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The theory of pseudo-differential operators is a powerful tool to deal withdifferential equations involving differential operators under the square rootsign. These type of equations are pivotal elements to treat problems inanomalous diffusion and in relativistic quantum mechanics. In this paper wereport on new and unsuspected links between fractional diffusion, quantumrelativistic equations and particular families of polynomials, linked to theCarlitz family, and playing the role of relativistic heat polynomials. Weintroduce generalizations of these polynomial families and point out theirspecific use for the solutions of problems of practical importance.

Theory of relativistic heat polynomials and one-sided L\'evy distributions