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Hermite polynomials in asymptotic representations of generalized Bernoulli, Euler, Bessel, and Buchholz polynomials. (English) Zbl 0979.33004

The paper first develops a general underlying structure for expansions in terms of Hermite polynomials, H n (z). For polynomials p n (z) with generating function

F(z,w)= n=0 p n (z)w n ,(1)

where F is analytic with respect to w in a domain that contains the origin, a Cauchy-type representation is therefore

p n (z)=1 2πi C F(z,w)dw w n+1 ·(2)

The generating function F(z,w) is written as

F(z,w)=e A(z) w-B(z)w 2 f(z,w),(3)

where A(z) and B(z) are independent of w, to create a form with an appearance similar to that for the Hermite polynomials

e 2zw-w 2 = n=0 H n (z) n!w n ·(4)

Since the function f is analytic at w=0,

t(z,w)= k=0 c k (z)w k ·(5)

Substituting (3) and (5) into (2), and then using the Hermite formula (4), one obtains a finite expression for each p n (z) for analysis. Subsequent choices of A(z) and B(z) are based on the requirement that c 1 (z)=c 2 (z)=0, in order to make t(z,w) “close” to 1 near the origin w=0, 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 n, complex order μ and (complex) argument z. For the Euler polynomials, E n μ (z), for example,

2 μ e wz (e w +1) μ = n=0 E n μ (z) n!w n ,|w|<π,

A(z)=z-μ/2, B(z)=μ/8, c 2n+1 =0 for n0 and the even-subscripted c’s are independent of z. As well,

lim μ 8 μ n/2 E n μ μ 2+μ 2z=H n (z),

and this can be used for approximation of zeros of the E n μ . Similar results are obtained in each of the other three situations.

33C45Orthogonal polynomials and functions of hypergeometric type
94A40Channel models (including quantum)