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\def\AnARAA{1}
\def\CaDLAA{2}
\def\FoWaAA{3}
\def\GaRaAA{4}
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%equation numbers
\def\AA{1}
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\def\po#1#2{(#1)_#2}
\def\coef#1{\left\langle#1\right\rangle}
\def\fl#1{\left\lfloor#1\right\rfloor}
\def\cl#1{\left\lceil#1\right\rceil}

\topmatter 
\title Asymptotic analysis of a Selberg-type integral via hypergeometrics
\endtitle 
\author C.~Krattenthaler$^{\dagger}$
\endauthor 
\affil 
Fakult\"at f\"ur Mathematik, Universit\"at Wien,\\
Nordbergstra{\ss}e~15, A-1090 Vienna, Austria.\\
WWW: \tt http://www.mat.univie.ac.at/\~{}kratt
\endaffil
\address Fakult\"at f\"ur Mathematik, Universit\"at Wien,
Nordbergstrasze~15, A-1090 Vienna, Austria.
WWW: \tt http://www.mat.univie.ac.at/\~{}kratt
\endaddress
\thanks $^\dagger$Research partially supported 
by the Austrian Science Foundation FWF, grants Z130-N13 and S9607-N13,
the latter in the framework of the National Research Network
``Analytic Combinatorics and Probabilistic Number Theory"%
\endthanks
\subjclass Primary 33C20;
 Secondary 33A15  33C52 60B20 82B05
\endsubjclass
\keywords Selberg-type integral, hypergeometric series, contiguous relation,
balanced hypergeometric series, transformation formula
\endkeywords
\abstract 
We show how to determine the asymptotics of a certain Selberg-type 
integral by means of tools available in the theory of
(generalised) hypergeometric series. This provides an alternative
derivation of a result of Carr\'e, Deneufch\^atel, Luque and Vivo
[{\tt ar$\chi$iv:1003.5996}]. 
\endabstract
\endtopmatter
\document

In \cite{\CaDLAA}, Carr\'e, Deneufch\^atel, Luque and Vivo consider
the Selberg-type integral  
$$
S_k(a,b)=\frac {1} {N!}\int _{[0,1]^N} ^{}x_1^k
\Bigg(\prod _{1\le i<j\le N} ^{}(x_i-x_j)^2\Bigg)
\Bigg(\prod _{i=1} ^{N}x_i^{a-1}(1-x_i)^{b-1}\,dx_i\Bigg),
$$
and they
determine its asymptotic behaviour when $N,a,b$ all tend to infinity
so that $a\sim a_1N$ and $b\sim b_1N$, where $a_1$ and $b_1$ are given
non-negative real numbers. The reader is referred to the introduction
of \cite{\CaDLAA} for information on motivation from random matrix
theory connected to random scattering theory to investigate this question.

It should be noted that $S_0(a,b)$ {\it is} a Selberg integral, which can be
evaluated in a product/quotient of gamma functions
(cf.\ \cite{\FoWaAA}). This being the case, the asymptotics of
$S_0(a,b)$ is easily determined by means of known asymptotic formulae
for the Barnes $G$-function (see \cite{\WhWaAA}). 
Thus, it suffices to consider the quotient 
$$J_k=\frac {S_k(a,b)} {S_0(a,b)}$$
(this quotient is denoted by $I_k/N$ in \cite{\CaDLAA}) and determine
its asymptotic behaviour. By (now) classical identities in the theory
of symmetric functions, it is shown in \cite{\CaDLAA, Cor.~II.3} that
$$
J_k=\frac {1} {N\cdot k!}\sum_{i=0} ^{k-1}(-1)^i \binom {k - 1} i
\frac { (N - i)_{ k}\, (a + N - i - 1)_{ k}}
{   (a + b + 2 N - i - 2)_{ k}},
\tag\AA
$$
where $(\alpha)_m$ is the usual Pochhammer symbol defined by
$(\alpha)_m=\alpha(\alpha+1) \cdots(\alpha+m-1)$ for $m\ge1$, and
$(\alpha)_0=1$.
In view of this rather simple expression, which, after substitution of
$a$ by $a_1N$ and of $b$ by $b_1N$, is a rational function in $N$, the
problem of determining the asymptotic behaviour of the resulting
expression looks innocent. 
However, inspection of the expression (\AA) shows that
this is not so innocent at all: the summands are all of the order of magnitude 
$N^k$ for large $N$, while (see Theorem and Corollary below), in fact, the
expression in (\AA) has a {\it finite} limit as $N\to\infty$ in the
described situation. Hence, what happens here is a large amount of 
cancellation of leading asymptotic terms. This is what one has to get under
control, and this constitutes the challenge in determining this limit.

