zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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.


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