%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% plain TeX
\magnification1200
\hsize = 6truein
\vsize = 9truein
\normalbaselines
\overfullrule=0pt

\font\smcp=cmcsc8 
\headline={\ifnum\pageno>1 
   {\smcp the electronic journal of combinatorics 3 (1996), \#R34
    \hfill\folio} \fi} 
\nopagenumbers

\newcount\seccount
\def\section#1\par{\vskip 30pt%
    \advance\seccount by 1
    \centerline{\bf\the\seccount. #1}
    \vskip12pt\noindent}
\def\nonumsection#1\par{\vskip 30pt%
    \centerline{\bf #1}
    \vskip12pt\noindent}

\def\betweenskip{\bigskip}

\newcount\excount  \newcount\lemmacount  \newcount\thmcount


\def\example #1{\betweenskip
    \advance\excount by 1\noindent
    {\bf Example \the\excount} (#1).~~}
\def\thmfont{\it}
\def\theorem #1{\advance\thmcount by 1
    \edef#1{\the\thmcount}
    \betweenskip\noindent
    {\bf Theorem~\the\thmcount}.\thmfont~~}
\def\lemma #1{\advance\lemmacount by 1
    \edef#1{\the\lemmacount}
    \betweenskip\noindent
    {\bf Lemma~\the\lemmacount}.\thmfont~~}
\def\endthing{\rm}
\def\Endthing{\rm\betweenskip}

\def\proof{\betweenskip\noindent
    {\bf Proof}: }
\def\Proof #1{\betweenskip\noindent
    {\bf Proof} (of #1): }
\def\qed{~~\vrule height1.4ex width1.2ex depth.1ex}

\newcount\eqcount \eqcount=0
\def\equno#1{\global\advance\eqcount by 1
    \xdef#1{(\the\eqcount)}
    (\the\eqcount)}
\def\eqlabel{\eqno\equno}

\def\itemskip{\smallskip}
\def\Item{\itemskip\item}
\def\Bitem{\Item{$\bullet$}}
\def\Itemitem{\itemskip\itemitem}
\def\Bitemitem{\Itemitem{$\bullet$}}

\def\vec#1{{\bf #1}}
\def\vectheta{{{\mathchoice
    {{\bf 0}\kern-1ex\vrule width.8ex height.8ex depth-.7ex}%
    {{\bf 0}\kern-1ex\vrule width.8ex height.8ex depth-.7ex}%
    {{\bf 0}\kern-.8ex\vrule width.6ex height.53ex depth-.43ex}%
    {\theta}}\,}}

\def\tr{^{\rm t}}
\def\tr{'}
\def\multint{\mathop{{\int\!\!\cdots\!\!\int}}\limits}
\def\inftysum#1{\sum_{#1=-\infty}^\infty}

\def\Exp  #1{\exp\left \{#1\right\}}
\def\Expg #1{\exp\bigl \{#1\bigr \}}
\def\Expgg#1{\exp\biggl\{#1\biggr\}}
\def\ExpG #1{\exp\Bigl \{#1\Bigr \}}
\def\ExpGG#1{\exp\Biggl\{#1\Biggr\}}

\def\mod#1{~({\rm mod}~#1)}

\def\set#1#2{\left\{#1\left|\; #2\right.\right\}}
\def\setg#1#2{\bigl\{#1\bigm| #2\bigr\}}
\def\setG#1#2{\Bigl\{#1\Bigm| #2\Bigr\}}

\def\Sqrt#1{\sqrt{\mathstrut\smash{#1}}}

\font\tenBB=msbm10
\font\sevenBB=msbm7
\font\fiveBB=msbm5
\def\Jap#1{\mathchoice{\hbox{\tenBB  #1}}{\hbox{\tenBB  #1}}%
                     {\hbox{\sevenBB #1}}{\hbox{\fiveBB #1}}}
\def\complex{\Jap C}
\def\reals{\Jap R}
\def\integers{\Jap Z}


\def\BG   {1}
\def\BR   {2}
\def\Dr   {3}
\def\FSe  {4}
\def\FSo  {5}
\def\GR   {6}
\def\Gar  {7}
\def\GJ   {8}
\def\Har  {9}
\def\Hay {10}
\def\LP  {11}
\def\Odl {12}
\def\Pre {13}

\null\vfill

\centerline{Admissible Functions and Asymptotics for}
\centerline{Labelled Structures by Number of Components}

\vskip12pt

\centerline{Edward A. Bender}
\centerline{Center for Communications Research}
\centerline{4320 Westerra Court}
\centerline{San Diego, CA 92121, USA}
\centerline{\tt ed@ccrwest.org}

\medskip

\centerline{L. Bruce Richmond}
\centerline{Department of Combinatorics and Optimization}
\centerline{University of Waterloo}
\centerline{Waterloo, Ontario N2L 3G1, Canada}
\centerline{\tt lbrichmo@watdragon.uwaterloo.ca}


\medskip

\centerline{Submitted: August 19, 1996; Accepted: November 27, 1996}

\bigskip

\nonumsection{Abstract}

Let $a(n,k)$ denote the number of combinatorial structures of size
$n$ with $k$ components.
One often has $\sum_{n,k} a(n,k)x^ny^k/n! = \Expg{yC(x)}$, where
$C(x)$ is frequently the exponential generating function for connected
structures.
How does $a(n,k)$ behave as a function of $k$ when $n$ is large and
$C(x)$ is entire or has large singularities on its circle of
convergence?
The Flajolet-Odlyzko singularity analysis does not directly apply in
such cases.
We extend some of Hayman's work on admissible functions of a single
variable to functions of several variables.
As applications, we obtain asymptotics and local limit theorems for
several set partition problems, decomposition of vector spaces,
tagged permutations, and various complete graph covering problems.

\vfill\vfill\vfill

\noindent
1991 AMS Classification Numbers.
\quad
Primary: 05A16

\noindent
$\phantom{\hbox{1991 AMS Classification Numbers.}}$
\quad
Secondary: 05A18, 15A03, 41A60

\eject


\section Introduction

A variety of combinatorial structures can be decomposed into
components so that the generating function for all structures is the
exponential of the generating function for components:
$A(x)=e^{C(x)}$.
(This is a single variable instance of the {\it exponential formula}.)
In this case, $A(x,y)=e^{yC(x)}$ is the generating function for
structures by number of components and is an ordinary generating
function in $y$.
For the present discussion, we assume $C(x)$ is an exponential
generating function.
One often wishes to study $a_{n,k}=[x^ny^k/n!]A(x,y)$, the number of
$k$-component structures of size $n$.
In particular, one may ask how $a_{n,k}$ varies with $k$ for fixed
large $n$.
{From} a somewhat different viewpoint, one may want to study the
probability distribution for the random variable $X_n$ given by
${\rm Pr}(X_n=k)=a_{n,k}\!\bigm/\!\sum_k a_{n,k}$ as $n\to\infty$.

One approach is to observe that $k!\,a_{n,k}=[x^n/n!](C(x))^k$.
Such methods are useful for estimating the larger coefficients of
$(C(x))^k$ as $n$ varies and $k$ is large, which is not the same as
studying the larger values of $a_{n,k}$ for fixed $n$.
Consequently, one may find that the method only yields estimates in
the tail of the distribution of $X_n$.
See Gardy~[\Gar] for a discussion of these methods.
However, it is sometimes possible to extend the range to include the
larger values of $a_{n,k}$.
See Drmota [\Dr], especially Section 3.

Working directly with $A(x,y)$ is likely to provide estimates for the
larger coefficients rather than tail probabilities.
Unfortunately, multivariate generating functions have proven to be
recalcitrant subjects for asymptotic analysis.
When $A(x,y)$ has small singularities, methods akin to Darboux's
Theorem may be useful.
See Flajolet and Soria~[\FSo] and Gao and Richmond~[\GR] for examples.
See Odlyzko~[\Odl] for an extensive discussion of asymptotic methods.

In order to study a variety of single-variable functions with large
singularities, Hayman~[\Hay] defined a class of admissible functions
in such a way that (a)~class members have useful properties and
(b)~class membership can easily be established for a variety of
functions. 
We refer to his functions as H-admissible.
Hayman's results include:
\Bitem
If $p$ is a polynomial and the coefficients of $e^p$ are eventually
strictly positive, then $e^p$ is H-admissible.
\Bitem
If $f$ is H-admissible, so is $e^f$.
\Bitem
If $f$ and $g$ are H-admissible, so is $fg$.
\smallskip\noindent
In~[\BR] we made a somewhat ill-considered attempt to extend his
notions to multivariate generating functions.
In this paper we present a simpler alternative definition which has
applications to the problems described in the first paragraph and
which includes H-admissible functions as a special single variable
case.

The next section contains our definition for a class of
admissible functions and an estimate for coefficients of such
functions.
Section~3 provides theorems for establishing the admissibility of a
variety of functions, especially those related to counting structures
by number of components of various types via the exponential formula.
Applications are presented in Section~4.
Proofs of the theorems are given in Section~5.



\section Definitions and Asymptotics

Let $\vec x$ be $d$-dimensional, let $\reals_+$ be the positive
reals, and let $\vec re^{i\vectheta}$ be the vector whose $k$th
component is $r_ke^{i\theta_k}$.
Suppose $f(\vec x)$ has a power series expansion 
$\sum a_{\vec n}\vec x^{\vec n}$ where $\vec x^{\vec n}$ is the
product of $x_k^{n_k}$.
The lattice $\Lambda_f\subseteq\integers^d$ is the $\integers$-module
spanned by the differences of those $\vec n$ for
which $a_{\vec n}\ne0$.
We {\it assume\/} that $\Lambda_f$ is $d$-dimensional.
Let $d(\Lambda_f)$ be the absolute value of the determinant of a
basis of $\Lambda_f$.
In other words, $d(\Lambda_f)$ is the reciprocal of the density of
$\Lambda_f$ in $\integers^d$.
The polar lattice $\Lambda^*_f\subseteq\reals^d$ is the
$\integers$-module of vectors $\vec v$ such that $\vec v\cdot\vec u$
is an integer for all $\vec u\in\Lambda_f$.
If $\vec v_1,\ldots,\vec v_d$ is a $\integers$-basis for
$\Lambda^*_f$, a {\it fundamental region\/} for $f$ is the
parallelepiped
$$
\Phi(f) = \Bigl\{c_1\vec v_1+\cdots+c_d\vec v_d \Bigm|
   -\pi \le c_k \le \pi ~~{\rm for}~~ 1 \le k \le d\Bigr\}.
$$
Since the basis for a lattice is not unique, neither is $\Phi(f)$.
If coefficients $a_{\vec n}$ are nonzero for all sufficiently large
$\vec n$, then $\Lambda^*_f=\Lambda_f=\integers^d$,
$d(\Lambda_f)=1$, and we may take $\Phi(f)=[-\pi,\pi]^d$.

