•
INTEGRALS OF SELBERG POLYNOMIALS
by
Donald St. P. Richards 1
Department of Statistics
University of North Carolina
Chapel Hill, NC 27514
•
1partially supported by NSF grant MCS-8403381.
•
Key words and phrases: Selberg's integral, Mehta's integral,
hypergeometric series, orthogonal polynomials, Dyson's conjecture,
mUltivariate beta integrals.
ABSTRACT
•
A polynomial p(t 1 ,
polynomial if (1) p(t 1 ,
(2) p(t 1,
(3) p(t 1 ,
P(1-t 1 ,
,t n)
,t n ) in n real variables is a Selberg
,t n) is homogeneous of (even) degree k,
= ( t 1t 2... t n) ~ p(-1
t1 ,
.
,t -1)
n for some lnteger
~,
,t n ) is symmetric in t 1,
,t n, and (4) p(t 1 , ... ,t n) =
,1-t n ). The definition of these polynomials was motivated
by a study of Selberg's (Norsk. Mat. Tidsskr. 26 (1944), 71-78)
derivation of a multivariate beta-type integral formula.
In this
paper, we derive analogs of Selberg's and Mehta's integral formula for
a large class of Selberg polynomials.
,
1.
INTRODUCTION
In this paper, we evaluate multidimensional beta-type integrals,
1/
.
involving the IISelberg polynomials" introduced in [7J.
For some necessary background recall that in
an important generalization of the beta integral.
~~,
Selberg computed
He proved the following
result.
1.1.
with Re(x)
Jheor~
~
(Selberg [10J).
0, Re(y)
>
Let x, y, and z be complex numbers
0 and Re(z)
>
max{-l/n, -Re(x)/(n-1), -Re(y)/(n-I)}.
Further, let
be the discriminant polynomial in n real variables tI, ... ,t n.
Then,
n
•
IT t.x-I( I-t,. )y-I dt .
Jo1••• JI0 ~ (tI, ... ,t n)z i=I
1
,
=
~
r(x+(j-1)z) r(y+(j-I)z) r(jz+I)
j=l
r(x+y+(n+j-2)z) r(z+l)
(1.1 )
Recent work of Andrews [IJ, Askey [2J, [3J, Macdonald [6J, Morris [9J
and others have related (1.1) to topics as diverse as generalized hypergeometric series, orthogonal polynomials, the Dyson conjecture and the
root systems of finite reflection groups.
Following from (1.1) as a limiting
case is the Mehta-Dyson integral [8; p. 42J:
(2Tr) -n/2
J:oo'" [00
if Re(z)
>
-I/n, then
n
exp( -;
L t~) ~(tI'''.' tn)z dt I .. · dt n
i=I
= ~ r(jz+I)
j=l r(z+I)
(1. 2)
-2-
In the course of proving (1.1), Selberg utilized the following
four properties of the polynomial p(!)
= ~(!) (we write t = (t l , ... ,t n)).
(51) p(t) is homogeneous:
p(st)
sk p(t)
=
Sf.
.
•
lR ,
for some nonnegative integer k;
(52) p(t) is translational:
p(t) = p(l-t)
where 1
(1,1, ... ,1) ;
=
~-reciprocal:
(53) p(t) is
for some nonnegative integer
~,
p(!) = (t l t 2·· .tn)~ p(E- ) ,
l
..
where t- l = (t-1l ,oo., t-n1 ) .'
•
(54) p(t) is symmetric:
p(t)
-
where ot
=
=
for any permutation
0
of {1,2, ... ,n},
p(at)
-
(t o (1), ... ,to (n))
A polynomial p(t) which satisfies (51)-(54) is a
of type
(n,~,k).
Thus,
~(t)
~elberg
polynomial
is a 5elberg polynomial of type (n,2(n-1),
n(n-l)).
The paper [7] contains a complete description of, and algorithms for
constructing, the 5elberg polynomials.
