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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?

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

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Afaik It's not called anything other than a first order recursive filter, aka single pole IIR...

And it's not functionally equivalent to the last line you have written. There's no constant term...

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