Information criteria formulation in identification tool box

Hi all,

I am working on information criteria. I observed that the general AIC and BIC formulation is AIC =-2 ln(L)+k and BIC= -2ln(L)+klog(N). Where L is the likelihood and can be replaced with mean square error.

I want to ask that that the formulation used in identification tool box is some what lengthy (https://www.mathworks.com/help/ident/ref/aic.html). Can some one guide me to the material/research articles that explain the derivation. Even for normalized AIC.

Thanking in advance!