To solve the problem,
Carr\'e, Deneufch\^atel, Luque and Vivo develop a difference
operator calculus in \cite{\CaDLAA}, which is of interest 
in its own right.

The purpose of the present note is to demonstrate that the
tools for solving this problem
are readily available in the literature on
({\it generalised}) {\it hypergeometric series},
and it is clear that they may serve as well to solve problems of
similar type. 

Namely, using the standard hypergeometric notation
$$
{}_p F_q\!\left[\matrix a_1,\dots,a_p\\
b_1,\dots,b_q\endmatrix; z\right]=\sum _{m=0} ^{\infty}\frac
{\po{a_1}{m}\cdots\po{a_p}{m}} {m!\,\po{b_1}{m}\cdots\po{b_q}{m}}
z^m \,
$$
where $\po{a_1}{m}$, etc.\ are again Pochhammer symbols,
Equation~(\AA) becomes
$$
J_k=
\frac {(N+1)_{ k-1}\,
  (a + N-1)_{ k}} {k!\,(2N+ a + b -2)_{ k}}
{}_4 F_3\!\left[\matrix 1 - N, 1 - k, 2 - a - N, 3 - a - b - k - 2N\\
   2 - a - k - N, 1 - k - N, 3 - a - b - 2N\endmatrix; 1\right].
\tag\AB$$
The point of this conversion is that one may now look in the
literature of hypergeometric series, which provides a large arsenal of
utilities, such as transformation formulae for hypergeometric series
etc., and see whether one (or a combination of more) of these formulae
enable us to convert (\AB) into a different form from which the
asymptotics of $J_k$ can be directly read off. As I show below, this
is indeed the case. To the informed reader it will come as no surprise that the
computations below have been carried out using the {\sl Mathematica}
package HYP \cite{\KratHYP}. 

The crucial observation to make is that the $_4F_3$-series in (\AB) 
is not just an ``arbitrary"
hypergeometric series, but that it is a {\it $(-1)$-balanced\/} series
(cf.\ \cite{\AnARAA, p.~140}), that is, that the sum of the lower
parameters of the hypergeometric series minus the sum of
the upper parameters is $-1$. Unfortunately, there are no transformation
formulae available for $(-1)$-balanced series, but there are plenty of
them available for $1$-balanced series (or, as is common to say,
for {\it
balanced series}), such as, for example
(see \cite{\GaRaAA, Appendix (III.16), $q\uparrow1$}; this is T4302 in HYP),
$$\multline
{} _{4} F _{3} \!\left [ \matrix { A, B, C, -n}\\ { E, F, 1 + A + B + C - E -
   F - n}\endmatrix ; {\displaystyle 1}\right ] \\=
{\frac {({ \textstyle A)_n\,(  E + F-A - B)_n\, (  E + F-A - C}) _{n}}
     {({ \textstyle E)_n\, ( F)_n\, ( E + F-A - B - C}) _{n}}}\\
\times
  {} _{4} F _{3} \!\left [ \matrix { -n, E-A,  F-A,  E + F-A - B - C}\\
    { E + F-A - B,  E + F-A - C, 1 - A - n}\endmatrix ; {\displaystyle
    1}\right ]   ,
\endmultline\tag\AC$$
where $n$ is a non-negative integer.
That is, we are just by ``$2$ off" in applying a transformation formula.

