 
Summary: SYMBOLIC COMPUTATION OF INTEGRALS BY RECURRENCE
MICHAEL P. BARNETTy
Abstract. We discuss (1) the construction of recurrence formulas for several types of inde nite
and de nite integral, (2) the conversion of some of the recurrence schemes to general closed formulas,
(3) the mechanization of these processes, (4) the outperformance of present automatic integration
software by the procedures that use our formulas, and (5) the interface between symbolic computation
(computer algebra) and mathematical discourse.
Key words. symbolic computation, computer algebra, automatic integration, recurrence for
mulas, potential theory
AMS subject classi cations. 68W30, 33E20, 35Q40
1. Introduction. The gamma functions
Rz
0 e;ttk;1dt 1] and the Gaunt coe 
cients
R1
;1 Pl(x)Pm(x)Pn(x)dx 2] comprise two sets of integrals that can be computed
by recurrence. These examples and thousands of others are used throughout the nat
ural sciences and engineering. A recent online literature search by the author found
hundreds of citations that mention recurrence formulas for integrals in the title and/or
the abstract. Typically, a formula for an integral, that recurs on an integer exponent