We say that $f(x)=o_{u(x)}(g(x))$ for $x$ in some set $\cal S$ if
there is a function $\lambda(t)\to0$ as $t\to\infty$ such that
$|f(x)/g(x)|\le\lambda(|u(x)|)$ for all $x\in\cal S$.
The extension to equations involving little-oh expressions is done in
the usual manner.

If $B$ is a square matrix, $|B|$ denotes the determinant of $B$.
We use $\vec v\tr$ and $S\tr$ to denote the transpose of the vector
$\vec v$ and the matrix $S$.

\betweenskip

\noindent
{\bf Definition of Admissibility}.~
Let $f$ be a $d$-variable function that is analytic at the origin
and has a fundamental region $\Phi(f)$.
When $\Lambda_f$ is $d$-dimensional, we say that $f(\vec x)$ is 
{\it admissible in ${\cal R}\subseteq\reals_+^d$ with angles
$\Theta$\/} if there are (i)~a function $\Theta$ from $\cal R$ to open
subsets of $\Phi(f)$ containing $\vec 0$ and (ii)~functions
$$
\vec a:\complex^d\to\complex^d
~~{\rm and}~~
B:\complex^d\to\complex^{d\times d}
$$
such that
\Item{(a)}
$f(\vec x)$ is analytic whenever $\vec r\in\cal R$ and  
$|x_i|\le r_i$ for all $i$;
\Item{(b)}
$B(\vec r)$ is positive definite for $\vec r\in\cal R$;
\Item{(c)}
the diameter of $\Theta(\vec r)$ is $o_u(1)$, where $u=|B(\vec r)|$;
\Item{(d)}
for $\vec r\in\cal R$, $u=|B(\vec r)|$, and 
$\vectheta\in\Theta(\vec r)$, we have
$$
f(\vec re^{i\vectheta})
= f(\vec r)\bigl(1+o_u\bigl(1)\bigr)
  \ExpG{i\vec a(\vec r)\tr\vectheta 
        - \vectheta\tr B(\vec r)\vectheta/2};
\eqlabel\eqAasymp
$$
\Item{(e)}
For $\vec r\in\cal R$, $u=|B(\vec r)|$, and $\vectheta$ in the
complement of $\Theta(\vec r)$ relative to $\Phi(f)$, we have
$$
f(\vec re^{i\vectheta}) 
= o_u\bigl(f(\vec r)\bigr)\bigm/|B(\vec r)|^{1/2}.
\eqlabel\eqAsmall
$$

\noindent
We say $f$ is {\it super-admissible\/} if \eqAsmall{} can be
replaced by
$$
f(\vec re^{i\vectheta}) 
= o_u\bigl(f(\vec r)\bigr)\bigm/|B(\vec r)|^t
\eqlabel\eqAsuper
$$
for all $t$, where $o_u$ may depend on $t$.
\par
Usually one can let $\vec a(\vec x)$ and $B(\vec x)$ be the gradient
and Hessian of $\log f$ with respect to $\log\vec x$; that is,
$$
a_i(\vec x) = {x_i\partial f\over f\partial x_i}
~~{\rm and}~~
B_{i,j} = x_j{\partial a_i\over\partial x_j} = B_{j,i}.
$$
We call these the {\it gradient\/} $\vec a$ and $B$.
\betweenskip

Since H-admissible functions satisfy $b(r)\to\infty$ as $r\to R$,
it is easily verified that this definition includes H-admissible
functions.
The asymptotic result for H-admissible functions holds for our
admissible functions:


\theorem\asympthm
Suppose $f(\vec x)$ is admissible in $\cal R$.
Let $\vec k$ be any vector such that $[\vec x^{\vec k}]f(\vec x)\ne0$,
let $u=|B(\vec r)|$, and let $\vec v = \vec a(\vec r)-\vec n$.
Then
$$
\left[\vec x^{\vec n}\right] f(\vec x)
= {d(\Lambda_f)f(\vec r)\vec r^{-\vec n}
   \over(2\pi)^{d/2}|B(\vec r)|^{1/2}}
\Bigl(\Expg{-\vec v^{\rm t}B(\vec r)^{-1}\vec v/2} + o_u(1)\Bigl)
\eqlabel\asymptotics
$$
for $\vec r\in\cal R$ and $\vec n-\vec k\in\Lambda_f$.
\endthing




\section Classes of Admissible Functions

In this section we state various theorems that allow us to establish
admissibility for generating functions for a variety of combinatorial
structures.
We begin with two theorems for multiplying admissible functions:
Theorem~2 allows us to combine structures of similar size and
Theorem~3 allows us to make (minor) modifications in our structures.
Theorem~4 allows us to do simple multisection of admissible functions;
that is, limit attention to structures with simple congruence properties.
As already remarked H-admissible functions are admissible (with
gradient $\vec a=a$ and $B=b$).
In addition, the exponentials of polynomials considered in Theorems~2
and~3 of [\BR] are superadmissible.
The proofs given there suffice, but the notation differs somewhat:
$\Theta(\vec r)$ is called ${\cal D}(\vec r)$.
It seems likely that one could extend the results in~[\BR] to larger
classes of polynomials and/or larger domains $\cal R$.
In Theorems~5--7 we construct a variety of admissible functions of the
form $\Exp{yC(x)}$.

Suppose $f$ is admissible in $\cal R$ with angles $\Theta$.
Suppose there are variables not appearing in $f$.
We extend $\cal R$ and $\Theta$ to include these variables by forming
the Cartesian product of $\cal R$ with copies of $(0,\infty)$ and the
Cartesian product of $\Theta$ with copies of $[-\pi,\pi]$.
We extend $\vec a$ and $B$ by adding entries of zeroes; however,
we ignore the appended coordinates when computing $|B|$ and when
determining admissibility.

\theorem\prodONEthm
We assume the various objects associated with $f$ and $g$ are extended
as described above so that they include the same set of variables.
Suppose that
\Bitem
$f$ is super-admissible in $\cal R$ with angles $\Theta_f$;
\Bitem
$g$ is super-admissible in $\cal R$ with angles $\Theta_g$;
\Bitem
$|B_f(\vec r)+B_g(\vec r)|$ is unbounded on $\cal R$;
\Bitem
there are constants $C$ and $k$ such that
$$
|B_f(\vec r)+B_g(\vec r)|
\le C \min\bigl(|B_f(\vec r)|^k,\,|B_g(\vec r)|^k\bigr)
~~\hbox{\thmfont for}~~
\vec r\in\cal R.
\eqlabel\eqBineq
$$
Then $fg$ is super-admissible in $\cal R$ with
angles $\Theta_{fg}(\vec r)=\Theta_f(\vec r)\cap\Theta_g(\vec r)$.
Furthermore, $\Lambda_{fg}=\Lambda_f+\Lambda_g$, the the set of
vectors $\vec u+\vec v$ where $\vec u\in\Lambda_f$ and  
$\vec v\in\Lambda_g$, and we may take
$$
\vec a_{fg} = \vec a_f + \vec a_g
~~\hbox{\thmfont and}~~
B_{fg} = B_f + B_g,
$$
\Endthing


\noindent
There are two important observations concerning Theorem~\prodONEthm:
\Bitem
In using it, one normally chooses ${\cal R}$ to be as big a subset as
possible of ${\cal R}_f\cap{\cal R}_g$ such that \eqBineq{} holds.
\Bitem
Hayman shows that, if $f(x)$ is H-admissible, then so is $f(x)+p(x)$
when $p(x)$ is a polynomial.
This is not true for admissible functions.
For example, if $f(x)=g(x^2)$ is admissible, $f(x)+x$ is not.
This problem could be avoided if we changed the definition of
$\Lambda_f$ to use only sufficiently large $n$ rather than all $n$.
Unfortunately Theorem~\prodONEthm{} would fail because, for example
$e^{x^2}$ and $e^{x^2}+x$ would be super-admissible but their product
would not be.


\theorem\prodTWOthm
Suppose that $f$ is admissible (resp.~super-admissible) in $\cal R$ with
angles $\Theta$ and that
$g(\vec re^{i\vectheta})$ is analytic for $\vec r\in\cal R$.
Let $u=|B_f(\vec r)|$.
Suppose that there are $\vec a_g$ and $B_g$ such that
\Item{(a)}
$\Lambda_g\subseteq\Lambda_f$;
\Item{(b)}
for $\vec r\in\cal R$ and $\vectheta\in\Theta(\vec r)$,
$$g(\vec re^{i\vectheta}) = g(\vec r)
\Expg{i\vec a_g(\vec r)\tr\vectheta
     - \vectheta\tr B_g(\vec r)\vectheta +o_u(1)};
\eqlabel\prodTWOb
$$
\Item{(c)}
there is a constant $C$ such that
$|g(\vec re^{i\vectheta})| \le Cg(\vec r)$ for $\vec r\in\cal R$;
\Item{(d)}
there is a constant $C$ such that
$|B_f(\vec r)+B_g(\vec r)|\le C|B_f(\vec r)|$ for $\vec r\in\cal R$.
\par\noindent
Then $fg$ is admissible (resp.~super-admissible) in $\cal R$ with
angles $\Theta$ and we may take 
$$
\vec a_{fg} = \vec a_f + \vec a_g
~~\hbox{\thmfont and}~~
B_{fg} = B_f + B_g,
$$
\Endthing

\noindent
There are three important observations concerning Theorem \prodTWOthm:
\Bitem
We do not assume that $g$ is admissible.
\Bitem
One may need to extend $\vec a_g$ and $B_g$ as described before
Theorem~\prodONEthm.
In this case, $\Lambda_g$ should also be extended by adding
components containing zeroes to its vectors.
\Bitem
If $\vec a_g$ and $B_g$ are so small that \prodTWOb{} reduces to
$g(\vec re^{i\vectheta})=g(\vec r)\bigl(1 + o_u(1)\bigr)$, the
contribution of $g$ to the asymptotics in Theorem~\asympthm{} is
simply a factor of $g(\vec r)$.

