Finding when a series terminates

Question

A recurrence relation $$\frac{b_{j+2}}{b_{j}} = \frac{j-\xi}{(j+2)(j+1)}$$ defines the general term for the power series where $\xi$ is a constant. $$g(x)=\sum^{\infty}_{j=0} b_{j}x^{j}.$$ I want to show this series then terminates when $\xi=m\in\mathbb{Z}$. So evidently if $m$ is even then the even part of the power series terminates but the odd part doesn't, similarly if $m$ is odd then the odd part terminates, but there is no number for which both odd and even parts terminate. What am I doing wrong?

Answer 1

If $m$ is even, there is a solution with non-zero even terms that terminates.
We can let the odd terms all equal zero.
The general solution for $m=0$ is $b_0+A(b_1x+b_3x^3+b_5x^5+...)$. The one with $A=0$ is a polynomial.
If $m$ is odd, there is a solution, with zero even terms, and non-zero odd terms that terminate.
The general solution for $m=1$ is $b_1x+A(b_0+b_2x^2+b_4x^4+...)$. Again, to get a polynomial, let $A=0$.