This monograph deals with the expansion properties, in the complex
domain, of sets of polynomials which are defined by generating
relations. It thus represents a synthesis of two branches of analysis
which have been developing almost independently. On the one hand there
has grown up a body of results dealing with the more or less formal
prop- erties of sets of polynomials which possess simple generating
relations. Much of this material is summarized in the Bateman compendia
(ERDELYI [1], voi. III, chap. 19) and in TRUESDELL [1]. On the other
hand, a problem of fundamental interest in classical analysis is to
study the representability of an analytic function f(z) as a series, Lc,
. p, . (z), where {p, . } is a prescribed sequence of functions, and the
connections between the function f and the coefficients c, . .
BIEBERBACH's mono- graph Analytische Fortsetzung (Ergebnisse der
Mathematik, new series, no. 3) can be regarded as a study of this
problem for the special choice p, . (z) =z", and illustrates the depth
and detail which such a specializa- tion allows. However, the wealth of
available information about other sets of polynomials has seldom been
put to work in this connection (the application of generating relations
to expansion of functions is not even mentioned in the Bateman
compendia). At the other extreme, J. M.