\theorem\multiADMISthm
Let $f(\vec x)=\sum a_{\vec n}\vec x^{\vec n}$ be a $d$-variable
admissible (resp. super-admissible) function.
Let $\Lambda$ be a sublattice of $\Lambda_f$ and suppose $\vec k$ is
such that $a_{\vec k}\ne0$.
Define
$$
g(\vec x) = \sum_{\vec n\in\Lambda}
a_{\vec k+\vec n}\vec x^{\vec k+\vec n}.
$$
We may take $\Phi(g)\subseteq\Phi(f)$.
The function $g$ is admissible (resp. super-admissible) with
$$\Lambda_g=\Lambda,~~
\vec a_g =\vec a_f,~~
B_g = B_f,~~
{\cal R}_g = {\cal R}_f,~~
\hbox{\thmfont and}~~
\Theta_g = \Theta_f.
$$
\endthing

\theorem\multiONEthm
Suppose that
\Bitem
 $f(x)=\sum a_nx^n$ is an H-admissible function with $a_0=0$ and
(possibly infinite) radius of convergence $R$;
\Bitem
$\cal K$ is a subset of $\{0,1,\ldots,m-1\}$;
\Bitem
$\lambda_k$ are nonnegative reals for $0 \le k < m$ with $\lambda_k>0$
if and only if $k\in\cal K$.
\itemskip\noindent
Define $\lambda_n=\lambda_k$ whenever $n\equiv k\mod m$,
$$
g(x) = \sum_{n=0}^\infty\lambda_n a_nx^n,
\eqlabel\eqmulti
$$
and $\overline\lambda 
= \bigl(\sum_{k=0}^{m-1}\lambda_k\bigr)\!\bigm/\!m$.
Then:
\Item{(a)}
For some $R_0<R$, the function $h(x)=e^{g(x)}$ is super-admissible in\break
${\cal R}=\set{r}{R_0<r<R}$ with angles
$$
\Theta(r)=\setG{\theta}{|\theta|<1/g(r)^{1/3+\epsilon}}
$$
and the gradient $\vec a$ and $B$, provided $\epsilon>0$ is
sufficiently small.
Also
$$
a_h(r)\sim \overline\lambda rf'(r)
~~\hbox{\thmfont and}~~
B_h(r)\sim \overline\lambda r(rf'(r))'.
$$
If $d$ denotes the greatest common divisor of $m$ and the elements of
$\cal K$, then $\Lambda_h$ is generated by $(d)$; that is,
$\Lambda_h=\integers(d)$.
\Item{(b)}
For some $R_0<R$ and all $\delta>0$, the function $h(x,y)=e^{yg(x)}$
is super-admissible in
$$
{\cal R} = \setG{(r,s)}{R_0<r<R~~{\rm and}~~
             g(r)^{\delta-1} < s < g(r)^{1/\delta}}.
$$
with angles
$$
\Theta(r,s)
= \setG{\vectheta}{|\theta_k|<1/(sg(r))^{1/3+\epsilon}}
$$
and the gradient $\vec a$ and $B$, provided $\epsilon>0$ is
sufficiently small.
Also
$$
\vec a_h(r,s) \sim \overline\lambda s
\pmatrix{rf'(r)\cr f(r)\cr},
~~
B_h(r,s) \sim \overline\lambda s
\pmatrix{r(rf'(r))'& rf'(r)\cr rf'(r)&f(r)\cr},
$$
and
$$
|B_h(r,s)| = {s^2\over2}
       \sum_{n,k}(n-k)^2\lambda_n a_n\lambda_k a_k r^{n+k}.
\eqlabel\eqBdet
$$
If $k\in\cal K$ and $d$ denotes the greatest common divisor of $m$
and differences of pairs of elements of $\cal K$, then $\Lambda_h$ is
generated by $(k,1)$ and $(d,0)$; that is 
$\Lambda_h=\integers(d,0)+\integers(k,1)$.
\endthing


\theorem\multiTWOthm
Suppose that 
\Bitem
$f(x)$ is analytic in $|x|<1$ with $f(0)=1$ and $f(x)\ne0$ for
$|x|<1$;
\Bitem
$x^{-k}\log f(x)$ has a power series expansion in powers of
$x^m$ for some integers $k$ and $m$ with $0\le k<m$;
\Bitem
$C(r)$ is a positive function on $(0,1)$ with
$$
(1-r){C'(r)\over C(r)}\to 0 ~~\hbox{\thmfont as}~~ r\to1;
$$
\Bitem
there exist positive constants $\alpha$ and $\beta$ with $\beta<1$
such that 
$$
\log f(x) \sim C(|x|)(1-x)^{-\alpha}
~~{\thmfont as}~~
x\to 1
$$
uniformly for $|\arg x|\le \beta(1-r)$ and such that
$$
\bigl|\log f(re^{i\theta})\bigr| 
\le \bigl|\log f(re^{i\beta(1-r)})\bigr|
~~\hbox{\thmfont for}~~
\beta(1-r) \le |\theta| \le \pi/m.
\eqlabel\eqcirclebound
$$
\itemskip\noindent
Then, with $g(r)=\log f(r)$:
\Item{(a)}
For some $R_0<1$, the function $f(x)$ is super-admissible in
${\cal R}=\set{r}{R_0<r<1}$ with angles
$$
\Theta(r)=\setG{\theta}{|\theta|<(1-r)/g(r)^{1/3+\epsilon}}
$$
and the gradient $\vec a$ and $B$, provided $\epsilon>0$ is
sufficiently small.
Also $\Lambda_f=\integers(d)$ where $d=\gcd(k,m)$.
\Item{(b)}
For some $R_0<1$ and all $\delta>0$, the function $h(x,y)=f(x)^y$
is super-admissible in
$$
{\cal R} = \setG{(r,s)}{R_0<r<1~~{\rm and}~~
             g(r)^{\delta-1} < s < g(r)^{1/\delta}}.
$$
with angles
$$
\Theta(r,s)
= \setG{(\theta,\varphi)}
  {|\theta|<(1-r)/(sg(r))^{1/3+\epsilon}
   ~~{\rm and}~~
   |\varphi|<1/(sg(r))^{1/3+\epsilon}}
$$
and the gradient $\vec a$ and $B$, provided $\epsilon>0$ is
sufficiently small.
Also $\Lambda_h=\integers(m,0)+\integers(1,k)$.
\endthing


\theorem\peakedthm
Suppose that $f(x)=\sum a_nx^n$ has radius of convergence $R>0$ and
that $a_n\ge0$ for all $n$.
Let $\nu(r)$ be the value of $n$ such that $a_nr^n$ is a maximum.
Suppose that, for every $\epsilon>0$, $\nu(r)=o(f(r)^\epsilon)$ as
$r\to R$.
Suppose that there exist $\rho<1$, $A$, a function $K(m)>0$ and an
$N$ depending on $\rho$, $A$, and $K$ such that, for all
$\nu=\nu(r)>N$ and all $k>0$,
$$
A\rho^k \ge {a_tr^t\over a_\nu r^\nu}
~~\hbox{\thmfont where}~~
t = \nu \pm k
\eqlabel\eqsmallterms
$$
and
$$
K(m) \le {a_jr^j\over a_\nu r^\nu}
~~\hbox{\thmfont whenever}~~
|j-\nu|\le m.
\eqlabel\eqlargeterms
$$
Then $f(x)$ is entire and the conclusions of Theorem~\multiONEthm{}
hold for it.
\endthing






\section Applications

Admissibility allows one to compute asymptotics for the coefficients
of a variety of generating functions, but the accuracy of the method
is limited by one's ability to estimate the solution of 
$\vec a(\vec r)=\vec n$ and then estimate $f(\vec r)$ and
$\vec r^{\vec n}$ accurately.
On the other hand, admissibility allows one to establish asymptotic
normality rather easily, and obtaining asymptotic estimates for the
means and covariances is usually fairly easy:
Suppose our generating function is of the form $f(\vec x,\vec y)$ and
is ordinary in $\vec y$.
Partition all vectors and matrices into block form according the the
two sets of variables $\vec x$ and $\vec y$.
Let $a_{\vec n,\vec k}$ be the coefficients of $f$.
Set $\vec a(\vec r, \vec 1)=(\vec n,\vec k^*)$, solve for
$\vec r$ asymptotically in terms of $\vec n$ and use this to compute
$\vec k^*$ and $B(\vec r,\vec 1)$ asymptotically as functions of
$\vec n$.
Let $\vec n$ go to infinity in a way that $(\vec r,\vec 1)\in\cal R$
and $|B|\to\infty$.
{From} Theorem~\asympthm{} and the formula
([\Pre,~pp.\thinspace25--26])
$$
\pmatrix{B_{1,1}&B_{1,2}\cr B_{1,2}\tr&B_{2,2}}^{-1}
= \pmatrix{A&C\cr C\tr&D^{-1}}
~~{\rm where}~~
D = B_{2,2}-B_{1,2}\tr (B_{1,1})^{-1}B_{1,2},
\eqlabel\eqvariance
$$
it follows that $a_{\vec n,\vec k}/\sum_{\vec k}a_{\vec n,\vec k}$
satisfies a local limit theorem with means vector and covariance
matrix asymptotic to $\vec k^*$ and $D$, respectively.
When $\vec x$ and $\vec y$ are 1-dimensional, $D=|B|/B_{1,1}$.


\example
{Stirling Numbers of the Second Kind}
With multivariate situations, it is important to know the range
of values of the subscripts of the coefficients (rather than the
variables in the generating function) for which the asymptotics
applies.
We examine $\Exp{y(e^x-1)}$, the generating function for $S(n,k)$,
the Stirling numbers of the second kind.
Let $|x|=r$ and $|y|=s$.
Since $f(x)=e^x-1$ is H-admissible, we can apply
Theorem~\multiONEthm(b) with $m=1$ and $\lambda_0=1$.
(There is no multisection.)
Then
$$
\vec a(r,s) = s\pmatrix{re^r\cr e^r-1},
~~
B(r,s) = s\pmatrix{(r^2+r)e^r & re^r\cr re^r & e^r-1\cr},
$$
and
$$
{\cal R} = \setG{(r,s)}
  {R_0<r~~{\rm and}~~e^{r(\delta-1)} < s < e^{r/\delta}}.
$$
Setting $\vec a=(n,k)$, we obtain
\Item{(i)}
$n/k\sim r$ and
\Item{(ii)}
the value of $r$ lies between the solutions of $n=re^{r\delta}$ and
$n=re^{r(1+1/\delta)}$.

