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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, ${H}_{n}\left(z\right)$. For polynomials ${p}_{n}\left(z\right)$ with generating function

$F\left(z,w\right)=\sum _{n=0}^{\infty }{p}_{n}\left(z\right){w}^{n},\phantom{\rule{2.em}{0ex}}\left(1\right)$

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

${p}_{n}\left(z\right)=\frac{1}{2\pi i}{\int }_{C}F\left(z,w\right)\frac{dw}{{w}^{n+1}}·\phantom{\rule{2.em}{0ex}}\left(2\right)$

The generating function $F\left(z,w\right)$ is written as

$F\left(z,w\right)={e}^{A\left(z\right)}w-B\left(z\right){w}^{2}f\left(z,w\right),\phantom{\rule{2.em}{0ex}}\left(3\right)$

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

${e}^{2zw-{w}^{2}}=\sum _{n=0}^{\infty }\frac{{H}_{n}\left(z\right)}{n!}{w}^{n}·\phantom{\rule{2.em}{0ex}}\left(4\right)$

Since the function $f$ is analytic at $w=0$,

$t\left(z,w\right)=\sum _{k=0}^{\infty }{c}_{k}\left(z\right){w}^{k}·\phantom{\rule{2.em}{0ex}}\left(5\right)$

Substituting (3) and (5) into (2), and then using the Hermite formula (4), one obtains a finite expression for each ${p}_{n}\left(z\right)$ for analysis. Subsequent choices of $A\left(z\right)$ and $B\left(z\right)$ are based on the requirement that ${c}_{1}\left(z\right)={c}_{2}\left(z\right)=0$, in order to make $t\left(z,w\right)$ “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, ${E}_{n}^{\mu }\left(z\right)$, for example,

$\frac{{2}^{\mu }{e}^{wz}}{{\left({e}^{w}+1\right)}^{\mu }}=\sum _{n=0}^{\infty }\frac{{E}_{n}^{\mu }\left(z\right)}{n!}{w}^{n},\phantom{\rule{1.em}{0ex}}|w|<\pi ,$

$A\left(z\right)=z-\mu /2$, $B\left(z\right)=\mu /8$, ${c}_{2n+1}=0$ for $n\ge 0$ and the even-subscripted $c$’s are independent of $z$. As well,

$\underset{\mu \to \infty }{lim}{\left(\frac{8}{\mu }\right)}^{n/2}{E}_{n}^{\mu }\left(\frac{\mu }{2}+\sqrt{\frac{\mu }{2}}z\right)={H}_{n}\left(z\right),$

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.

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