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In the literature different so-called $\kappa$-distribution functions arediscussed to fit and model the velocity (or energy) distributions of solar windspecies, pickup ions or magnetospheric particles. Here we introduce ageneralized (isotropic) $\kappa$-distribution as a "cookbook", which admits asspecial cases, or "recipes", all the other known versions of $\kappa$-models. Adetailed analysis of the generalized distribution function is performed,providing general analytical expressions for the velocity moments, Debyelength, and entropy, and pointing out a series of general requirements thatplasma distribution functions should satisfy. From a contrasting analysis ofthe recipes found in the literature, we show that all of them lead to almostthe same macroscopic parameters with a small standard deviation between them.However, one of these recipes called the regularized $\kappa$-distributionprovides a functional alternative for macroscopic parameterization without anyconstraint for the power-law exponent $\kappa$.

The $\kappa$-cookbook: a novel generalizing approach to unify $\kappa$-like distributions for plasma particle modeling