Higher order asymptotics from multivariate generating functions

Mark C. Wilson, University Of Auckland(Joint With Robin Pemantle, Alex Raichev)

Description

Higher order asymptotics from multivariate generating functions Mark C. Wilson, University of Auckland (joint with Robin Pemantle, Alex Raichev) Rutgers, 2009-11-19

• dz1 ?


• linear partial


• differential operators


• higher order


• siam review


• multivariate generating


• power series


• analytic function

Sujets

• Cambridge University
• Differential operator
• Power series
• Analytic function

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 9
• Langue: English

Extrait

Higher order asymptotics from multivariate
generating functions
Mark C. Wilson, University of Auckland
(joint with Robin Pemantle, Alex Raichev)
Rutgers, 2009-11-19Higher order asymptotics from multivariate generating functions
OutlineHigher order asymptotics from multivariate generating functions
Preliminaries
References
I Our papers at mvGF site:
www.cs.auckland.ac.nz/~mcw/Research/mvGF/ .
I P. Flajolet and R. Sedgewick, Analytic Combinatorics,
Cambridge University Press, 2009.
I A. Odlyzko, survey on Asymptotic Enumeration Methods in
Handbook of Combinatorics, Elsevier 1995, available from
www.dtc.umn.edu/~odlyzko/doc/asymptotic.enum.pdf.
I E. Bender, survey on Asymptotic Enumeration, SIAM Review
16:485-515, 1974.
I L. H ormander, The Analysis of Linear Partial Di erential
Operators (Ch 7), Springer, 1983.dI A (multivariate) sequence is a function a :N !C for some
xed d. Usually write a instead of a(r).r
I The generating function of the sequence is the formal powerP
rseries F (z) = a z .rr
dI If the series converges in a neighbourhood of 02C , then F
de nes an analytic function there.
Higher order asymptotics from multivariate generating functions
Preliminaries
Notation
I Boldface denotes a multi-index: z = (z ;:::;z ),1 d
r rr 1 dr = (r ;:::;r ), z =z :::z , dz =dz ^dz ^^dz .1 d 1 2 d1 dI The generating function of the sequence is the formal powerP
rseries F (z) = a z .rr
dI If the series converges in a neighbourhood of 02C , then F
de nes an analytic function there.
Higher order asymptotics from multivariate generating functions
Preliminaries
Notation
I Boldface denotes a multi-index: z = (z ;:::;z ),1 d
r rr 1 dr = (r ;:::;r ), z =z :::z , dz =dz ^dz ^^dz .1 d 1 2 d1 d
dI A (multivariate) sequence is a function a :N !C for some
xed d. Usually write a instead of a(r).rdI If the series converges in a neighbourhood of 02C , then F
de nes an analytic function there.
Higher order asymptotics from multivariate generating functions
Preliminaries
Notation
I Boldface denotes a multi-index: z = (z ;:::;z ),1 d
r rr 1 dr = (r ;:::;r ), z =z :::z , dz =dz ^dz ^^dz .1 d 1 2 d1 d
dI A (multivariate) sequence is a function a :N !C for some
xed d. Usually write a instead of a(r).r
I The generating function of the sequence is the formal powerP
rseries F (z) = a z .rrHigher order asymptotics from multivariate generating functions
Preliminaries
Notation
I Boldface denotes a multi-index: z = (z ;:::;z ),1 d
r rr 1 dr = (r ;:::;r ), z =z :::z , dz =dz ^dz ^^dz .1 d 1 2 d1 d
dI A (multivariate) sequence is a function a :N !C for some
xed d. Usually write a instead of a(r).r
I The generating function of the sequence is the formal powerP
rseries F (z) = a z .rr
dI If the series converges in a neighbourhood of 02C , then F
de nes an analytic function there.I the sequencefag is aperiodic;r
I the directions r :=r=jrj of interest for which we seek
asymptotics of a are generic, so nothing changes qualitativelyr
in a small neighbourhood;
I F =G=H with G;H entire functions but F is not itself
entire. Key examples: rational function that is not a
polynomial.
Higher order asymptotics from multivariate generating functions
Preliminaries
Standing assumptions
To avoid too many special cases, we restrict until further notice to
the following, most common, case:
I a 0 (the combinatorial case);rI the directions r :=r=jrj of interest for which we seek
asymptotics of a are generic, so nothing changes qualitativelyr
in a small neighbourhood;
I F =G=H with G;H entire functions but F is not itself
entire. Key examples: rational function that is not a
polynomial.
Higher order asymptotics from multivariate generating functions
Preliminaries
Standing assumptions
To avoid too many special cases, we restrict until further notice to
the following, most common, case:
I a 0 (the combinatorial case);r
I the sequencefag is aperiodic;rI F =G=H with G;H entire functions but F is not itself
entire. Key examples: rational function that is not a
polynomial.
Higher order asymptotics from multivariate generating functions
Preliminaries
Standing assumptions
To avoid too many special cases, we restrict until further notice to
the following, most common, case:
I a 0 (the combinatorial case);r
I the sequencefag is aperiodic;r
I the directions r :=r=jrj of interest for which we seek
asymptotics of a are generic, so nothing changes qualitativelyr
in a small neighbourhood;