COEFFICIENTS IN EXPANSIONS OF CERTAIN
RATIONAL f-.1ULTIVARIABLE RJNCTIONS
by
Ibnald Richards
University of North Carolina
Chapel Hill, NC 27514
AM6 1980 Subject Classification:
Primary 05AlO
KEY WORDS AND PHRASES: coefficients, series expansions, rational function,
basic composition formula, Schur function, q-binomial coefficient,
Clebsch-Gordan numbers, stable measures, hypergeometric functions.
ABSTRACT
Motivated by the work of Evans, Ismail and Stanton (Canad. J. Math. 34
(1982),1011-1024), we obtain the coefficients appearing in the series
expansion of certain rational multivariate functions .
•
1.
INTRODUCfION
In a recent article, Evans, Ismail and Stanton [3] considered the problem
of evaluating the constant terms of some interesting rational functions.
work was motivated by a problem of Mallows [8].
Their
One of the main results in
[3] is
Theorem 1.
Let A and B be positive integers.
For each pair of non-negative
integers u,v, the coefficient K(u,v) of yUzV in the power series expansion of
(1.1)
g(y,z) = (l-yz)
A+B
(l-y)
-A
(l-z)
-B
is
(1.2)
K(u,v) =
(A+U~V-I) (B+~-U)
_
(A+~-V) (B+~=~-l)
•
This paper is concerned with the problem of obtaining multivariable analogues
of Theorem 1.
Along the way, there are at least two obstacles to deriving
suitable extensions.
One is that the natural extensions of some of the tech-
niques used in [3], for example, Lagrange inversion in several variables, do
not seem to be particularly helpful.
A second difficulty is that one is not
quite certain of what will constitute a "suitable" extension of g(y, z) ;
further remarks are directed at this problem in Section 5.
Despite
these problems, it turns out that some interesting results can
be obtained for the function
A n
(l-xlxZ ...x) IT (I-x.)
n j=l
J
which reduces to (1.1) when n=2.
-B.
J
Here and throughout, A,B l ,B 2 , ... ,Bn are
positive integers.
Actually, even this choice seems recondite (cf. section 5).
So we shall
choose the integers B. to have a common value B, and consider not f(x) but
J
-2-
n
f(x) fl(x), where flex) = II (x. -xk) is the usual Vandennonde determinant.
j<k
With
J
the above restriction on the B., f(x) becomes a synunetric function of the variJ
abIes x ,x 2 , ... ,x ' and we are able to use various results from the theory of
n
l
synunetric functions.
Interestingly, Theorem 2 below shows that the coefficients in the power
series expansion of f(x)fl(x) can be expressed as linear combinations of
determinants of binomial coefficients.
This result is already valid when
n= 2 by virtue of (1.2), since K(u,v) is actually a determinant.
However,
the difference in the methods used is probably of independent interest.
In a
sense, we reverse the techniques of Evanset al [3] by using integrals to
evaluate the series coefficients.
The layout of the paper goes as follows; Section 2 lists some preliminary
material, while Section 3 derives the coefficients of f(x)fl(x).
Section 4
considers two q-analogues of f(x), and expresses the related coefficients in
terms of determinants of q-binomia1 coefficients.
Finally, Section 5 relates
the problems of obtaining the coefficients of f(x) itself and evaluating
terminating balanced generalised hypergeometric function.
Some remarks are
also directed towards the problem of developing suitable multivariate analogues
of g(y,z), and deriving combinatorial interpretations for some integrals similar
to some of those in [3].
2•
PRELIMINARY MATERIAL
Throughout, we use the usual notation for rising factorials, viz., for any
non-negative integer k,
(2.1)
(a)k = f(a+k)jf(a) .
Let A = (A 1 ,A 2 , ... ,An ) be a partition, i.e. a set of non-negative integers
arranged in non-increasing order,
-3 -
The swn k = Al + 1.. 2 + ••• + An of the parts of A, denoted by IAI , is called
the weight of A; for brevity, we write A I- k. The nwnber of non-zero Ai'
denoted by 9,(1..), is called the length of A.
