Chapter3 2 3. Obtaining Explicit Formulas and Identities for Polynomials Defined by Generating Functions of the Form F(t)x ⋅ G(t)α

Abstract In this chapter, we study properties of polynomials defined by generating functions of the form A t x α = F t x ⋅ G t α . Based on the Lagrange inversion theorem and the theorem of logarithmic derivative for generating functions, we obtain new properties related to the compositional inverse generating functions of those polynomials. Also we study the composition of generating functions R tA t , where A t is the generating function of the form F t x ⋅ G t α . We apply those results for obtaining explicit formulas and identities for such polynomials as the generalized Bernoulli, generalized Euler, Frobenius-Euler, generalized Sylvester, generalized Laguerre, Abel, Bessel, Stirling, Narumi, Peters, Gegenbauer, and Meixner polynomials.