It is easily shown, for example,
that (51)-(53) entail (i) k is even, and (ii) 2k
= n~. A much deeper
result is that the dimension of V(n,£,k), the vector space of 5elberg
polynomials of type (n,£,k), equals par(n,£,k) - par(n,£,k-1); here,
-~-
par(n,~,k)
equals the number of partitions [5] of k into at most n parts
with no part exceeding~. In particular, when n=2, 6(t)k/2 is (up to
•
•
constant multiples) the unique Selberg polynomial of degree k.
In point of fact, [7J required as an alternative to (52), the criterion
(52)1
p(t) = p(l+t) .
-
~
~
We chose in [1] not to work with (S2) since it is, in conjunction with
(51), more restrictive than (52)1.
In this paper, we show that Selberg's proof of Theorem 1.1 extends to
an evaluation of the integral (1.1) with 6(t) replaced by a monomially
bounded Selberg polynomial (Definition 2.3).
Consequently, we obtain a
plethora of Selberg-type integrals.
In Section 2, we abstract from [7] some preliminary material on the
Selberg polynomials.
Then, Section 3 contains the appropriate generalizations
of (1.1) and (1.2).
Although we do not pursue any applications of our results, it is clear
that we can relate them to the derivation of constant term identities as in
[6] and [9].
-4-
2.
SELBERG POLYNOMIALS
2.1. Lemma ([7; Theorem 2.3J).
A polynomial p(t) is translational
•
if and only if there exists a polynomial q(t 1 , ... ,t _ ) such that
n 1
p(~) = q(t 1-t 2 ,···,t 1-t n )·
Let us sketch the proof.
First, we note that the translational
property (52) entails (indeed, is equivalent to) the criterion
p(sl-t)=
p(t)
..............
(2.1)
..... ,
for all
SE
R.
Indeed, for fixed t, (52) shows that the polynomial
r(s) = p(s 1 - t) - p(t) has a zero at every positive integer.
Hence,
r(s) is identically zero.
direction.
Substituting s=t 1 completes the proof in one
Since the converse is trivial, then the proof is complete.
Thus, every Selberg polynomial is (by symmetry) a polynomial in the
differences t.-t.,
1 -< i < j -< n. Using these observations, we are able to
1
J
construct a large number of Selberg polynomials. The following result is
a simple version of some general construction procedures developed in [7J.
2.2.
Theorem.
Let A = (a .. ) be a symmetric nxn matrix with nonlJ
negative, even, integer entries. Assume thatAhas zero on the main diagonal
n
and
L
j=1
a · =Q,for all i=I, ... ,n, where nQ,-::O (mod 4).
iJ
n
Then, the polynomial
a ..
p(t) = n (t.-t.) lJ + symm
~
i<j
1
J
(2.2)
is a nontrivial Selberg polynomial of type (n,R"
Here and throughout, the term symm
II
ll
nR,/2).
in (2.2) denotes all terms
required to ensure that the polynomial p(t) is symmetric.
Then p(t) is
•
-5-
clearly homogeneous (of degree
Further, p(t) is
symmetric and translational.
since each variable t i appears exactly
a ..
times in the product .n. (ti-t j ) lJ
-
Q,
k=n~/2),
~-reciprocal
n
l<J
In order to describe the class of Selberg polynomials to be treated
in the next section, we recall some basic facts on symmetric functions [5].
A partition \ of a nonnegative integer k is a set of nonnegative
integers A1, ... ,A n such that A1+A2+ ... +An=k. Customarily, we write
A=(A1, ... ,A n) where A1~ ... ~An and refer to the Ai as the parts of A.
The monomial symmetric function, in the variables ~t
(t 1 , ... ,t n ),
=
corresponding to A is the polynomial
,
.
mA(~)
A1+... +A n . It is
well known that every symmetric, homogeneous polynomial can be expressed
Thus,
is symmetric and homogeneous of degree k
=
as a linear combination of monomial symmetric functions.