\noindent
Thus $r$ is between roughly $\delta\log n$ and $\log n/\delta$.
It follows from this and (i) that we have admissibility as
long as $(k\log n)/n$ is bounded away from 0 and $\infty$.
Consequently, for any positive constants $c$ and $C$,
Theorem~\asympthm{} provides uniform asymptotics for
$S(n,k)$ when
$$
{cn\over\log n} < k < {Cn\over\log n}.
\eqlabel\eqSTIRLING
$$

If, instead, we set $\vec a(r,1)=(n,k^*)$, we obtain the equations
$n=re^r$ and $k^*=e^r-1$.
Hence $r\sim\log n$ and $k^*\sim n/\log n$.
Using \eqvariance, we obtain
$$
D = (e^r-1) - (re^r)^2/(r^2+r)e^r \sim e^r/r \sim n/(\log n)^2
$$
and so $S(n,k)$ satisfies a local limit theorem with mean and
variance asymptotic to $n/\log n$ and $n/(\log n)^2$,
respectively, a result obtained by Harper~[\Har].\qed


\example
{Other Set Partitions}
The coefficient of $y_1^{k_1}y_2^{k_2}\cdots x^n/n!$ in
$$
f(x,\vec y) = \Expgg{\sum_{k=1}^\infty y_k x^k/k!}
\eqlabel\partitions
$$
is the number of partitions of an $n$-set with exactly $k_i$ blocks
of size $i$.
In the previous example, we set $y_i=y$ for all $y$.
Other results are possible, particularly when one is interested in
residue classes modulo $m$.
Some illustrative examples follow.

Let ${\cal K}\subset\{0,1,\ldots,m-1\}$ and set $y_i=1$ when $i$
modulo~$m$ is in $\cal K$ and 0 otherwise.
Since $e^x-1$ is H-admissible, $g(x)=f(x,\vec y)$ is admissible by
Theorem~\multiONEthm(a).
The coefficient of $x^n/n!$ is the number of set partitions of a
$n$-set with block sizes congruent modulo $m$ to elements in 
$\cal K$.

Suppose, instead, we set $y_i=y$ when $i$ modulo $m$ is in $\cal K$
and 0 otherwise.
Then Theorem~\multiONEthm(b) applies and the coefficient of
$x^ny^k/n!$ in $g(x,\vec y)$ is the number of partitions of an $n$-set
with exactly $k$ blocks all of whose sizes are congruent modulo~$m$
to elements in $\cal K$.
Asymptotic normality follows as it did for the Stirling numbers and
the mean and variance are asymptotically the same as we found there.

If all but a finite number of $y_i=0$ and the rest are equal to $y$,
$f(x,\vec y)$ is the exponential of a polynomial and admissibility
follows by the methods in~[\BR] unless the polynomial is a monomial.

Not every choice of which $y_i$ are zero leads to an admissible
function.
For example, it can be shown that $f(x)=\Exp{\sum x^{n_k}/(n_k)!}$ is
not admissible if the $n_k$ grow sufficiently rapidly since
$f(re^{i\theta})/f(r)$ is not sufficiently small when $r$ is near
$n_k$ and $\theta$ is a multiple of $2\pi/n_k$.

{From} \partitions, $\bigl[x^ny_e^{k_e}y_o^{k_o}/n!\bigr]
\Bigl(\Expg{y_e(\cosh x-1)}\Expg{y_o\sinh x}\Bigr)$ is the number of
partitions of an $n$-set that have $k_e$ blocks of even size and
$k_o$ blocks of odd size.
By Theorem~\multiONEthm(b), $f(x,y_e)=\Expg{y_e(\cosh x-1)}$ and
$g(x,y_o)=\Expg{y_o\sinh x}$ are super-admissible and
$$
\eqalign{
{\cal R}_f = {\cal R}_g
&= \setG{(r,s)}{R_0<r~~{\rm and}~~
    e^{(\delta-1)r} < s < e^{r/\delta}}\cr
\Theta_f = \Theta_g
&= \setG{\vectheta}
        {|\theta_k|<(e^{-r}/s))^{1/3+\epsilon}}\cr
B_f(r,s_e)
&= s_e\pmatrix{r^2\cosh r + r\sinh r & r\sinh r\cr
                            r\sinh r & \cosh r-1\cr}\cr
B_g(r,s_o)
&= s_o\pmatrix{r^2\sinh r + r\cosh r & r\cosh r\cr
                            r\cosh r & \sinh r\cr}.\cr}
\eqlabel\eqOddEven
$$
Hence
$$
|B_f|=s_e^2r(\sinh r-r)(\cosh r -1)\sim s_e^2re^{2r}/4
$$
and
$$
|B_g|=s_o^2r(\sinh r\cosh r - r)\sim s_o^2re^{2r}/4.
$$
We now apply Theorem~\prodONEthm.
Since
$$
B_f+B_g = \pmatrix{
r(rs_e+s_o)\cosh r + r(rs_o+s_e)\sinh r 
             & rs_e \sinh r    & rs_o \cosh r\cr
rs_e \sinh r &  s_e(\cosh r-1) & 0           \cr
rs_o \cosh r & 0               &  s_o \sinh r\cr},
$$
we have
$$
\eqalign{
|B_f+B_g|
&= rs_es_o(\cosh r-1)
 \bigl(s_e\sinh r(\sinh r-r) + s_o(\cosh r \sinh r -r)\bigr)\cr
&\sim s_es_o(s_e+s_o)re^{3r}/8.\cr}
$$
It follows that $fg$ is super-admissible in
$$
{\cal R}
= \setG{(r,s_e,s_o)}{R_0<r~~{\rm and}~~
       e^{(\delta-1)r} < s_e,s_o < e^{r/\delta}}
$$
with angles
$$
\Theta(r,s_e,s_o)
= \setG{\vectheta}
  {|\theta_k|<(e^{-r}/\max(s_e,s_o)))^{1/3+\epsilon}}.
$$
Consequently we obtain asymptotics for the coefficients provided
$k_e\log n/n$ and $k_o\log n/n$ are bounded away from 0 and $\infty$.

Suppose we want to count partitions by the number of non-singleton
blocks.
The generating function is $f(x,y)g(x)$ where
$$
f(x,y) = \Expg{y(e^x-x-1)}
~~{\rm and}~~
g(x) = e^x.
$$
Apply Theorem~\multiONEthm(b) without multisection to show that $f$
is super-admissible with angles
$$
\Theta(r,s) 
= \setG{\vectheta}{|\theta_k|<(e^{-r}/s)^{1/3+\epsilon}}.
$$
Now apply Theorem~\prodTWOthm.
The conditions on $g$ are easily checked.
In particular, one must verify \prodTWOb{} for
$|\theta|<e^{-\delta r}$. 
In this range
$$
\Exp{re^{i\theta}} = \Expg{r\bigl(1+O(\theta)\bigr)} \sim e^r.
$$
Unfortunately, the theorems do not allow us to do the complementary
problem---count partitions by number of singleton blocks using the
generating function\break
$e^{xy}\Exp{e^x-1-x}$.

Fix integers $k$ and $m$.
Let $a_{n,j}$ be the number of partitions of an $n$-set into $j$
blocks such that the total number of elements in blocks of odd
cardinality is congruent to $k$ modulo $m$.
The generating function is $fh$ where
$$
f(x,y) = \Exp{y(\cosh x -1)},~~
g(x,y) = \Exp{y\sinh x},
$$
and $h(x,y)$ is the sum of those terms in $g$ for which the power of
$x$ modulo $m$ is $k$.
By Theorem~\multiONEthm, $f$ and $g$ are super-admissible with the
$\cal R$, $\Theta$ and $B$ given by \eqOddEven.
By Theorem~\multiADMISthm{} with $\Lambda=m\integers\times \integers$,
$h$ is super-admissible.
By Theorem~\prodONEthm{}, $fh$ is super-admissible and, furthermore,
we may take $\cal R$ and $B$ to be as in Example~1.
It follows that asymptotics are obtainable for $a(n,j)$ whenever
\eqSTIRLING{} holds.\qed


\example
{Decompositions of Vector Spaces}
Let $D_{n,k}(q)$ be the number of decompositions of an
\hbox{$n$-dimensional} vector space over ${\rm GF}(q)$ as a direct sum
of $k$ nonzero subspaces where the order of the subspaces is
irrelevant. 
It follows from Example~11 of Bender and Goldman~[\BG] that
$$
h(x,y)
= 1 + \sum_{n=1}^\infty\sum_{k=1}^n
     {D_{n,k}(q)x^ny^k\over c_n}
= e^{yf(x)}
~~{\rm where}~~
f(x) = \sum_{n=1}^\infty {x^n\over c_n}
\eqlabel\vectoreqn
$$
and $c_n = (q^n-1)\cdots(q^n-q^{n-1})$.
Let $C_i$ stand for some positive constant.
We apply Theorem~\peakedthm{} without multisection.
Note that $c_n\sim Qq^{n^2}$ where $Q=\prod(1-q^{-k})$.
The largest term in $f(r)$ is near the solution $\nu$ of
$r=q^{2\nu}$.
If $m=\nu\pm t$ is a positive integer, then a simple calculation
shows that
$$
C_1 q^{-t^2} < {r^m/c_m\over r^{\nu}/Qq^{\nu^2}} < C_2 q^{-t^2}.
$$
Thus Theorem~\peakedthm{} applies.
We obtain $n/k\sim\nu=(\log_q r)/2$.
Since $C_3q^{\nu^2} < f(r) <  C_4q^{\nu^2}$ and the theorem requires 
$f(r)^\delta < sf(r) < f(r)^{1/\delta}$, it follows that
$\epsilon(\log n)^{1/2} < \nu < (\log n)^{1/2}/\epsilon$.
Thus asymptotics are obtained when $k(\log n)^{1/2}/n$ is bounded away
{from} 0 and $\infty$.

