I have the differential equation $(1-x^2)y'' -2xy' +\lambda y = 0$. If I solve it with a power series of the form $\displaystyle \sum_{n=0}^{\infty} a_n (x+1)^n $ centered at $x=-1$ I got a recurrence relation. Should I get the same recurence if I solve my differential equation with a power series of the form $\displaystyle \sum_{n=0}^{\infty} a_n x^n$ centered at $x=0$? Should it give the same coefficients $a_j$?
Thanks in advance,
Best regards.