2.3.
Definition.
Let the symmetric polynomial p(t) be homogeneous
of degree k, P be a set of partitions of k, and
(2.3)
be the expansion of p(t) in monomial symmetric functions.
.
aAfO for all A in P).
nonnegative integers
(We assume that
Then p(E) is monomially bounded if there exist
a1~ ... ~an' Sl~ ... ~Bn
such that
-6-
n
(i i )
L
i=l
2.4.
n
B·1 -
L
i=l
Examples.
Ct
=k
i
(i)
•
Selberg [9] showed that A(t) is monomially
bounded with a.1 = n-i, B.1 = 2n-i-1; i=l, ... ,n .
(ii)
Let n be even and
n/2
p(t) = n
i=l
(ti-tn_i+1)~ + symm
It may be shown that only partitions A which satisfy A = A,
~ = (~-A n , ~-A n- 1"'" ~-Al)' can appear in the decomposition (2.3);
moreover, every such partition appears. Therefore, p(t) is monom;ally
bounded with a; =
(iii)
(~~n+;
=
~£,
6i =
£,
and
(t~n+i
= 0, ;=1,2, ... ,!n .
A Selberg polynomial (of type (4,6,12)) which is not monomially
are both monomially bounded Selberg polynomials.
The computations
underlying this and other examples were carried out using the symbolic
mani pul ator REDUCE [4].
As the following result shows, we can readily construct monomially
bounded Selberg polynomials of arbitrary type.
.
../
bounded is
-7-
2.5.
Proposition.
(i)
Let Pl{!) and P2{!) be monomially
bounded Selberg polynomials of types (n'~l,kl) and (n'~2,k2)
respectively.
•
Then the product
Pl{~)P2{~)
is a monomially bounded
Selberg polynomial of type (n'~1+~2,kl+k2) .
(ii)
Let Pl{t1, ... ,tm) and P2{tm+1 , ... ,t n ) be monomially bounded,
homogeneous (of even degrees k1 and k2 , respectively), translational,
~-reciprocal
polynomials.
Then the polynomial
p{!) = Pl{t1,···,tm) P2{tm+1 , ... ,t n ) + symm
is monomially bounded Selberg polynomial of type
Proof.
(i)
It is evident, from (SI)-{S4), that
Selberg polynomial of type
•
(n,~,kl+k2).
(n,kl+k2'~).
Pl{~)P2{~)
is a
Therefore we need only check
the monomial boundedness. Since Pl{t) (resp. P2{t)) is monomially bounded,
A
A
- ~
~
then every monomial t 11... t nn (resp. t11 ... t nn) appearing in the expansion
satisfies a.<A·<S.
(resp. a~<~.<S~),
i;l, ... ,n, for
of Pl{t)
(resp. P2{t))
1- 1- 1
1- 1- 1
n
some fixed set of integers a., S· (resp. a~, S~), where >(S.-a.)
n
11
11
t 1 1 = k1
(resp. L:(r~~-Cl.~) = k2 ). In the expansion of PI (t)P2{t), ~very monomial
1 1 1
will then be of the form
where
p
and
0
are permutations on
(~ p1
(')
+
c< 0
I ( . ) < A (.)
1-01
{l
,2, ... ,n}.
Since
+ ~ 0(')
< r~ (.) + G01
1-p1
I (
.)
i=l, ... ,n, and obviously,
then Pl{!)P2{!) is monomially bounded.
The proof of (ii) follows from similar arguments.
,
-8-
3.
INTEGRALS OF SELBERG POLYNOMIALS
Throughout, we set
1
c(p) =
1
J ••• J
p(t) dt 1 ·· .dt n,
(3.1)
00-
where p(t) is a Selberg polynomial.
When evaluating c(p), we shall use
an expansion of p(t) which follows directly from Lemma 2.1.:
(3.2)
3.1.
Theorem (Selberg's Integral).
Selberg polynomial, and the integers
Definition 2.3.
