Higher order asymptotics from multivariate generating functions

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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

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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;