Volume 138, 1992
Special Functions on Finite Upper Half Planes
JEFF ANGEL, NANCY CELNIKER, STEVE POULOS, AUDREY
TERRAS, CINDY TRIMBLE AND ELINOR VELASQUEZ
The plan is to define some graphs which are finite analogs of the familiar
Poincare upper half plane. Then we will study some special functions living on
these graphs- mostly eigenfunctions of the adjacency operator. Here we give a
brief summary of our motivation. More details will be given later in the paper.
See also the earlier papers  and .
1) There are many applications of the finite circle 7J.,fn7J.,; that is, the quotient
ring of integers modulo
In particular, Fourier analysis on the finite circle
is necessary for swift computation of approximations to functions on the usual
circle J'Rf7J.,. So why not study a finite analog Hq of the Poincare upper half
with hopes of casting some light on the darker corners of
Here q is normally a power of an odd prime p. In fact, we will find that there
are finite counterparts for all the special functions on
which we considered
in Terras ; e.g., spherical, Bessel functions and modular forms. Much of our
work concerns eigenfunctions of combinatorial analogs of the Laplacian. Equiva-
lently we study eigenfunctions of adjacency operators of upper half plane graphs.
These operators may also be viewed as finite versions of the Radon transform
(see Velasquez ) as well as finite convolution operators. The K-Bessel func-
tion analogs were considered by Velasquez . The spherical functions theory
presented here was given by Poulos [26). The analogs of modular forms and
other examples in Sections 2.3.1 and 2.3.2 below were developed by Trimble.
Many authors have considered finite analogs of classical special functions. See
for example Evans [10).
Mathematics Subject Classification.
Primary 33C80, 43A90, Secondary05E30, 11F37.
/( ey words and phrases.
Modular form, spherical function, finite field, highly regular graph,
Ramanujan graph, Bessel function, Eisenstein series, Kloosterman sum.
This paper is in final form and no version of it will be submitted for publication elsewhere .
1992 American Mathematical Society
$.25 per page