1
If Re(x)
1
Jo••• J0p(t)
"
> -ex. ,
n
Re(y)
Let p(t) be a monomially bounded
, , (i=I, ... ,n)
cy..,(L
> -n ,
n
be specified by
•
then
n
IT
i=1
=
c( p)
n r(x+a.) r(y+a) r(B.+2)
- - _J
~_~L
--L_
2
j =1
r ( x+y+ R.) l' ((t . +1)
11
J
(3.3)
J
Further, .
c(p} = n~k
Proof.
I c.
IJT=k ~
n
(j.+l
i=2 '
IT
)-1
(3.4)
.
Denote the integral in (3.3) by I(p).
By expanding p(t,), we
see that I(p) is a linear combination of integrals of the form
=
n
r(xH.) r(y)
II
---,--------
,
j=1 r(x+y+/t.)
-9-
where, without loss of generality,
A1~A2~
monomially bounded, then a.<A.<S.
(i=1,
1- 1- 1
r(x+A i )
r(x+a i )
1
r(x+a. )
1
r(x+Y+Si)
is
Hence
S.-a.-1
1
1
1
II
. 0
J=
r(x+Y+6 i )
=
,n).
p(~)
Since
A. -a. -1 '
- - - ' - = ---'--f(X+Y+A i )
~An.
(X+A.+j)
(X+y+A.+j)
II
1
1
j=O
(x,y)
q
Ai
. where qA.(X,y) is a polynomial of degree at most 6i-Ai in y and at
1
most 6i-Ai in x.
1
Therefore,
1 n
x+A.-1
n r(x+a.)r{y)
t. 1 (l-t.)y-1 dt . = Q
(x,y) I I ·
1
0 i=1 1
1
1
A1,···,A n
j=1 r(x+y+s.)
1
J ... J
o
II
n
n
I
(I
where Q,
, (x,y) is a polynomial of degree at most (S.-A.) =
6.) - k
1
11
1 1
1\1,···,l\n
•
in y.
Therefore,
I(p) = Q(x,y)
n r(x+a i ) r(y)
j=l r(x+Y+6 i )
II
n
where Q(x,y) is a polynomial of degree at most
(I
Si) - k in y.
Let
1
R(y) =
n
II
i=1
y(y+l) ... (y+a.-1); then
1
I(p)
= Q(x,y)
R(y)
Replacing t by
-
1~t
--
Q(x,y)
R(y)
=
n r(x+a.) r(y+a.)
II
1
j=1 r(x+y+s j )
in (3.3), we see that
Q(y,x)
R(x)
J
(3.5)
-10-
Since Q(y,x)/R(x) is a polynomial in y, then so is Q(x,y)/R(y); hence,
R(y) divides Q(x,y).
Further,
n
I
i=1
n
(Xi = deg(R(y))
_<:
deg(Q(x,y))
_<
(I
1
n
fl'i) - k = L (Xi'
i=1
the last equality holding since p(t) is monomially bounded.
Therefore,
deg(R(y)) = deg(Q(x,y)) and hence Q(x,Y)/R(y) is independent of y.
By symmetry Q(y,x)/R(x) is independent of x, so that
Q(x,l) = Q(x,y) = Q(y,x)_ = Q(y,l)
R(1)
R(y)
R(x)
R(I)
is independent of both x and y; say, Q(x,y)/R(y) = c1(p), a constant.
Substituting x=y=1 in (3.5) we obtain
r 2 (cx.+l)
J ...J p(t)dt ... dt = c (p) n ------'.-o 0 - 1
n
1 i=1 1'(0.+2)
1
1
n
,
•
which leads immediately to (3.3).
Since p(t) is symmetric, then
(3.6)
Replacing t., by tIt.,
i=2, ... ,n, (3.6) becomes
1
-11-
Substituting the expansion (3.2) into the above integral and integrating
termwise. we obtain (3.4).
3.2.
)
Remarks.
