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A professor is writing the general form of a Transfer Function as:
$$H(s)=k\cdot s^{l}\cdot \frac{\prod\limits_{i=1}^{m}\left(1+\frac{s}{z_{i}}\right)}{\prod\limits_{i=1}^{n}\left(1+\frac{s}{p_{i}}\right)}$$
but in every book and in every source that I have found is never mentioned that way. The poles and zeros are never in fractions in the numerator and denominator. Am I missing something? Has anyone any source to explain the logic behind this specific form of Transfer Function?
There is nothing to be surprised at. A transfer function can be shown in one of the two following notations.
$$ H(s) = K\dfrac{(s-z_1)(s-z_2)\dots(s-z_n)}{(s-p_1)(s-p_2)\dots(s-p_m)} $$
$$ H(s) = \dfrac{b_ns^n + b_{n-1}s^{n-1}+ \dots + b_0s^0}{a_ms^m + a_{m-1}s^{m-1}+ \dots + a_0s^0} $$
The first one is good for seeing the locations of poles and zeros at first glance, and for performing partial fraction expansion. The second one is good for easily extracting the differential equation between input and output, also for transforming into state space form. Each notation has their own advantages and handicaps under every topic.
