Preface

This volume is largely based on the lectures presented during a special ses-

sion of the 865th meeting of the American Mathematical Society, convened in

Tampa, Florida, during the period March 22- 23, 1991. This special session, en-

titled "Hyper geometric functions on domains of positivity, Jack polynomials and

applications," was centered on a branch of research initiated some forty years

ago by Bochner. The initial impetus for Bochner's work came from questions

in analytic number theory.

It

is remarkable that, since then, these hypergeo-

metric functions have been found to be important in areas as diverse as combi-

natorics, harmonic analysis, molecular chemistry, multivariate statistics, partial

differential equations, probability theory, representation theory and mathemati-

cal physics. In addition, the scope of these functions has been broadened consid-

erably. While the initial investigations of these hypergeometric functions were

carried out within the context of matrix spaces, the articles within this volume

relate these functions to the study of domains of positivity and root systems.

At this stage, some brief, historical remarks are in order. Given a real, positive

definite (symmetric)

n

X

n

matrix, A, with

integer

entries, let r(A) denote the

number of k x

n

matrices T, with integer matrices, and where k

~

n,

such that

T'T

=

A. A problem considered by Bochner (and others) is to investigate the

asymptotic behavior of r(A) as "A

----

oo." A natural approach to studying the

asymptotic behavior of r(A) is to study a generating function for r(A). Thus if

Z

is also positive definite and

n

x

n,

and etr(Z) ::::: exp(tr Z), define the theta

function

G(Z)

=

L

etr( -7rTZT').

T

It

is not difficult to derive Jacobi's formula

G(Z)

=

(detz)kf 2

e(z-

1

),

hence

(1) :Z::r(A) etr(-7rAZ)

=

(detz)kf 2 :Z::r(A) etr(-7rAZ-

1

).

A A

vii

This volume is largely based on the lectures presented during a special ses-

sion of the 865th meeting of the American Mathematical Society, convened in

Tampa, Florida, during the period March 22- 23, 1991. This special session, en-

titled "Hyper geometric functions on domains of positivity, Jack polynomials and

applications," was centered on a branch of research initiated some forty years

ago by Bochner. The initial impetus for Bochner's work came from questions

in analytic number theory.

It

is remarkable that, since then, these hypergeo-

metric functions have been found to be important in areas as diverse as combi-

natorics, harmonic analysis, molecular chemistry, multivariate statistics, partial

differential equations, probability theory, representation theory and mathemati-

cal physics. In addition, the scope of these functions has been broadened consid-

erably. While the initial investigations of these hypergeometric functions were

carried out within the context of matrix spaces, the articles within this volume

relate these functions to the study of domains of positivity and root systems.

At this stage, some brief, historical remarks are in order. Given a real, positive

definite (symmetric)

n

X

n

matrix, A, with

integer

entries, let r(A) denote the

number of k x

n

matrices T, with integer matrices, and where k

~

n,

such that

T'T

=

A. A problem considered by Bochner (and others) is to investigate the

asymptotic behavior of r(A) as "A

----

oo." A natural approach to studying the

asymptotic behavior of r(A) is to study a generating function for r(A). Thus if

Z

is also positive definite and

n

x

n,

and etr(Z) ::::: exp(tr Z), define the theta

function

G(Z)

=

L

etr( -7rTZT').

T

It

is not difficult to derive Jacobi's formula

G(Z)

=

(detz)kf 2

e(z-

1

),

hence

(1) :Z::r(A) etr(-7rAZ)

=

(detz)kf 2 :Z::r(A) etr(-7rAZ-

1

).

A A

vii