Appel polynomials associated with Fourier transforms and their applications to differential equations

Formulas for the coefficients of the expansion into a series of Appel polynomials associated with a differential equation of parabolic type are obtained. It has been established that Appel polynomials are involved in the formulas for the expansion of the solution to the Cauchy problem for equations of parabolic type into a series of derivatives of the fundamental solution. A new method for solving the Cauchy problem is proposed, the essence of which is to use series expansion in Appel polynomials. The results generalize the method for solving the heat equation on the real axis by expanding it into a series of Hermite polynomials. The connection between the Fourier transform and series in associated Appel polynomials is studied. The issue of using Hermite polynomials for the Laplace transform has been studied.