CONSTANT TER M IDENTITIE S xii i

Jacobi polynomial s attache d t o root system s

The Jacob i polynomial s P\(k) o f Heckma n an d Opda m (loc . cit.) attache d

to a roo t syste m R ar e a limitin g cas e o f our P\(q

J

t)J wher e th e parameter s q

and t (though t o f as real o r complex numbers ) bot h ten d t o 1 in such a way that

(1 — t)/(l — q) tends t o a definit e limi t k. The y wer e define d i n loc. cit. a s the

simultaneous eigenfunctions o f a second-order differential operato r that extrapolate s

the radia l par t o f the Laplace operato r o n a symmetric spac e G/K, wher e G is a

connected noncompact semisimpl e Lie group and K is a maximal compact subgrou p

of G; the root syste m R i s the restricted roo t syste m o f G/K, an d the parameter

k i s hal f th e roo t multiplicit y (assume d constant , fo r simplicit y o f description) .

When th e root syste m i s of type A, the Jacobi polynomial s coincid e wit h the Jack

symmetric function s o f Chapter I , §9 .

Constant ter m identitie s

The constant ter m identities now to be described constitute anothe r anteceden t

of th e orthogonal polynomial s o f Chapter II , of a mor e combinatoria l flavour . I n

1962 F. J. Dyson [D ] was led by considerations of statistical mechanics to conjecture

that th e constant ter m i n the expansion o f the produc t

flil-XiXj1)",

where x\ ,..., jCfi ar e independent variable s an d A; is a positive integer , shoul d be

(1) (nk)\/(kl) n.

This conjectur e wa s soon proved true , by J. Gunson [G2 ] and K. Wilson [W] , who

showed mor e generall y tha t th e constant ter m i n the expansion of

(2) Jlil-XiXj

1)"',

where k\ ,..., k

n

ar e nonnegative integers , i s the multinomial coefficien t

(3)

fci!-..fcn! '

Next, in 1975 G. E. Andrews [A ] conjectured a ^-analogue of this result. Writ e

faq)n = H(l-q

i-1x),

%=i

where q,x ar e independent variable s an d n i s a n intege r 0 . The n Andrews '

conjecture wa s that th e polynomial ( G

Z[xf1,...,

x^

1q]),

(4)

YlfajXiXj

1'^)*;.,

where Sij — 1 or q according a s i j o r i j, should hav e constan t ter m (i.e. ,

independent o f x\,..., x n) equa l to

/

5

N (g;g)fci+-+fc

n

te^Hi --(q',q)k

n

when g = l , thi s reduce s t o the previous result . However , (5 ) proved a harder nut

to crack , an d was finally prove d correc t b y Zeilberger an d Bressoud [ZB ] i n 1985.