The paper first develops a general underlying structure for expansions in terms of Hermite polynomials, . For polynomials with generating function
where is analytic with respect to in a domain that contains the origin, a Cauchy-type representation is therefore
The generating function is written as
where and are independent of , to create a form with an appearance similar to that for the Hermite polynomials
Since the function is analytic at ,
Substituting (3) and (5) into (2), and then using the Hermite formula (4), one obtains a finite expression for each for analysis. Subsequent choices of and are based on the requirement that , in order to make “close” to 1 near the origin , i.e., to make the generating function form (3) close to that for the Hermite polynomials. Four cases are then considered: generalized Bernoulli, Euler, Bessel and Buchholz polynomials of degree , complex order and (complex) argument . For the Euler polynomials, , for example,
, , for and the even-subscripted ’s are independent of . As well,
and this can be used for approximation of zeros of the . Similar results are obtained in each of the other three situations.