On polynomials connected to powers of Bessel functions (Q2876602)

Starting with the power series expansion NEWLINE\[NEWLINEf(z)=\sum_{n=0}^{\infty}\frac{1}{a_n}\frac{z^n}{n!},\quad a_n\neq 0,NEWLINE\]NEWLINE the corresponding expansion for \(\{f(z)\}^r\) with \(r\in\mathbb{N}\), NEWLINE\[NEWLINE\{f(z)\}^r=\sum_{n=0}^{\infty}A_n(r)\frac{1}{a_n}\frac{z^n}{n!}NEWLINE\]NEWLINE defines the coefficients \(A_n(r)\). These coefficients are studied in detail in the case that \(f(z)\) is a normalized modified Bessel function of the first kind. Using an identity of Euler a recurrence relation is obtained for these coefficients. Furthermore, the coefficients are identified as Bell polynomials evaluated at values of the Bessel zeta function. This leads to additional recurrence relations for these polynomials. Finally it is shown that the polynomials are of binomial type and a connection to the umbral formalism on Bessel functions introduced by Cholewinski is established.