# What is it called when you have a single-pole recursive digital lowpass filter where the coefficients don't add up to unity?

Is there a name for a single-pole recursive digital lowpass filter where the coefficients don't quite add up to 1?

For example: with the filter y[n]=a0*x[n]+b1*y[n-1] instead of using coefficients

a0=0.1
b1=0.9


What if I used

a0=0.11
b1=0.9


Is there a name for this, or is it just functionally equivalent to something else like y[n]=a0*x[n]+b1*y[n-1]+c0

Also, at what point will the filter become unstable? The example above where the coefficients add up to 1.01 looks stable to me but I think a0=0.009 and b1=1.001 would be unstable. Where's the line where it crosses from stable to unstable? b1=1?

Stability only depends on $$b_1$$, and it needs to satisfy $$|b_1|<1$$ for stability of the recursion.