(i)
As we noted earlier. the proof of (3.3) abstracts
an argument due to Selberg [9] for the polynomial 6(t).
(ii)
Since products of monomially bounded polynomials are also monomially
bounded. (3.3) carries over to integral powers of p(t) and even (via
Carlson's theorem) to complex powers of p(t).
However. the evaluation of
c(p) may become far more difficult.
Next. we derive the appropriate extension of (1.2).
Jheore~
3.3.
(Mehta-Dyson Integral).
bounded Selberg polynomial.
(2rr)-!n
rex:
00
r:oo
Then.
n
exp(-i i~1 t~) p(!)dt 1 ... dt n =
n r (r~ .+2)
= c(p)
Proof.
Let p(E) be a monomially
(3.7)
~---­
II
j=1 1'2(ltj +1)
We use the standard procedure [2; p. 938] for deriving Mehta's
formula from Selberg's integral.
Set x=y and 2t i =1+(2X)-! si(i=l •...• n) in
(3.3); using the homogeneity and translationality of p(t). we obtain
(2x)'
(
-(2x)~
(2x)'
J
-(2x)!
p(s)
.~
n
J1
i=l
s~
(l - - ' )x-1 ds .
2x
= x~(k+n) 22xn +,(3k-n) c(p)
'
n
II
j=l
,
2
r(2x+r~i) l' (a.+1)
,
1'(S·+2)
(3.8)
-12-
Now we let x-+ oo ; then, the left-hand-side of (3.8) converges to
r: ... r:
-00
-<Xl
n
2
exp(-! i:::1
l: s.)
p(s)ds
1 ···ds n .
1
_.
«
Denoting the right-hand-side of (3.8) by r(x), Stirling's formula
shows that as x -+ w,
(3.9)
n
where
o(p) ::: tk -
as x-+ w
•
l: ~ ..
1
Necessarily, o(p)
1
~
0 since (3.9) remains finite
If o(p) > 0, then l,)(p2) ::: 28(p) >0 and (3.9) entails
(2n)
- .1
2
which is absurd.
n
JO)_w··· fXJ exp(-i
_00
n
'I
i~l
2
2
si) (p(~)) ds 1 ·· .ds n ::: 0 ,
Therefore, 6(p) ::: 0 and (3.7) follows from (3.9).
Acknowledgement.
We owe much to Dennis Stanton.
The initial stimulus
which led to this paper arose during conversations with him.
•
Further,
he provided us with the first example of a Selberg polynomial which
was distinct from A(t).
REFERENCES
e
.
1.
ANDREWS, G.E., Notes on the Dyson conjecture, SIAM J. Math. Anal., 11
(1980), 787-792.
2.
ASKEY, R., Some basic hypergeometric extensions of integrals of Selberg
and Andrews, SIAM J. Math. Anal., 11 (1980), 938-951.
3.
ASKEY, R., Computer algebra and definite integrals, preprint.
4.
HEARN, A.C. REDUCE 2 User's Manual, Second Edition, University of Utah
Computational Physics Group, Report No. UCP-19, March 1973.
5.
r1ACDONALD, I.G., Symmetric Functions and Hall Polynomials, Clarendon
Press, Oxford, 1979 .
6.
MACDONALD, I.G., Some conjectures for root systems and finite reflection
groups, SIAM J. Math. Anal., 13 (1982), 988-1007.
7.
MENA, R., BRIDGES, W., ISSACSON, E. and RICHARDS, D., Selberg polynomials,
in preparation.
8.
MEHTA, M.L., Random Matrices and the Statistical Theory of Energy Levels,
Academic Press, New York, 1967.
9.
MORRIS, W.G. II, Constant term identities for finite and affine root
systems: conjectures and theorems, Ph.D. thesis, University of WisconsinMadison, 1982.
10. SELBERG, A. Bemerkninger om et multipelt integral, Norsk. Mat. Tidsskr.,
26 (1944), 71-78.