Entire groups generated by fractional powers of operators

Let $T$ be a power-bounded operator on a Banach space $X$. We treat the sequence of polynomials $(p_{n;a}{)}_{n\ge 0}$ such that the entire group generated by the fractional power operator $-(I-T{)}^a$ is given by $$e^{-t(I-T{)}^a}=e^{-t}\sum _{n=0}^{\mathrm{\infty }}{p}_{n;a}(t)T^n,t\in \text{CC},\mathrm{\Re }a>0.$$ We provide a self-contained introduction to the polynomial family $(p_{n;a}{)}_{n\ge 0}$, for $a\in \text{CC}$, whose coefficients are determined by means of a suitable recurrence relation. The sequence $(n!p_{n;a}{)}_{n\ge 0}$ forms a family of Sheffer polynomials. For $\mathrm{\Re }a>0$ and $t\in \text{CC}$, the sequence $(p_{n;a}(t){)}_{n\ge 0}$ belongs to the Lebesgue sequence space ${\mathrm{\ell }}^1$ of absolutely summable sequences. Moreover, these polynomials are closely related to the Lévy density functions $(f_{t,\alpha }{)}_{t>0}$ defined for $0<\alpha <1$. Finally, we discuss several particular cases corresponding to specific values of $a\in \text{CC}$, as well as applications to fractional powers in the Banach algebra ${\mathrm{\ell }}^1$, multiplication operators, and Cesàro means.