The conj'ugate of A is the
partition A' =(Ai,A2, ... ,A~) such that
A! = Card {j: A.
J
1
~
i}
i
For any partition A = (Al ,A , ... ,An) with 9,(A)
2
defined by
= 1,2, ... ,n .
~
n, the Schur functions sA are
(2.2)
It is clear that sA(x) is symmetric in xl ,x 2 ' ... ,xn . Further, since the
numerator of (2.2) is divisible by each of the factors x.-x., 1 ~ i < j ~ n,
1
J
then it is divisible also by the Vandermonde determinant
6(X) = IT (x.-x.) = det(xn.- j ) .
(2.3)
.
.
l<J
J
1
1
Thus, the SA(X) are symmetric polynomials in xl ,x 2 , ... ,xn ' and it can be shown
that they form a Ll-basis for the vector space of homogeneous symmetric polynomials in xl' ... 'xn .
Schur flIDctions were first defined as in (2.2) by Jacobi, and later related
to the theory of group representations by Schur.
A recent treatment of Schur
functions is given by Macdonald [7].
Two useful series expansions, which are to be regarded as purely fonnal,
without regard to questions of convergence, are ([7], pp. 33-35):
n
(2.4)
IT
m
IT
(l-x.y.)
1
i=l j=l
n
(2.5)
IT
m
IT
00
= L L SA(X)SA(y)
k=O AI-k
00
(l-x.y.)
1
i=l j=l
Setting Yl = Y2 =
J
-1
...
J
=
L L
k=O AH<
SA(X)SA'(y}
= ym = t in (2.4), we obtain
-4-
In fact, (Z.6) holds more generally for any real number m.
number of the generalised binomial coefficients
negative integer.
Only a finite
(~) are non-zero if m is a
An explicit formula for (~) is ([7], Exercises 3.4 and 4.1)
(Z.7)
To evaluate certain sums whose terms contain determinants, we shall use
the basic composition formula ([6],p.17).
This result states that if S is a
sigma-finite measure on the real line JR., and {fj}j=l and {gj}j=l are subsets
of Ll(S), then
(Z.8)
= (nl)-l
f··· f
IR
IR
det(f.(tk))det(g.(tk))dS(t1)···dS(t )
J
J
n
= det(f f.(t)gk(t)dS(t)) .
IR J
The basic composition formula is attributed by Muir [10] to Andreief [1].
shall apply it when 8 is supported by an interval or by a discrete set.
3.
COEFFICIENTS IN 1HE EXPANSION OF f(x)fl(x)
Theorem Z.
For each n-tuple (ul'uZ' ... 'un ) of non-negative integers, the
u
l
z...
z
U
coefficient C(ul,uZ' ... 'un ) of Xl X
(3.1)
is
g(x) = (l-xlxZ ...x)
n
A
n
IT
j=l
u
n
xn in the expansion of
(I-x.)
J
-B n
IT
(x.-x )
j<k J k
We
-5-
(3.2)
(-l)n£(-A)
C(u ,u , ... ,u ) = (_1)!2fi(n-l)+v ~
£'
£ det((.+ B n ) )
n
l 2
£~O·
J uk -)(,-n
where
Proof.
(3.3)
Applying the expansion (2.6), we obtain
A n
-B
(1-x l x2 ···x) IT (I-x.)
n j =1
J
k
(-1) (-A)£ B
£
= I I
I
£!
(A)(xl·"xn) SA(X).
£=0 k=O At-k
00
00
To compute C(ul , ... ,un)' which is the constant tenn in the expansion of
-u -u
-u
i8.
xl lX 2... x n g(x), we set x. = e J, 0 ~ 8- < 2n, 1 ~ j ~ n, and
2
n
J
J
integrate over [0,2n)n. Then the integral to be evaluated is
n i8.(£-u.)
i8
i8 n i8. i8
(2n)-nJ ••• J ( IT e J
J )s,(e l , ... ,e n) IT(e J- e k)d8 ... d8
n
l
. 1
A
. k
[0, 2n) n J =
J<
(3.4)
n i8.(£-u.)
i8.].1k
J
J )det(e J )d8 ... d8
n
l
[0,2nt j=l
.
