Let me rephrase the question, considering the case for sigma
F(x) = sigma( k=0, x) f(k)
So therefore:
F(1) = f(0) + f(1)
F(2) = f(0) + f(1) + f(2)
I was just wondering how we could find rational and complex evaluations of F(x). The only requirement I believe it should have is:
F(x) + f(x+1) = F(x+1)
For the case of products
P(x) = E( k=0, x) f(k)
and therefore:
P(1) = f(0)*f(1)
P(2) = f(0)*f(1)*f(2)
etc etc
The only requirement P(x) requires is:
f(x+1)*P(x) = P(x+1) ; which as I was saying is not dissimilar to the gamma function. when f(x) = x, it is the gamma function plus one. For F(x), when f(x) = x, F(x) should equal Gauss' formula for sum of a series.
F(x) = sigma( k=0, x) f(k)
So therefore:
F(1) = f(0) + f(1)
F(2) = f(0) + f(1) + f(2)
I was just wondering how we could find rational and complex evaluations of F(x). The only requirement I believe it should have is:
F(x) + f(x+1) = F(x+1)
For the case of products
P(x) = E( k=0, x) f(k)
and therefore:
P(1) = f(0)*f(1)
P(2) = f(0)*f(1)*f(2)
etc etc
The only requirement P(x) requires is:
f(x+1)*P(x) = P(x+1) ; which as I was saying is not dissimilar to the gamma function. when f(x) = x, it is the gamma function plus one. For F(x), when f(x) = x, F(x) should equal Gauss' formula for sum of a series.
