What is the best concise mathematical notation for applying an IIR filter to a signal?

Question

For denoting the application of an FIR filter with impulse response $\mathbf{k}$ to a signal $\mathbf{x}$ in time domain, we can use the convolution operation, i.e., $\mathbf{x}*\mathbf{k}$. What is the best concise way to denote the application of an IIR filter to a signal in time domain? An IIR filter has, by definition, an impulse response that is infinitely long. Using convolution to denote its application seems odd to me. I am not asking about what an IIR filter is or how it is defined but how to denote it mathematically in a visually appealing way.

Answer 1

Standard notation for the time domain is the linear difference equation. See for example: https://ccrma.stanford.edu/~jos/filters/Difference_Equation_I.html

If you write it in the form

$$ \sum_i a_i y[n-1] = \sum_i b_i x[n-i] $$

it has a one to one correspondence with the z-transform.

$$H(z) = \frac{\sum_i b_i z^{-i}}{\sum_i a_i z^{-i}} $$

That's however NOT the best way to actually implement a time domain IIR filter. That would be a topic for a different question.