= (2n)-n J... J ( IT e
where].1k = Ak+n-k, k=1,2, ... ,n.
Letting Sn denote the group of pennutations
on n symbols, we expand the detenninant in the integrand of (3.4) and obtain
2n
2n n
L sgn(o)(2n)-nJ ••• J (IT exp(i8.(£-u.+].1 )))d8 ... d8
l
o j=l
J
J OJ
n
OES
0
(3.5)
n
With o(a,b) denoting Kronecker's delta symbol, the orthogonality properties of
i8.
the {e J} show that (3.4) equals
(3.6)
L sgn(o)
OE Sn
n
IT 0(£+].1 ,u.) = det(o(£+].1.,~ )).
·=l
oJ- J
J K
J
Hence, along with (2.7) and (3.3), (3.6) shows that
-6-
00
(3.7)
00
(-l)k(-A)
.
ILL
=
£=0 k=O Af- k
£!
£ det((A ~·+k))det(o(£+~.,~))
j J
J
n
00
= I
(-A)£
00
£=0 £!
~
1
I
>~
2
>... ~ >0
(-1)
I(ll·-n+j)
1 J
B
det((~._n+k))
J
n
the second equality following from the substitutions
~.=A.+n-j,
J
J
1
j
~
~
n.
In the second series, notice that we may replace the limits
~l > ~2 > ••• > ~n > 0
by
~l ~ ~2 ~ ••• ~ ~n ~ 0,
both determinants in the summand are zero.
since if
~r
=
~s
for some r,s,
Then, we apply the discrete
version of the basic composition formula (2.8), and obtain (3.2) after some
o
simplifications are made.
Since A is a positive integer, it follows that the sum in (3.2)
terminates, at latest, when £ = A.
We next consider several interesting special cases of Theorem 2.
If any two of the u. are equal, then C(ul ,u 2 , ... ,un) = o.
.
J
In this event, it is easy to see that all the detenninants in
Corollary 3.
Proof.
o
(3.2) are identically zero.
Corollary 4.
Proof.
_
~(n+ 1) (B)n
+1
C(1,2,3, ... ,n) - (-1)
{Ill + (_l)n A} •
Let D(m,p,r) = det(( ~ .)), 1 ~ i, j ~ r + 1, where m,p and rare
p-l+J
non-negative integers.
Section 733, pp. 681-2.
The evaluation of D(m,p,r) may be located in Muir [11],
By reversing the order in which the rows appear, it
B
)
1, det((.+k
n
)
J -x'-n
([11], Section 732, pp. 680-1)
follows that when £
~
_
-
(-1) ~(n-l) D(B,l-£,n-l).
Since
-7-
D(m,p,r) = D(m,r+l,p)
(3.8)
with the convention that D(m,p,r) = 0 if r < 0, then the result follows from
(3.8).
0
1vbre generally, we can obtain an explicit result if the {uj } form an
arithmetic progression, uj = u + jd, j=1,2, ... ,n. The result following can be
deduced from Theorem 2 using the formulae in Muir [11], Section 734, pp. 683-5.
For any positive integer d, C(u+d,u+2d, ... ,u+nd) =
Corollary 5.
00
(_l)nu+~n(n+l)d L
£=0
J
Il~2
~~0(U-t;1j+l)d)
~J~·~_o
in-I
(u~;~d)
n-2-i B-u+£-d- (d-I) i +j
_i=_O
(
d_-_l
)
n-l
II
j=l
(jd-l )n-j
(j-l)d
Although Corollary 3 shows that C(u ,u ,u , ... ,u ) = 0, this does not seem
l l 3
n
to be a general phenomenon for ftmctions similar to f(x). The proof of the
following result is similar to that of Theorem 2.
Theorem 6.