In order to relate series with (small) integral differences between
the parameters, the hypergeometric literature offers {\it contiguous
relations}. One of the simplest such relations is (this is C14 in HYP),
$$
  {} _{4} F _{3} \!\left [ \matrix A, B, C, D\\E, F, G
\endmatrix ; {\displaystyle
    z}\right ]  
= z\frac {BCD} {EFG}
  {} _{4} F _{3} \!\left [ \matrix A, B+1, C+1, D+1\\E+1, F+1, G+1
\endmatrix ; {\displaystyle z}\right ] + 
  {} _{4} F _{3} \!\left [ \matrix  A-1, B, C, D\\E, F, G\endmatrix ; {\displaystyle
    z}\right ]   .
$$
If we iterate this contiguous relation, then we arrive at
$$\multline 
  {} _{4} F _{3} \!\left [ \matrix A,B,C,D\\E,F,G
\endmatrix ; {\displaystyle z}\right ] =
        z^r \frac {(B)_r\, (C)_r\, (D)_r} {(E)_r\, (F)_r\, (G)_r} 
  {} _{4} F _{3} \!\left [ \matrix 
A,B+r,C+r,D+r\\E+r,F+r,G+r\endmatrix ; {\displaystyle z}\right ]\\ +
\sum_{s=0} ^{r-1} z^s \frac {(B)_s\, (C)_s\, (D)_s} 
{(E)_s\, (F)_s\, (G)_s} 
   {} _{4} F _{3} \!\left [ \matrix 
A-1,B+s,C+s,D+s\\E+s,F+s,G+s
\endmatrix ; {\displaystyle z}\right ].
\endmultline
\tag\AD$$
We now apply (\AD) with $r=k-1$ to the $_4F_3$-series in (\AB). 
The reader should observe that thereby all the $_4F_3$-series in the
sum over $s$ will be $0$-balanced, while the single $_4F_3$-series
outside of the sum will reduce to $1$ because one of the upper
parameters is $1-k+r=0$. To be precise, 
after writing the result in compact form,
application of (\AD) to (\AB) yields
$$\align 
J_k&=
 \sum _ {s=0} ^{k-1}
\frac {(N+1)_{k-1}\, (a + N-1)_k\, 
      (1 - k)_s\, (2 - a - N)_s\, (3 - a - b - k - 2 N)_s}
   {k!\,( a
   + b + 2 N-2)_k\,(2 - a - k - N)_s\, (1 - k - N)_s\, (3 - a - b - 2 N)_s}\\
&\kern40pt
\times
  {} _{4} F _{3} \!\left [ \matrix 
-N, 1 - k + s, 2 - a - N + s, 3 - a - b - k - 2 N + s\\
       2 - a - k - N + s, 1 - k - N + s, 3 - a - b - 2 N + s
\endmatrix ; {\displaystyle 1}\right ]
\endalign$$
Continuing in the same spirit, if we apply the contiguous relation
(\AD) a second time, this time with $r=k-s-1$, then this will produce
a double sum in which all appearing $_4F_3$-series are balanced. To be
precise, we obtain
$$\multline 
J_k\\=
\sum _{s=0} ^{k-1} \sum _{t=0} ^{k-s-1}
\frac { (N+1)_{k-1}\, (a + N-1)_k\,
(1 - k)_{s+t}\,   (2 - a - N)_{s+t}\, (3 - a - b - k - 2 N)_{s+t}}
 {k!\,(a + b + 2 N-2)_k\,
  (2 - a - k - N)_{s+t}\,(1 - k - N)_{s+t}\,(3 - a - b - 2 N)_{s+t}  
    }\\
\times
  {} _{4} F _{3} \!\left [ \matrix 
3 - a - b - k - 2 N + s + t, -1 - N, 2 - a - N + s + t, 
      1 - k + s + t\\3 - a - b - 2 N + s + t, 1 - k - N + s + t, 
      2 - a - k - N + s + t\endmatrix ; {\displaystyle 1}\right ].
\endmultline$$
Now the transformation formula (\AC) can be applied, and, after some
simplification, the resulting expression collapses to
$$\align
J_k&=
\sum _{s=0} ^{k-1} \sum _{t=0} ^{k-s-1}
\frac { (a-1)_{ k - s - t-1}\, (1 - a - N)_{ s + t+1}\, 
  (k - s - t)_{ s + t}\, (s + t+2)_{ k - s - t-1}} 
{k!\,(2 - a - b - 2 N)_{ k}}\\
&\kern3cm
\cdot
  {} _{4} F _{3} \!\left [ \matrix 
1 - k + s + t, k, a + b + N-2, a + N\\s + t+2, a-1, 
    a + b + 2 N-1\endmatrix ; {\displaystyle 1}\right ]\\
&=\sum _{m=0} ^{k-1} 
\frac { (a-1)_{ k - m-1}\, (1 - a - N)_{ m+1}\, 
  (k - m)_{ m}\, (m+1)_{ k - m}} 
{k!\,(2 - a - b - 2 N)_{ k}}\\
&\kern3cm
\cdot
  {} _{4} F _{3} \!\left [ \matrix 
- k + m+1, k, a + b + N-2, a + N\\m+2, a-1, 
    a + b + 2 N-1\endmatrix ; {\displaystyle 1}\right ].
\endalign$$
Now the limit $N,a,b\to\infty$, in which $a\sim a_1N$ and $b\sim
b_1N$, can be safely performed. 
Hence, we have proven the following result.