By solving $(n,k^*) = \vec a(r,1) = (rf'(r),f(r))$ for $r$ and $k^*$,
the asymptotic formula gives us a local limit theorem for
$D_{n,k}(q)$ as $n\to\infty$.
We now study the asymptotic mean and variance.
Define $\nu$ and $\delta$ as functions of $r$ by
$$
\nu = \bigl[(\log_q r)/2\bigr] = (\log_q r)/2 - \delta.
$$
Using $\nu\to\infty$, \eqsmallterms, \eqlargeterms, and \eqBdet, we
have
$$
\eqalignno{
f(r)
&\sim {1\over Q}\inftysum t {r^{\nu+t}\over q^{(\nu+t)^2}}
= {q^{\nu^2+2\delta\nu}\over Q}
  \inftysum t {1\over q^{t^2-2\delta t}}\cr
rf'(r)
&\sim {\nu q^{\nu^2+2\delta\nu}\over Q}
  \inftysum t {1\over q^{t^2-2\delta t}}
 \sim \nu f(r)\cr
r(rf'(r))'
&\sim {\nu^2 q^{\nu^2+2\delta\nu}\over Q}
  \inftysum t {1\over q^{t^2-2\delta t}}
 \sim \nu^2 f(r)\cr
|B(r,1)|
&\sim {q^{2\nu^2+4\delta\nu}\over2Q^2}
  \inftysum{t,u} {(t-u)^2\over q^{t^2+u^2-2\delta(t+u)}}
\sim {q^{2\nu^2+4\delta\nu}(S_2S_0-S_1^2)\over Q^2},\cr}
$$
where
$$
S_k = S_k(\delta,q) = \inftysum t {t^k\over q^{t^2-2\delta t}}.
$$
{From} $n=r'f(r)$ we have $\log_q n\sim\nu^2$ and so
$\nu\sim\Sqrt{\log_q n}$.
Thus the mean $k^*$ is asymptotic to $n/\Sqrt{\log_q n}$.
Since the variance is given by $|B|/B_{1,1}$, we have
$$
{\rm variance} \sim {C(\delta,q)n\over(\log_q n)^{3/2}}
~~{\rm where}~~
C(\delta,q) = {S_2S_0-S_1^2\over S_0^2}.
$$
To evaluate the sums $C(\delta,q)$, one needs to know $\delta$ and
this depends on more detailed knowledge of $\nu$ and $r$ than we have
obtained.
However, we can say something about it:
\Bitem
We have $C(\delta+1,q)=C(\delta,q)=C(-\delta,q)$ from which it follows
that $C(\delta,q)$ is determined by its values on $0\le\delta\le1/2$.
\Bitem
By using the $t=0$ and $\pm1$ terms in $S_k$ we find that,
$$
\hbox{for fixed $\delta$ and $q\to\infty$,} ~~ 
C(\delta,q) \sim \cases{
2/q,             & if $\delta=0$\cr
1/q^{1-2\delta},\phantom{\Bigl)} & if $0<\delta<1/2$\cr
1/4,             & if $\delta=1/2$.\cr}
$$
\Bitem
Since $r\sim q^{2\nu}$ and $n=rf'(r)$, which is between
$C\nu q^{\nu^2}$ and $C'\nu q^{\nu^2}$, we have
$r \sim \Exp{2\log q\Sqrt{\log_q n}}$.
Hence, as $n\to\infty$, $C(\delta(n),q)$ approaches a periodic
function of $\Exp{2\log q\Sqrt{\log_q n}}$.
Since the period of $\delta$ as a function of $r$ is 1, its period in
terms of $n$ is about
$$
n\Sqrt{\log_q n}\,\Exp{-2\log q\Sqrt{\log_q n}}.
$$

If we use the Eulerian generating function with
$c_n=(q^n-1)\cdots(q-1)$ in \vectoreqn, we obtain similar results
with $q$ replaced by $q^{1/2}$ and $D_{n,k}(q)$ counts direct sum
decompositions into orthogonal subspaces.\qed


\example
{Tagged Permutations}
A {\it tagged permutation\/} is a permutation written in one-line form
together with a distinguished increasing subsequence.
Following Flajolet and Sedgewick~[\FSe], the generating function is
given by
$$
h(x,y) = {1\over1-x}\Exp{{xy\over1-x}},
$$
where the exponential variable $x$ keeps track of permutation length
and the ordinary variable $y$ keeps track of distinguished subsequence
length.
Lifschitz and Pittel~[\LP] and Flajolet and Sedgewick~[\FSe] obtained
asymptotics for the coefficients of $h(x,1)$ using real and complex
analysis, respectively.
Using Theorem~\multiTWOthm(b) with $f(x)={x\over1-x}$ and $C(r)=1$, we
see that $f(x,y)=\Exp{{xy\over1-x}}$ is super-admissible.
One easily computes
$$
\vec a_f(r,s) = s\pmatrix{{r\over(1-r)^2}\cr {r\over1-r}\cr},
~~
B_f(r,s) = s\pmatrix{{r(1+r}\over(1-r)^3 & {r\over(1-r)^2}\cr
                   {r\over(1-r)^2} & {r\over1-r}\cr},
$$
and $|B_f(r,s)|={r^3s^2\over(1-r)^4}$. 

We now apply Theorem~\prodTWOthm{} with $g(x,y)={1\over1-x}$ to
conclude that $h(x,y)$ is super-admissible.
Only \prodTWOb{} requires any effort.
For $(\theta,\varphi)\in\Theta_f(r,s)$, where $\Theta_f$ is given by
Theorem~\multiTWOthm(b), we have
$$
\log(1-re^{i\theta})
= \log(1-r) + O(\theta/(1-r))
= \log(1-r)
  + O\left(\Bigl({\textstyle{1-r\over s}}\Bigr)^{1/3+\epsilon}\right).
$$
Using the definition or $\cal R$ in Theorem~\multiTWOthm{} and the
above formula for $|B_f(r,s)|$, one easily verifies that the big-oh is
$o_u(1)$.

Let $t(n,k)$ be the number of $n$-long tagged permutations with tags
of length $k$.
It follows from the above work that $t(n,k)$ is asymptotically
normal as $n\to\infty$, with mean and variance asymptotic to $\Sqrt n$
and $\Sqrt n/2$, respectively.
It also follows from the formula for $\cal R$ that asymptotics can be
obtained for tagged permutations whenever
$(1-r)^{1-\delta}<s<(1-r)^{-1/\delta}$.
Since $k\sim{s\over1-r}$ and $n\sim{s\over(1-r)^2}$, some algebra
shows that we can obtain asymptotics whenever
$n^{\delta/(1+\delta)} < k < n^{(1+\delta)/(1+2\delta)}$; that is,
we can obtain asymptotics for $t(n,k)$ as $n\to\infty$ provided
$n^\epsilon < k < n^{1-\epsilon}$ for some $\epsilon>0$.\qed


\example
{Covering Complete Graphs}
A cover of the complete graph with graphs of some specified type is
simply the number of sets of graphs of that type such that the total
number of vertices is $n$.
The exponential formula $f(x,y)=e^{yg(x)}$ applies, where $g(x)$ is the
exponential generating function for graphs of the desired type.
Here are some examples taken from problems 3.3.5--7 in Goulden and
Jackson's text [\GJ,~p.\thinspace 187].
\Bitem
The generating function for coverings with complete graphs is
$\Expg{y(e^x-1)}$, which was studied in Example~1.
\Bitem
The generating function for coverings with complete bipartite graphs
having at least one vertex in each part of the bipartition is
$\Expg{y(e^x-1)^2/2}$ and Theorem~\multiONEthm(b) applies.
\Bitem
The generating function for coverings with star graphs is
$\Expg{y(xe^x-x^2/2)}$ and Theorem~\multiONEthm(b) applies.
(A star graph on $k\ge1$ vertices is a tree consisting of one vertex
of degree $k-1$ to which the remaining $k-1$ vertices are attached.)
\Bitem
The generating function for coverings with paths is
$\Exp{{yx(2-x)\over2(1-x)}}$ and Theorem~\multiTWOthm(b) applies.\qed





\section Proofs of Theorems

Throughout the proofs, $\epsilon$ and $C$ stand for positive
constants, {\it not necessarily the same at each occurrence}.
The value of $\epsilon$ is intended to be small whereas $C$ need not
be.
References to results in [\Hay] have an $H$ prefixed as in
Theorem~H.II.


\Proof{Theorem \asympthm}
We follow essentially the same argument as in~[\Hay] and~[\BR].
With $f(\vec x)=\sum a_{\vec n}\vec x^{\vec n}$ and $d$ the dimension
of $\vec x$, we have
$$
a_{\vec n}\vec r^{\vec n} = {1\over(2\pi)^d}\multint_{[-\pi,\pi]^d}
f(\vec re^{i\vectheta})\Exp{-i\vec n\tr\vectheta}d\vectheta.
$$
Suppose that $a_{\vec n}\ne0$.
Let $\vec u\in\Lambda^*_f$.
The integrand is invariant when $\vectheta$ is replaced by
$\vectheta+2\pi\vec u$ because $\vec u\tr(\vec m-\vec n)$ is an
integer whenever $a_{\vec m}\ne0$.
It follows that we can restrict the integral to $\Phi(f)$ and
multiply the result by $(2\pi)^d/{\rm vol}(\Phi(f))=d(\Lambda_f)$.

