Can You Solve This Non-Constant Coefficient Difference Equation?

[Poirot1]

I wish to solve a non constant coefficents difference equation $(x+1)y_{x+1}-(r+x)y_{x}+ry_{x-1}=0$ where r is a constant. Is there a characteristic equation and generic solution for this ?

 

[HallsofIvy]

Actually, the standard way of solving that would be to write $$y_{x+1}= [(r+x)y_x- ry_{x-1}]/(x+1)$$ and solve in "blocks". You would have to be given an "initial value" of y on, say, $$0\le x\le 2$$. For example, if you we given that y(x)= x for $$0\le x\le 2$$ then , for $$2\le x\le 3$$ we would have $$y(x)= (r+x-1)(x-1)- r(x-2)/(x)$$.