\proclaim{Theorem}
The limit of the quantity $J_k$ as $N,a,b\to\infty$ such that $a\sim
a_1N$ and $b\sim b_1N$ is equal to
$$\multline 
\lim_{N\to\infty}J_k=
\sum _{m=0} ^{k-1} 
(-1)^{k-m-1}\binom {k-1}m
\left(\frac {a_1} {a_1+b_1+2}\right)^k
\left(\frac {a_1+1} {a_1}\right)^{m+1}\\
\cdot
\sum _{\ell=0} ^{k-m-1}
(-1)^\ell\binom {k-m-1}\ell
\frac {(k+\ell-1)!(m+1)!} {(k-1)!(m+\ell+1)!}
\left(\frac {(a_1+1)(a_1+b_1+1)} {a_1(a_1+b_1+2)}\right)^\ell.
\endmultline
\tag\AE$$
\endproclaim

This theorem does not give the same expression as in \cite{\CaDLAA,
Theorem~IV.4}. However, once one has found {\it some} expression, it
is easy to pass to an equivalent one by using hypergeometric
transformation and summation formulae. (Since, in the above derivation,
there is plenty of room for variation --- for example, the transformation
formula (\AC) could be applied in different ways, or we could have applied
a different transformation formula for balanced $_4F_3$-series, see
\cite{\SlatAC, Sec.~4.3.5} ---, it is conceivable that there is a direct 
route leading to the corollary below, using the above ideas. We did not
pursue this, however; in particular, we did not do a systematic search
of such an alternative route.)

\proclaim{Corollary \smc(\cite{\CaDLAA, \smc Theorem~IV.4})}
The limit of the quantity $J_k$ as $N,a,b\to\infty$ such that $a\sim
a_1N$ and $b\sim b_1N$ is equal to
$$
\lim_{N\to\infty}J_k=
\frac {1} {k}\sum _{j=0} ^{k-1}(-1)^j 
\binom {k+j-1}j
\frac {(a_1+1)^{j+1}} {(a_1+b_1+2)^{k+j}}
\sum_{i=0} ^{k-j-1}\binom ki\binom k{i+j+1}
(a_1+1)^i.
$$