Let $\Theta^*(\vec r)$ be the largest set of $\vectheta$ such that
$c\vectheta\in\Theta(\vec r)$ when $0<c<1$.
Note the following:
\Bitem
The interior of $\Theta^*(\vec r)$ is contained in
$\Theta(\vec r)$.
\Bitem
$\Exp{-\vectheta\tr B\vectheta/2}=o_u(1)/|B(\vec r)|^{1/2}$ on
the boundary of $\Theta^*(\vec r)$ because no points on the boundary
of $\Theta^*(\vec r)$ are in $\Theta(\vec r)$.
\Bitem
For every $\vectheta$, there is an $\epsilon(\vec r)$ such that
$\epsilon \vectheta\in\Theta^*(\vec r)$ because the origin is in the
interior of $\Theta(\vec r)$.
\smallskip\noindent
Since $B$ is positive definite, replacing $\vectheta$ by $c\vectheta$
with $c>1$ increases $\vectheta\tr B\vectheta$ and so
$$
\hbox{%
$\Exp{-\vectheta\tr B\vectheta/2}=o_u(1)/|B(\vec r)|^{1/2}$
~~for all~~$\vectheta\not\in\Theta^*(\vec r)$.}
\eqlabel\eqsmallExp
$$
It follows that
$$
a_{\vec n}\vec r^{\vec n}\; = \;
{d(\Lambda_f)\over(2\pi)^d}\multint_{\Theta^*(r)~}
f(\vec re^{i\vectheta})\Exp{-i\vec n\tr\vectheta}d\vectheta
\; + \;
{o_u(f(\vec r))\over|B(\vec r)|^{1/2}}.
$$
Using \eqAasymp{} for $\vectheta\in\Theta^*(\vec r)$ gives us
$$
f(\vec re^{i\vectheta})\Exp{-i\vec n\tr\vectheta}
= f(\vec r)\bigl(1+o_u\bigl(1)\bigr)
  \ExpG{i\vec v\tr\vectheta
        - \vectheta\tr B(\vec r)\vectheta/2}.
$$
Since $B$ is positive definite, we can write $B=S\tr S$ for some
real $d\times d$ matrix $S$.
With $\vec y=S\vectheta$ and $\vec w^2 = \vec w\tr\vec w$,
$$
\eqalign{
i\vec v\tr\vectheta
        - \vectheta\tr B\vectheta/2
&= i\vec v\tr S^{-1}\vec y - \vec y^2/2\cr
&= - \bigl((S\tr)^{-1}\vec v\bigr)^2/2 
   - \bigl(\vec y-i(S\tr)^{-1}\vec v\bigr)^2/2\cr
&= - \vec v\tr B^{-1}\vec v/2
   - \bigl(\vec y-i(S\tr)^{-1}\vec v\bigr)^2/2.\cr}
$$
Hence
$$
\eqalign{
\multint_{\Theta^*~}&
  \Exp{i\vec v\tr\vectheta - \vectheta\tr B(\vec r)\vectheta/2}
  d\vectheta\cr
=\;& {\Expg{- \vec v\tr B(\vec r)^{-1}\vec v/2}\over|B(\vec r)|^{1/2}}
\multint_{S\Theta^*~}
  \Expg{-\bigl(\vec y-i(S\tr)^{-1}\vec v\bigr)^2/2}d\vec y\cr
=\;& {\Expg{- \vec v\tr B(\vec r)^{-1}\vec v/2}\over|B(\vec r)|^{1/2}}
\multint_{\reals^d~}
  \Expg{-\bigl(\vec y-i(S\tr)^{-1}\vec v\bigr)^2/2}d\vec y\cr
&+ {O(1)\Expg{- \vec v\tr B(\vec r)^{-1}\vec v/2}\over|B(\vec r)|^{1/2}}
\multint_{\cal T~}
  \Expg{-\vec x^2/2}d\vec x,\cr}
$$
where, by \eqsmallExp{}, $\cal T$ is a set of $\vec x\in\reals^d$ for
which $\Exp{-\vec x^2/2}=o_u(1)/|B|^{1/2}$.
It follows that the integral over $\cal T$ is $o_u(1)$.
The integral over $\reals^d$ is the product of $d$ integrals of the form
$$
\int_{-\infty}^\infty \Exp{-(y-ic)^2/2} dy
$$
and so equals $(2\pi)^{d/2}$.\qed

\Proof{Theorem \prodONEthm}
Let $h=fg$.
As already described before Theorem~\prodONEthm, we can extend the
$\cal R$, $\Theta$, $\vec a$ and $B$ values for $f$ and $g$ to
include all the variables in $h$.
We can expand $\Lambda$ as well by adding components which equal 0
to the vectors in $\Lambda$.
Then $\Lambda^*$ will no longer be a lattice---the corresponding
components of vectors there can be any real numbers since a real
number times 0 is 0.

For the function $h$, we must verify (a)--(d) and \eqAsuper{} in the
definition of super-admissibility in Section~2.
Property (a) is immediate.

We now prove (b).
Since
$\vec v\tr B_h\vec v = \vec v\tr B_f\vec v + \vec v\tr B_g\vec v$
and since each summand is nonnegative by the positive
semidefiniteness of the extended $B_f$ and $B_g$, it follows that
$B_h$ is positive semidefinite.
Suppose that $\vec v\tr B_h\vec v=0$.
Then
$\vec v\tr B_f\vec v = 0$ and $\vec v\tr B_g\vec v = 0$.
Since the original $B_f$ is positive definite, the components of 
$\vec v$ associated with the variables of $f$ must be 0.
Similarly, the components of $\vec v$ associated with the variables of
$g$ must be 0.
Hence $\vec v=\vec 0$ and so $B_h$ is positive definite.

Using \eqBineq{} and $\Theta_h=\Theta_f\cap\Theta_g$, we obtain (c)
and~(d).

Before proving \eqAsuper, we prove the
claim concerning $\Lambda_h$.
Clearly $\Lambda_h \subseteq \Lambda_f+\Lambda_g$, but equality may
fail due to cancellation of terms when computing $fg$.
Note that
$$
(\Lambda_f+\Lambda_g)^* = \Lambda_f^*\cap\Lambda_g^*
$$
and the operator ${}^*$ reverses inclusion.
Hence it suffices to prove that
$\Lambda^*_h \subseteq \Lambda^*_f\cap\Lambda^*_g$.
Suppose to the contrary that $\vec v \in \Lambda^*_h$ and
$\vec v \not\in \Lambda^*_f\cap\Lambda^*_g$, say
$\vec v \not\in \Lambda^*_f$.
We may choose $\vec r$ so that $|B_h|$ is as large as we wish and
hence also $|B_f|$ by \eqBineq.
{From} (c) in the definition of admissibility, it follows that
$\Theta_f+\vec v$ will be disjoint from $\Theta_f+\Lambda^*_f$ and
so, by \eqAsuper{} for $f$ and \eqBineq, we have
$f(\vec re^{2\pi i\vec v})=o_u(f(\vec r))$.
Since $\vec v\in\Lambda^*_h$, $h(\vec re^{2\pi i\vec v})=h(\vec r)$,
we have the contradiction
$$
f(\vec r)g(\vec r) = h(\vec r) = h(\vec re^{2\pi i\vec v})
= f(\vec re^{2\pi i\vec v})g(\vec re^{2\pi i\vec v})
= o_u(1)f(\vec r)g(\vec r).
$$
This proves $\Lambda_h = \Lambda_f+\Lambda_g$ and also
$$
\Lambda^*_h = \Lambda^*_f\cap\Lambda^*_g. 
\eqlabel\eqfgLstar
$$

We now turn to \eqAsuper.
Since $\Lambda^*_f + \Lambda^*_g$ is a lattice, it follows that,
whenever the diameters of $\Theta_f(\vec r)$ and $\Theta_g(\vec r)$
are sufficiently small,
$$
\Bigl(\Theta_f(\vec r) + 2\pi\Lambda^*_f\Bigr)
\cap
\Bigl(\Theta_g(\vec r) + 2\pi\Lambda^*_g\Bigr)
=
\Bigl(\Theta_f(\vec r) \cap \Theta_g(\vec r)\Bigr)
+ 2\pi\Bigl(\Lambda^*_f \cap \Lambda^*_g\Bigr)
=
\Theta_h(\vec r) + 2\pi\Lambda^*_h,
$$
by \eqfgLstar{} and the definition of $\Theta_h$.
Consequently, when $\min(|B_f(\vec r)|,\,|B_g(\vec r)|)$ is
sufficiently small and $\vectheta$ is in the complement of 
$\Theta_h(\vec r)$ relative to $\Phi(h)$, \eqAsuper{} must hold for at
least one of $f$ and $g$.
This implies \eqAsuper{} for $h$.\qed


\Proof{Theorem \prodTWOthm}
Since $\Lambda_g\subseteq\Lambda_f$, it follows that
$\Lambda_{fg}=\Lambda_f$.
The remainder of the proof is straightforward and will be omitted.\qed


\Proof{Theorem \multiADMISthm}
By multidimensional multisection of series,
$$
g(\vec x)
= {1\over d(\Lambda)}\sum_{\vec v\in \Lambda^*/\integers^d}
f(\vec xe^{2\pi i\vec v})e^{-2\pi i\vec v\tr\vec k},
$$
where the sum makes sense since $e^{-2\pi i\vec v\tr\vec k}$ and the
vector $e^{2\pi i\vec v}$ are constant on a coset of
$\Lambda^*/\integers^d$.
Noting that, when $a_\vec n\ne0$, the value of
$e^{2\pi i\vec v\tr(\vec n-\vec k)}$ is constant on a coset of
$\Lambda^*/\Lambda_f^*$, we have
$$
g(\vec x)
= {1\over d(\Lambda)}\sum_{\vec v\in \Lambda^*/\Lambda_f^*}
f(\vec xe^{2\pi i\vec v})e^{-2\pi i\vec v\tr\vec k}.
$$
When the diameter of $\Theta_f(\vec r)$ is sufficiently small it
follows that, for $\arg(\vec x)\in\Theta_f(\vec r)$, only the 
$\vec v = \vec 0+\Lambda_f^*$ term is large.
Let $\Phi(g)$ be a fundamental region for $\Lambda^*$ contained in
$\Phi(f)$.
If $\arg(\vec x)$ is in the complement of $\Theta_f(\vec r)$ in
$\Phi(g)$, then none of $\arg(\vec x)+2\pi\vec v$ is in 
$\Theta_f(\vec r)+\Phi(f)$.
Hence $g(\vec x)$ is small in this case.\qed

\betweenskip

The following two lemmas lay the foundation for proving
Theorem~\multiONEthm.


\lemma\multiSIZE
In the notation of Theorem~\multiONEthm{} with $F=e^f$ and $G=e^g$,
we have $g(r)\sim \overline\lambda f(r)$,
$$
\eqalignno{
a_G(r) 
&= rg'(r)\sim \overline\lambda rf'(r)
= \overline\lambda a_F(r) = o_u(g(r)^{1+\epsilon}),\cr
B_G(r) 
&= r(rg'(r))' \sim \overline\lambda r(rf'(r))' 
= \overline\lambda B_F(r)
= o_u(g(r)^{1+\epsilon}),\cr
g(re^{i\theta})
&= g(r) + i\theta a_G(r) - \theta^2B_G(r)/2
+ o_u(\theta^3 g(r)^{1+\epsilon})
&\equno\eqTaylor\cr}
$$
for all $\epsilon>0$.
\endthing

\proof
Using the asymptotic formula for the coefficients of admissible
functions and an argument like that in Hayman's proof of
Theorem~H.II, the results for $a$ and $B$ follow in a straightforward
manner.
The last equation follows from Taylor's Theorem with remainder:
$$
H(\theta) = H(0) + H'(0)\theta + H''(0)\theta^2/2
+ \int_0^\theta (t-\theta)^2H'''(t)\,dt/2
$$
with $H(\theta)=g(re^{i\theta})$ and the observation that for $r$
sufficently near $R$,
$$
|H'''(\theta)| \le H'''(0) = O(r^3f'''(r)) = O(f(r)^{1+\epsilon}),
$$
where we used Theorem~H.III for growth of derivatives.\qed
\endthing


\lemma\multiHsix
Suppose $f$ is H-admissible in $|x|<R$, $g$ is given by \eqmulti,
and $\cal C$ is a compact subset of $(0,\infty)$.
Then there is an $R_1<R$ depending on $f$, $\cal C$, and $\epsilon$
such that:
\Item{(a)}
When $d$ is as in Theorem~\multiONEthm(a),
$$
\Re\bigl(g(re^{i\theta})\bigr) \le g(r) - g(r)^{1-2c-\epsilon}
$$
whenever $R_1<r<R$, $c\in\cal C$, and 
$g(r)^{-c} \le |\theta| \le \pi/d$.
\Item{(b)}
When $d$ is as in Theorem~\multiONEthm(b),
$$
\bigl|g(re^{i\theta})| \le g(r) - g(r)^{1-2c-\epsilon}
$$
whenever $R_1<r<R$, $c\in\cal C$, and
$g(r)^{-c} \le |\theta| \le \pi/d$.
\endthing


\proof
To prove the existence of $R_1$, it suffices to consider a fixed
$c\in\cal C$ since compactness of $\cal C$ allows us to take the
maximum $R_1$.

