Abstract
We provide a full asymptotic analysis of the N?rlund polynomials and Stirling polynomials. We give a general asymptotic expansion-to any desired degree of accuracy-when the parameter is not an integer. We use singularity analysis, Hankel contours, and transfer theory. This investigation was motivated by a need for such a complete asymptotic description, with parameter 1/2, during this author's recent solution of Wilf's 3rd (previously) Unsolved Problem.
Highlights
The Norlund polynomials have been studied in many contexts
Many connections have been identified with Bernoulli and Stirling numbers; see, e.g., [2], [3], [5], [6]
One feature of the following singularity analysis is the contrast between the structure of (B(z))α when α is an integer versus a non-integer
Summary
The Norlund polynomials have been studied in many contexts. They were introduced by Norlund [10]. Don Knuth wrote to the author recently [8] (upon seeing an abstract of the author’s talk based on a preprint of this paper), urging the use of Stirling polynomials σn(x) as described on page 272 of [7]: zez ez − 1 x. With this definition, we associate Norlund and Stirling polynomials by the identity bnα n!. Because of the extensive use of the Bernoulli numbers in the Euler-MacLaurin formula and in many other asymptotic expansions, it is natural to investigate the asymptotic properties of the Norlund and Stirling polynomials, when α is fixed, and as n → ∞
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