\endproclaim

\demo{Proof}
In the expression on the right-hand side of (\AE) we interchange the
summations over $m$ and $\ell$, and we write the (now) inner sum over $m$ in
hypergeometric notation. This leads to
$$\multline 
\lim_{N\to\infty}J_k=
\sum _{\ell=0} ^{k-1}(-1)^{k-\ell-1}
\frac {a_1^{k-\ell-1}\,(a_1+1)^{\ell+1}\,(a_1+b_1+1)^{\ell}}
{(a_1+b_1+2)^{k+\ell}}
    \frac {(k+\ell-1)!} {
    \ell !\,(\ell +1)!\,(k-\ell-1)!}\\
\cdot
  {} _{2} F _{1} \!\left [ \matrix 
2, - k + \ell +1\\\ell +2\endmatrix ; {\displaystyle  \frac
{a_1+1} {a_1} }\right ].
\endmultline$$
To the $_2F_1$-series we apply the transformation formula
(see \cite{\SlatAC, (1.7.1.3), sum
reversed at the right-hand side}; this is T2106 in HYP)
$$
{} _{2} F _{1} \!\left [ \matrix { A, -n}\\ { C}\endmatrix ; {\displaystyle
   z}\right ] = {z^n}
  {{  ({ \textstyle C-A}) _{n} }\over {({ \textstyle C}) _{n} }}
  {} _{2} F _{1} \!\left [ \matrix { -n, 1 - C - n}\\ {
       1 + A - C - n}\endmatrix ; {\displaystyle -{{1 - z}\over z}}\right ]  ,
$$
where $n$ is a non-negative integer.
Writing the resulting $_2F_1$-series as a sum over $i$, this gives
$$\multline 
\lim_{N\to\infty}J_k=
\sum _{\ell=0} ^{k-1}
\sum _{i=0} ^{k-\ell-1}
(-1)^{k-\ell-i-1}
\binom {k-\ell-1}i
\frac {(k+\ell-1)!  }
{(\ell-1)!\,\ell!\,(k-\ell-1)!\,(k-i-1)(k-i)
   }\\
\cdot
\frac {(a_1+1)^{k-i}\,(a_1+b_1+1)^{\ell}}
{(a_1+b_1+2)^{k+\ell}}.
\endmultline$$
Now we replace the term $(a_1+b_1+1)^{\ell}$ by
$\sum _{j=0} ^{\ell}\binom \ell {j}(-1)^j(a_1+b_1+2)^{\ell-j}$, we
make the sum over $\ell$ the inner-most sum, and finally write the sum
over $\ell$ in hypergeometric notation. In this way, the last
equation becomes
$$\multline 
\lim_{N\to\infty}J_k=
\sum _{j=0} ^{k-1}
\sum _{i=0} ^{k-j-1}
(-1)^{k-i-1}
\binom {k-j-1}i
\frac {(k+j-1)!  }
{(j-1)!\,j!\,(k-j-1)!\,(k-i-1)(k-i)
   }\\
\cdot
\frac {(a_1+1)^{k-i}}
{(a_1+b_1+2)^{k+j}}
  {} _{2} F _{1} \!\left [ \matrix {-k+i+j+1,k+j }\\ {
j       }\endmatrix ; {\displaystyle 1}\right ]  .
\endmultline$$
The $_2F_1$-series can be evaluated by the Chu-Vandermonde summation
(see \cite{\SlatAC, (1.7.7); Appendix (III.4)}; this is S2101 in HYP)
$$
{} _{2} F _{1} \!\left [ \matrix { A, -n}\\ { C}\endmatrix ; {\displaystyle
   1}\right ] = {{({ \textstyle C-A}) _{n} }\over 
    {({ \textstyle C}) _{n} }},
$$
where $n$ is a non-negative integer. After finally replacing $i$ by
$k-i-j-1$, the claimed result follows immediately upon minor
rearrangement of terms.\quad \quad \qed
\enddemo




\Refs

\ref\no \AnARAA\by 
G.~E.~Andrews, R.~A.~Askey and R.~Roy\book Special
functions\publ
The Encyclopedia of Mathematics and Its Applications, vol.~71,
Cambridge University Press\publaddr Cambridge \yr 1999\endref

\ref\no \CaDLAA\by C. Carr\'e, M. Deneufch\^atel, J.-G. Luque and 
P. Vivo\paper 
Asymptotics of Selberg-type integrals: The unitary case and Newton's
interpolation formula
\jour preprint; {\tt ar$\chi$iv:1003.5996}\vol \yr \pages \endref

\ref\no \FoWaAA\by P. J. Forrester and S. O. Warnaar\paper
The importance of the Selberg integral\jour
Bull\. Amer\. Math\. Soc\. (N.S.)\vol 45 \yr2008\pages 489--534\endref

\ref\no \GaRaAA\by G.    Gasper and M. Rahman \yr 1990 
\book Basic hypergeometric series
\publ Encyclopedia of Mathematics And Its Applications~35, Cambridge University Press
\publaddr Cambridge\endref

\ref\no \KratHYP \yr 1995 \paper HYP and HYPQ --- {\sl Mathematica\/}
packages for  
the manipulation of binomial sums and hypergeometric series, respectively 
$q$-binomial sums and basic hypergeometric series\jour J. Symbol\.
Comput\.\vol 20\pages 737--744\finalinfo the packages are freely
available at {\tt http://www.mat.univie.ac.at/\~{}kratt}\endref

\ref\no \SlatAC\by L. J. Slater \yr 1966 
\book Generalized hypergeometric functions
\publ Cambridge University Press
\publaddr Cambridge\endref

\ref\no \WhWaAA\by
E. T. Whittaker and G. N. Watson\book A course of modern analysis
\bookinfo fourth edition\publ 
Cambridge University Press\publaddr Cambridge\yr 1927\finalinfo
reprinted by Cambdridge University Press, 1996\endref

\endRefs





\enddocument


  