The coefficient of (xl x 2... xn)u in the power series expansion of
f(x) [6(x)]2 is zero if u < n-l, and for u c n-l is
(_l)un+~n(n-l)(n!)
u-n+l (-l)n£(-A) u-n+l-£ (B+j)
L ------------£ II ~
£=0
£!
j=O
(n+ J )
n
In this situation, it seems far more difficult to compute the most
general coefficients.
4.
lWO q-ANALOGUES
In this section, we use the standard notation [12]
r
k+r
(a;q)k = II (l-aq )j(l-aq ) , k = 0,1,2, ... ,
r=O
when working with basic hypergeometric series.
00
-8-
For the case of two variables, the results in [3] pertain to expansions of
the functions (wx;q)A+B/(w;q)A (qx;q)B and (wx;q)A+B/(w;q)A(x;q)B'
This
section treats two multivariate analogues,
n
.II (xOxi ; q) A+B
1=1
------n
(4.1)
(xO;q)A i~l (qxi;q)B
and
(4.2)
_
~
Since both f q and gq are symmetric functions in the n variables xl ,x 2 , ... ,xn '
it makes sense to ask not for the coefficient of the monomial
U 11
1l
U
o
n
1
o
X xl ... x , but instead for the coefficient of X sA(x ,x , ••• x ), where
l 2
o
u
o
n
A is a partition with ~(A) $ n. It turns out that, analogous to the results
of Section 3, the coefficients may be expressed in terms of linear combinations
of determinants of the q-binomial coefficients,
(4.3)
First some notation is required.
For any partition A with
~(A) ~
define
(4.4)
n(A) =
L (i-l)A.
·>1
l~
1
The significance of n(A) arises from the formula ([7], Exercise 3.1)
= qn(A) [mAI ]
SA (1 ,q,q 2m-I)
, ... ,q
(4.5)
k
where [A]' the generalised q-binomial coefficient may be defined by
(4.6)
m, we
-9-
We also recall the Clebsch-Gordan numbers
s~(x)s (x) =
(4.7)
l\
1\
mA~(v),
defined by the relation
L m~I\ll (v)s \) (x)
\1
where the sum is over all partitions v with 9,(v) :::; n, and Ivl =
IAI+I~I.
An
introductory treatment of Clebsch-Gordan numbers and related topics is given
by Humphreys [5].
Theorem 7.
For any non-negative integer j, and any partition v with 9,(v) :::; n,
the coefficient of xiSv(x) in the expansion of fq(xO;x) is
mA~(V)qn(A')+I~I+n(~)+n(p) [A~B] [~,] [~,](~)
I
(4.8)
A,~,p
where the sum is over all partitions A,~,p such that IAI:::; A+B, IA'I+lpl = j
and
max{9,(A),9,(~),
Proof.
As
9,(p)} : :; n .
. 1
in [7], set Yj = xoqJ-
for all j and m = A+B in (2.5).
After
applying (4.5), we obtain
n
(4.9)
i~l (xOxi;q)A+B =
In a similar way, (2.4) and (2.6) imply, respectively,
(4.10)
and
(4.11)
-n
(xO;q)A =
It is straightforward to verify that (4.11) reduces to Heine's q-binomial
theorem [12] when n=l.
SA(X)S~(x)
Now, multiplying the series in (4.9)-(4.11), expanding
as in (4.7), and collecting terms completes the proof.
0
-10-
It seems difficult to simplify (4.8), not only because the Clebsch-Gordan
numbers are difficult to manipulate, but also because the set of admissible
partitions
A,~,p
is somewhat complicated.
To complete this section, we obtain
a similar result for gq(XO;x); the proof is similar to that of Theorem 7, and
is omitted.
Theorem 8.
For any non-negative integer j, and any partition v, .Q,(v)
$
n,
the coefficient of x~sv(x) in the expansion of gq(xO;x) is
(4.12)
L mA~(V)qn(A ')+n(~)+n(p) [A~B] [~,] [~,] (~)
A,~,p
where the sum is over the same set of partitions listed earlier.
5.
