) confirms the identity.  

Posted by xdog on 2015-05-02 11:19:14