ABSTRACT

Path functions and their basic properties are obtained by extending

the constructive theory of partition generating functions developed by

Sylvester, Durfee, Andrews and others. Path functions also arise when we

have an expansion of a function satisfying a linear difference equation.

We give infinite families of generalizations of the , . f

0

summation

formula and the q-analogs of Gauss' theorem and the limiting form of

Jackson's theorem. Our expansions provide an interesting trade-off

between complexity and rate of convergence. We add one free parameter to

the q-analog of the limiting form of Jackson's theorem and generalize this

result.

* Partially supported by NSF grant MCS78-07244A02.

1980AMS subjectclassifications(1985Revision).

Primary: 33A30, 05A15.

Secondary: 05A17, 11P57, 11P72, 39B40, 41A25.

Key words:

basic hypergeometric function, partition, generating function,

q-difference equation, rate of convergence, q-analog, binomial

theorem, Gauss' theorem, Jackson's theorem.

Library of Congress Cataloging-in-Publication Data

Kadell, Kevin W. J., 1950-

Path functions and generalized basic hypergeometric

functions.

(Memoirs of the American Mathematical Society,

ISSN 0065-9266; no. 360)

"Volume 65 number 360 (third of 5 numbers)."

Bibliography: p.

1. Functions, Hypergeometric. 2. Partitions (Mathematics)

3. Generating functions. I. Title. II. Series.

QA3.A57 no. 360 [QA353.H9] 510s [515'.55] 86-28866

ISBN 0-8218-2420-1

iv