CONCLUDING REMARKS
There seems to be some connection between the functions considered here,
and balanced, terminating, generalised hypergeometric series.
To illustrate,
if each factor in (1.3) is expanded and the powers of x. 's are collected, we
u
u
1
find that the coefficient of XII ...x n is the terminating
n
Stml
(B . ) .
(-A.
)
n
1 u·-J
L
J II
1
j=O -J-.!--"- i=l CUi -J) !
00
(5.1)
Without losing generality, let u l be the smallest of the u .
i
in the form
(-A)
Reversing the
.(B )·
l J
ul-J
So in the case Bl = B2 = ... = Bn , (5.2) is reminiscent of terminating balanced
generalised hypergeometric series. Unfortunately, what is presently known
about these series seems to be not adequate for our pUrPOses.
-11-
As noted in Section 1, there still remains the questions of what
constitutes a suitable multivariable extension of g(y,z) in (1.1).
motivation for [3] was a problem posed by Mallows [8].
The original
In turn, Mallows'
problem was derived from certain properties of stable probability measures [4]
on the real line.
Therefore, it seemed natural that the theory of multivariate
stable measures on
:n:f
should give rise to generalisations of Mallows' integral
and hence to suitable extensions of g(y,z).
However, multivariate stable laws
are far more complicated than their one-dimensional counterparts; see Nrrller [9]
and Cambanis [2] for references to the literature.
situations when the stable laws on
:nf
In the more tractable
have radial (i.e. orthogonally
invariant) characteristic functions, the results of Zolotarev [14] can be used
in attempts to find extensions of g(y,z).
Disappointingly, they seem to lead
•
precisely to Mallows' integrals.
Finally, we obtain a combinatorial interpretation for some integrals
similar to the ones considered earlier in [3].
Let N,ml , ..• ,mA,nl, ... ,nB be
given positive integers such that mj IN and nklN for all j=l, ... ,A, k=l, ... ,B.
Consider the equation
(5.3)
where the p. and q. are non-negative integers.
J
1
Theorem 9.
The number I(N) of lIDordered solutions of (5.3) may be represented
as
I (N) =
Proof.
t
~ B
it (N+nk) ~
~ J2n·
e- 1tN A
IT [ (l-e it(N+m.)
J) IT ( I - e ) dt
2n
0
j=l
(l-e
ltm.
J)
k=l
ltn
( I - e ·k)
.
Our method of proof is based on Vinogradov's method ([13], p. 167).
First, it is clear that
-12-
I(N)
=
L ... L
PI~O
qB~O
where for each j ,k, the
respectively.
1
2TI
z- fa
•
A
B
I J J
I J J
exp(lt(-N+Lm.p. + Ln.q.))dt.
TI
over Pj and qk are tnmcated at N/mj and N/n j ,
Since the sum over PI is
SlUllS
l-e
it (N+m )
l
= ----..-==--l-e
It~
with similar expressions for the other sums, the results follows.
0
It is hoped that Theorem 9 will eventually lead to a combinatorial
interpretation of Corollary 4 in [3].
ACKNOWLEDGEMENTS
Thanks are due to Richard Askey and Dennis Stanton. They hosted a recent,
seminal, visit to Madison and Minneapolis. R.A. must be also thanked for
pointing out how beautiful the q-disease can be .
•
-13REFERENCES
..
1.
C. Andreief, Note sur lUle relation entre inteyrales definies des produits
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S. Carnbanis, Complex st98itric stable variables and processes, Tech.
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3.
R. Evans, M. Ismail and D. Stanton, Coefficients in expansions of certain
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W. Feller, An Introduction to Probabilit
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5.
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I. Macdonald,
S~tric
Oxford, 197
•
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lications,
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.
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C. Mallows, A fonnula for expected values, .Arner. Math. Mmthly, 87
(1980), 584.
--
9.
G. Miller, Properties of certain symmetric stable distributions, J.
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T. MUir, A Treatise on the Theory of Detenninants, (Albany, 1930).
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•
14.
V.M.