Series solutions near a regular singular point

[phrankle]

For solving a series solution near a regular singular point with the Frobenius method, why is it that the indices of summation derivatives aren't shifted?

For example, in my textbook and lecture notes

y = \sumA_{}nx^{}n+r from n=0 to infinity

y' = \sum(n+r)A_{}nx^{}n+r-1 from n=0 to infinity

y'' = \sum(n+r)(n+r-1)A_{}nx^{}n+r-2 from n=0 to infinity


But shouldn't the index for y' be from n=1 to infinity because it shifts up when you take the derivative of a summation? Shouldn't the index for y'' be from n=2 to infinity?

Thanks.

 

[tiny-tim]

Welcome to PF!

Hi phrankle! Welcome to PF!

No, because then your An wouldn't match your x^(n+r-1), would it?

Of course, you could change it to ∑A(n+1)x^(n+r), and do it from n = -1 to ∞ …

but that would be unnecessarily complicated, and you could easily make a mistake …