Let $x=re^{i\theta}$.
We assume that $r$ is sufficiently near $R$ for various asymptotic
estimates given below.
By H-admissibility, the coefficients of all sufficiently high powers
of $x$ in $f(x)$ are nonzero and $a_f(r)\to\infty$ as $r\to R$.
Let $r$ be so close to $R$ that all coefficients of $f(x)$ with
$n\ge a_f(r)$ are nonzero.
Let $t$ be the least integer such that $mt\ge a_f(r)$ and define
$\alpha_k=a_{mt+k}x^{mt+k}$.
By H-admissibility, we have 
$$
|\alpha_k| \sim f(r)/\sqrt{\mathstrut\smash{2\pi b_f(r)}}.
$$
Hayman proves that $b(r)=o(a(r)^2)=o(f(r)^\epsilon)$ for admissible
functions.
Using Lemma~\multiSIZE, it follows that $mt=o(g(r)^\epsilon)$ and
$|\alpha_k|>Cg(r)^{1-\epsilon}$.
Let $\theta_k$ be $(mt+k)\theta$ reduced modulo $2\pi$ so that
$|\theta_k|\le\pi$.
Then
$$
|\alpha_k| - \Re\alpha_k
= (1-\cos\theta_k)|\alpha_k|
> C g(r)^{1-\epsilon}\theta_k^2.
$$
It suffices to show that there is some $k$ for which $\lambda_k\ne0$
and $|\theta_k| \ge g(r)^{-c-\epsilon}$.

Suppose there is no such $k$.
By the gcd condition, there are integers $\mu$ and $\mu_k$
for $0\le k < m$ such that $\mu_k=0$ when $\lambda_k=0$ and
$$
d = \mu m + \sum_k \mu_k k.
$$
Let $j$ be such that $\lambda_j\ne0$ and define
$$
\varphi 
= \mu (\theta_{m+j}-\theta_j) + \sum_{k=0}^{m-1}\mu_k\theta_k.
$$
Since $|\theta_k| = O(g(r)^{-c-\epsilon})$, we have
$|\varphi| = O(g(r)^{-c-\epsilon})$.
Modulo $2\pi$,
$$
\varphi
\equiv\mu(m\theta) + \sum_{k=0}^{m-1}\mu_k(mt+k)\theta
\equiv d\theta 
+ \biggl(t\sum_{k=0}^{m-1}\mu_k\biggr)m\theta
\equiv d\theta
+ \biggl(t\sum_{k=0}^{m-1}\mu_k\biggr)(\theta_{m+j}-\theta_j)
$$
and so
$$
d\theta \equiv \varphi
- \biggl(t\sum_{k=0}^{m-1}\mu_k\biggr)(\theta_{m+j}-\theta_j)
~\mod{2\pi}.
$$
Since $t=o(g(r)^\epsilon)$, the right side of this congruence is
$O(g(r)^{-c-\epsilon})$.
Hence $\theta$ differs from a multiple of $2\pi/d$ by
$O(g(r)^{-c-\epsilon})$, a contradiction to the assumption\break
$g(r)^{-c} \le |\theta| \le \pi/d$.
This proves (a).

The proof of (b) is similar to that for (a) except that we now
want to estimate
$$
\delta 
= |c_i\alpha_i| + |c_k\alpha_k| - |c_i\alpha_i + c_k\alpha_k|.
$$
For two complex numbers $z$ and $w$ with $\beta=\arg(z\overline w)$,
we have
$$
(|z|+|w|)^2 - |z+w|^2 = 2|zw|\cos\beta
$$
whence
$$
|z| + |w| - |z+w|
= {2|zw|\cos\beta\over|z| + |w| + |z+w|}
\ge {|zw|\cos\beta\over|z| + |w|}.
$$
Hence
$$
\delta \ge C\cos\bigl((i-k)\theta\bigr)g(r)^{1-\epsilon}.
$$
The remainder of the proof is nearly the same as that for (a),
with $(i-k)\theta$ modulo $2\pi$ in place of $\theta_k$.\qed


\Proof{Theorem~\multiONEthm}
We begin by deriving the description of $\Lambda_h$.
Let $\cal S$ be the set of indices $n$ for which $x^n$ has a nonzero
coefficient in $g(x)$.
Since $f$ is admissible, its coefficients are positive for all
sufficiently large indices.
Hence, for some sufficiently large~$J$,
$$
\setg{k+jm}{k\in{\cal K},~~j\ge J}
\subseteq {\cal S} \subseteq
\setg{k+jm}{k\in{\cal K},~~j\ge 0}.
\eqlabel\eqSform
$$
The powers of $h(x)$ with nonzero coefficients are precisely those
which are sums of elements of $\cal S$.
{From} this and \eqSform, the proof that $\Lambda_h=\integers(d)$ is
now straightforward.
The powers of $h(x,y)$ which have nonzero coefficients are precisely
those of the form $(n,j)$ where $n$ is the sum of $j$ elements of
$\cal S$.
This can be rewritten as $j(k,1)+(n^*,0)$ where $k\in\cal S$ and
$n^*$ is a sum $j$ numbers of the form $s-k$ where $s\in\cal S$.
{From} this and \eqSform, the formula for $\Lambda_h$ is
straightforward.

To prove Theorem~\multiONEthm(a), one need only follow Hayman's proof
of Theorem~H.VI with his use of Lemmas~H.5 and~H.6 replaced by our
\eqTaylor{} and Lemma~\multiHsix(a), respectively.

We now prove Theorem~\multiONEthm(b).
Let $x=re^{i\theta}$, let $y=se^{i\varphi}$, and let $R$ be the
radius of convergence of $g$.
%Since $g(r)\to\infty$ as $r\to R$ by Lemma~H.2 and Lemma~\multiSIZE,
%it follows that $sg(r)\to\infty$ as $(x,y)$ moves to the boundary of
%the region given in Theorem~\multiONEthm(b).

One easily computes $\vec a$ and $B$ in terms of $g$ and its
derivatives and then applies Lemma~\multiSIZE{} to obtain the
asymptotics in the theorem.
With $g(x)=\sum c_nx^n$, one has
$$
\eqalign{
2|B(r,s)|/s^2
&= 2r(rg'(r))'g(r) - 2(rg'(r))^2\cr
&= \sum_{n,k=0}^\infty n^2c_nc_kr^{n+k}
 + \sum_{n,k=0}^\infty c_nk^2c_kr^{n+k}
 - 2\sum_{n,k=0}^\infty nc_nkc_kr^{n+k}\cr
&= \sum_{n,k=0}^\infty (n-k)^2c_nc_kr^{n+k}.\cr}
$$
This proves \eqBdet.

Since $B$ has positive diagonal entries, it will be positive definite
if $|B|>0$, which is the case for all $r<R$ if $c_n\ge0$ for all $n$;
however, a finite number of coefficients of the H-admissible function
$f$ may be negative.
Let $g^*$ be $g$ with these negative terms removed.
The previous argument shows that
$$
\eqalign{
2|B^*(r,s)|/s^2
&= \sum_{n,k=0}^\infty (n-k)^2c^*_nc^*_kr^{n+k}\cr
&\ge \sum_{n,k=0}^\infty c^*_nc^*_kr^{n+k}
 - \sum_{n=0}^\infty (c^*_nr^n)^2\cr
&\ge g^*(r)^2 - \sup_n(c^*nr^n)g^*(r)
= g^*(r)^2 - O(g^*(r)/b_f(r)^{1/2})g^*(r)\cr
&= g^*(r)^2(1+o(1))~~~
\hbox{as $r\to R$.}\cr}
$$
Since the entries in $B(r,s)/s$ and $B^*(r,s)/s$ differ by at most a
polynomial in $r$, the determinants differ by at most a polynomial in
$r$ times the largest entry in $B(r,s)/s$.
Since $f$ is H-admissible, Lemma~H.2 and Theorem~H.III tell us that
this difference is $O(f(r)^{1+\epsilon})$.
Since $|B^*|$ grows like $g(r)^2s^2$ and $\cal R$ requires that
$s>g(r)^{\delta-1}$, it follows that $r\to R$ and $|B(r,s)|\to\infty$
are uniformly the same condition in $\cal R$.
We also have $|B(r,s))|>0$ provided $r$ is sufficiently close to $R$;
that is, $R_0<r<R$ for some $R_0$.

By Lemma~\multiSIZE,
$$
g(x) = g(r)\Bigl(1+i\alpha(r)\theta-\beta(r)\theta^2/2
 + O(g(r)^\epsilon\theta^3)\Bigr)
$$
where $\alpha(r)=a_G(r)/g(r)$ and $\beta(r)=B_G(r)/g(r)$.
Hence
$$
\eqalignno{
yg(x)
=\;& sg(r)\Bigl(\cos\varphi +  i(\alpha(r)\theta+\sin\varphi)
   -\beta(r)\theta^2(\cos\varphi)/2 + O(g(r)^\epsilon\theta^3)\Bigr)
&\equno\eqygONE\cr
=\;& sg(r)\bigl(1 + i\vec a\tr\vectheta
       - \vectheta\tr B\vectheta\bigr)
+ sg(r)^{1+\epsilon}\sum_{k=0}^3 O\bigl(\varphi^{3-k}\theta^k\bigr)
&\equno\eqygTWO\cr}
$$
where $\vectheta=(\theta,\varphi)$ and $\epsilon$ is any positive
number.
Let $\eta$ be a small positive number.
When 
$$
|\theta|\le(1/sg(r))^{2\eta+1/3}
~~{\rm and}~~
|\varphi|\le(1/sg(r))^{2\eta+1/3},
$$
\eqygTWO{} establishes \eqAasymp.
When
$$
|\theta|\le(1/sg(r))^{\eta+1/3}
~~{\rm and}~~
|\varphi|\ge(1/sg(r))^{2\eta+1/3},
$$
$\alpha(r)\theta=o(\varphi)$ and so \eqygONE{} gives us
$$
|yg(x)| < sg(r)\bigl(1-C\varphi^2) \le sg(r) - C(sg(r))^{1/3-4\eta}
$$
for $r$ sufficiently near $R$, the radius of convergence of $g$.
This establishes \eqAsuper{} in that range of $\vectheta$.

