Causality of a discrete recursive system I am new in this site and this my first question.
How can I mathematically prove that the system with a transfer function like the below one is causal?
$$H(z)=\frac{B(z)}{A(z)}=\frac{b_{0}+b_{1} z^{-1}+b_{2} z^{-2}+\cdots+b_{N} z^{-N}}{1+a_{1} z^{-1}+a_{2} z^{-2}+\cdots+a_{M} z^{-M}}$$
P.S:
I know that for a system to be causal there has to be past inputs in the equation of difference and that the inpulse resoponse has to be zero for negative $n$ ($h[n]=0 , n&lt;0$) but the thing is that when i try to prove the second one by putting values into $n$ and writing down the result of $H[n]$, I always get a reslut that depents on a past value of $H[n]$
How can I mathematically prove that the system with a transfer function like the below one is causal?

You can't.
The transfer function $H(z)$ in itself doesn't tell you if the filter is causal or not.
More precisely. An LTI is causal if $h[n] =0$ for $n&lt;0$. But $H(z)$ is not enough to determine $h[n]$. You need also the ROC (region of convergence of the Z transform).
For example, $H(z)=\frac{1}{1+z^{-1}}$ (a particular case of your transfer function) is the Z transform both of a causal response function  and an anti-causal one (details here).
Specifically: for a LTI to be causal, the ROC of the $Z$ transform must extend &quot;to the outside&quot;, ie., must be of the form $|z| &gt; R_0$ or $|z| \ge R_0$.
For a given rational funtion, $R_0$ would be the largest absolute value of all poles.
Just write down the corresponding difference equation. Then it is obvious that the system is causal. There is no need to compute the pulse response. The difference equation reads

$y(n) + a_1 y(n-1) + \ldots = b_0 x(n) + b_1 x(n-1) + \ldots$

Clearly, if you solve for $y(n)$, the right hand side only depends on previous $y(n)$ and previous $x(n)$ so it is causal.

It would not be causal, if any exponent in the numerator or denominator of the transfer function is positive (e.g. there is some term with $z^2$, instead of $z^{-2}$ for instance). The difference equation would then contain $x(n+2)$ in the right hand side for $y(n)$. That is clearly not causal.

In an abstract way, all systems are causal that have transfer functions which is rational polynomial in $z^{-1}$ with a nonzero lowest order coefficient in the denominator.