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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 $$F(z,w)=\sum_{n=0}^\infty p_n(z)w^n,\tag 1$$ where $F$ is analytic with respect to $w$ in a domain that contains the origin, a Cauchy-type representation is therefore $$p_n(z)={1\over 2\pi i}\int_CF(z,w){dw\over w^{n+1}}.\tag 2$$ The generating function $F(z,w)$ is written as $$F(z,w)=e^{A(z)}w-B (z)w^2 f(z,w),\tag 3$$ where $A(z)$ and $B(z)$ are independent of $w$, to create a form with an appearance similar to that for the Hermite polynomials $$e^{2zw-w^2}=\sum_{n=0}^\infty {H _n(z) \over n! }w^n.\tag 4$$ Since the function $f$ is analytic at $w=0$, $$t(z,w)=\sum _{k=0}^\infty c_k(z)w^k.\tag 5$$ 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 $\mu$ and (complex) argument $z$. For the Euler polynomials, $E_n^\mu(z)$, for example, $${2^\mu e^{wz} \over(e ^w+1)^\mu}= \sum_{n=0}^\infty {E_n^\mu(z) \over n!} w^n, \quad |w|<\pi,$$ $A(z)=z-\mu/2$, $B(z)=\mu/8$, $c_{2n+1}=0$ for $n\ge 0$ and the even-subscripted $c$’s are independent of $z$. As well, $$\lim_{\mu \to \infty} \left({8 \over \mu}\right)^{n/2} E_n^\mu \left({\mu\over 2} +\sqrt {\mu\over 2}z\right) =H_n(z),$$ and this can be used for approximation of zeros of the $E_n^\mu$. Similar results are obtained in each of the other three situations.
Reviewer: A.G.Law (MR 2000k:33019)

##### MSC:
 33C45 Orthogonal polynomials and functions of hypergeometric type 94A40 Channel models (including quantum)
Full Text:
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