We finish establishing the asymptotic requirements on
$\Expg{yg(x)}$ by proving \eqAsuper{} for
$|\theta|\ge(1/sg(r))^{\eta+1/3}$.
Let $\lambda=\lambda(r)=\log s/\log f(r)$.
We are given $\delta-1\le\lambda\le1/\delta$.
Let $c=2(\lambda+1)/5$ and note that
$$
g(r)^c = g(r)^{2(\lambda+1)/5} = (sg(r))^{2/5}
 > (sg(r))^{\eta+1/3}.
$$
Apply Lemma~\multiHsix(b) to obtain
$$
\eqalign{|yf(x)|
\le |y|\Bigl(g(r)-g(r)^{1-2c-\epsilon}\Bigr)
&= sg(r) - sg(r)(g(r))^{-2c-\epsilon}\cr
&= sg(r) - (sg(r))^{1/5} g(r)^{-\epsilon}.\cr}
$$
Since $\epsilon$ is arbitrarily small and $sg(r)>g(r)^\delta$,
condition \eqAsuper{} follows.\qed


\Proof{Theorem \multiTWOthm}
This proof uses ideas from the proofs of Theorems~\multiONEthm{}
and~H.XII.
All conditions in the definition of super-admissibility are easily
established except for \eqAasymp{} and \eqAsuper.
By (H.17.2), when $|\theta|<{\beta(1-r)\over16r}$,
$$
g(re^{i\theta}) = g(r) + i \theta a_f(r) - \theta^2B_f(r)/2
+ E(r,\theta)
\eqlabel\eqTWOsmalltheta
$$
where $|E(r,\theta)| < A(\alpha,\beta)|\theta|^3g(r)(1-r)^{-3}$
for some function $A(\alpha,\beta)$.
{From} this, one easily establishes \eqAasymp{} as in the proof of
Theorem~\multiONEthm.

The proof of \eqAsuper{} for small $\theta$ and large $\varphi$ is
similar to the proof in Theorem~\multiONEthm.

The following discussion is intended for part (b) of the theorem.
Setting $s=1$ allows one to prove part~(a).

Suppose
$(1-r)/(sg(r))^{1/3+\epsilon}<|\theta|<\eta(1-r)$ for some small
$\eta$ to be specified later.
Let $\rho=\theta/(1-r)$.
Hayman shows that 
$$
{a_f(r)\over g(r)} \sim {\alpha\over1-r}
~~{\rm and}~~
{B_f(r)\over g(r)} \sim {\alpha(\alpha+1)\over(1-r)^2}
$$
{From} \eqTWOsmalltheta,
$$
\eqalign{
\left|{g(re^{i\theta})\over g(r)}\right|^2
&= \Bigl|1 - i\theta a_f/g -\theta^2 B_f/2g
  + O(\rho^3)\Bigr|^2\cr
&= \Bigl(1 -\theta^2 B_f/2g + O(\rho^3)\Bigr)^2
  + \Bigl(\theta a_f/g + O(\rho^3)\Bigr)^2\cr
&= 1 - \alpha\rho^2(1+o(1)) + O(\rho^3).\cr}
$$
It follows that with $\eta$ (and hence $\rho$) sufficiently small we
have
$$
\left|{g(re^{i\theta})\over g(r)}\right|^2
< 1 - \alpha\rho^2/2.
$$
Since $\rho \ge (1/sg(r))^{1/3+\epsilon}$, it follows that
$$
\bigl|sg(re^{i\theta})\bigr| < |sg(r)| - |sg(r)|^{1/3-\epsilon}.
$$

Next suppose that $\eta(1-r)\le|\theta|\le\beta(1-r)$.
Hayman proves that $\bigl|g(re^{i\theta})/g(r)\bigr|$ is bounded
above by a constant which is strictly less than 1 and so
$\bigl|g(re^{i\theta})\bigr|\le g(r) -\epsilon g(r)$.

For $\beta(1-r)<|\theta|\le\pi/m$, apply the previous paragraph and
\eqcirclebound.\qed


\Proof{Theorem \peakedthm}
To prevent complexity of argument from obscuring the underlying
ideas, we give a proof without multisection of $f$; that is, we assume
$g(\vec x)=f(\vec x)$.
The proof can be adapted for multisection by following the proof of
Theorem~\multiONEthm.

As can be seen from the proof of Theorem~\multiONEthm, it suffices to
establish 
\Bitemitem
some estimates of $r^kf^{(k)}(r)$ for $k=1,2$,
\Bitemitem
Lemma~\multiHsix{} for dealing with angles outside $\Theta$, and
\Bitemitem
Equation \eqTaylor{} for dealing with angles in $\Theta$.
\itemskip
\noindent
Using \eqsmallterms, one easily has that
$r^kf^{(k)}(r)=O(\nu^kf(r))$, which is $o(f(r)^{1+\epsilon})$ since
we are given $\nu=o(f(r)^{\epsilon})$ for all $\epsilon>0$.

Lemma~\multiHsix{} is easily established using \eqlargeterms.

We now prove \eqTaylor.
Let $F=e^f$, 
let $H(r,\theta) = f(r) + i\theta a_F(r) - \theta^2B_F(r)/2$,
and let $t$ be an integer to be specified later.
By \eqsmallterms, we have
$$
\eqalignno{
f(re^{i\theta})
=\;& \sum_{|k|<t} a_{\nu+k}r^{\nu+k}e^{i\theta(\nu+k)}
  + O\biggl(a_\nu r^\nu\sum_{k\ge t}\rho^k\biggr)\cr
=\;& \sum_{|k|<t} a_{\nu+k}r^{\nu+k}
    \Bigl(1 + i\theta(\nu+k) - \theta^2(\nu+k)^2/2
        + O(\theta^3(\nu+k)^3)\Bigr)
 + O(f(r)\rho^t)\cr
=\;& H(r,\theta)
 + \sum_{|k|\ge t} O(f(r)\rho^k)
    \Bigl(1 + |\theta|(\nu+k) + \theta^2(\nu+k)^2\Bigr)\cr
&+ O\biggl(a_\nu r^\nu\theta^3\sum_k (\nu+k)^3\rho^k\biggr)
 + O\bigl(f(r)\rho^t\bigr)\cr
=\;& H(r,\theta) + \sum_{j=0}^2 O\bigl(f(r)\theta^j(\nu+t)^j\rho^t\bigr)
 + O(f(r)\theta^3\nu^3) + O\bigl(f(r)\rho^t\bigr)\cr
=\;& H(r,\theta) + O\bigl(f(r)\rho^t\bigr)
 + O\bigl(f(r)\theta^3(\nu+t)^3\rho^t\bigr)
 + O\bigl(f(r)\theta^3\nu^3\bigr).\cr}
$$
Using the assumption that $\nu=o(f(r)^\epsilon)$ for all $\epsilon>0$
and setting $t=\log(f(r))/\epsilon|\log\rho|$, \eqTaylor{}
follows.\qed






\frenchspacing

\nonumsection{References}

\item{\BG.}
E. A. Bender and J. R. Goldman, Enumerative uses of generating
functions, {\it Indiana Univ. Math. J.} {\bf 20} (1971) 753--765.

\Item{\BR.}
E. A. Bender and L. B. Richmond, A generalisation of Canfield's
formula, {\it J. Combin. Theory Ser.~A\/} {\bf 23} (1986) 50--60.

\Item{\Dr.}
M. Drmota, A bivariate asymptotic expansion of coefficients of powers
of generating functions, {\it Europ. J. Combinat.} {\bf 15} (1994)
139--152.

\Item{\FSe.}
P. Flajolet and R. Sedgewick, The average case analysis of algorithms:
Saddle point asymptotics, INRIA Rpt.~No.~2376 (1994).

\Item{\FSo.}
P. Flajolet and M. Soria, Gaussian limiting distributions for the
number of components in combinatorial structures, {\it J. Combin.
Theory Ser.~A\/} {\bf 53} (1990) 165--182.

\Item{\GR.}
Z. Gao and L. B. Richmond, Central and local limit theorems applied
to asymptotic enumeration IV: multivariate generating functions,
{\it J. Comput. and Appl. Math.} {\bf 41} (1992) 177--186.

\Item{\Gar.}
D. Gardy, Some results on the asymptotic behaviour of coefficients of
large powers of functions, {\it Discrete Math.} {\bf 139} (1995)
189--217.
See also the review by A. Meir (MR~96f:41038).

\Item{\GJ.}
I. P. Goulden and D. M. Jackson, {\it Combinatorial Enumeration},
J. Wiley and Sons, New York (1983).

\Item{\Har.}
L. H. Harper, Stirling behavior is asymptotically normal,
{\it Ann. Math. Statist.} {\bf 38} (1967) 410--414.

\Item{\Hay.}
W. K. Hayman, A generalisation of Stirling's formula, 
{\it J. Reine Angew. Math.} {\bf 196} (1956) 67--95.

\Item{\LP.}
V. Lifschitz and B. Pittel, The number of increasing subsequences of a
random permutation, {\it J. Combin. Theory Ser.~A\/} {\bf 31} (1981)
1--20.

\Item{\Odl.}
A. M. Odlyzko, Asymptotic enumeration methods.
In R. L. Graham,\break
M. Gr\"otschel, and L. Lov\'asz (eds.), {\it
Handbook of Combinatorics, Vol.~II}, Elsevier, Amsterdam (1995)
1063--1229.

\Item{\Pre.}
S. J. Press {\it Applied Multivariate Analysis}, Holt, Rinehart and
Winston, New York (1972).


\bye