18 
Indiana University Studies 
This idea was not entirely a novel one since Pincherle and 
Bourl^t* in 1897 had shown that the inversion of a linear functional 
operator led to the integration of a differential equation of in- 
finite order. However, no use of the idea was made until Lalesco’s 
paper established its connection with integral equations. Thus he 
considered a Yolterra equation of first kind (1) with a kernel devel- 
opable in the following series: 
(t -x) (t — x) n 
K(x,t) = a Q (t) + ai(t) (- + a n (t) 
1! n! 
Assuming that lim f (n) (x) = F(x) existed, Lalesco showed that 
n — »- 00 
equation (1) could be replaced by a differential equation of infinite 
order : 
„JToo{[a 0 (x) u(x)]'" ) -[a 1 (x)u(x)] (n - 1) +. . . . + ( -l)"[a n (x) u(x)]}= F(x), 
with initial conditions on u(x) and its derivatives determined from 
an auxiliary infinite system of linear equations in an infinite number 
of unknowns. 
The ideas developed in this paper were later extended to include 
equations of Fredholm type 289 and. to the closely related subject of 
integral equations of infinite order, 288 i.e. equations of the form, 
u(x) + ai (x) f u(t) dt + a 2 (x) f u(t) dt 2 + = F(x). 
J 1 J 2 
This new method in integral equations lay dormant, however, 
until 1918 when, in an extensive paper, F. Schiirer 291 elaborated the 
ideas of Pincherle, Bourlet, and Lalesco and applied them to an 
extended class of functional equations. This paper was the starting- 
point for a series of papers by E. Hilbf and 0. Perron J which con- 
sidered the integration of special types of equations and their 
application to the solution of linear difference equations. 
Hilb introduced the novel idea of connecting the solution of a 
differential equation of infinite order with the solution of a set of 
equations in an infinite number of variables. Thus he derived a 
linear system by means of an infinite number of differentiations of the 
original equation and then made use of the methods of the Hilbert- 
Schmidt theory to obtain a solution. This method of attack has 
been employed by H. T. Davis 286 in discussing the Euler differential 
equation of infinite order which is associated with various types of 
singular integral equations. 
*Loc. cit. See Bourlet’s memoir, p. 178. 
fMathematische Annalen, vol. 82 (1920-21), pp. 1-39; vol. 84 (1921), pp. 16-30, 43-52; vol. 85 (1922), 
pp. 89-98. 
Jlbid., vol. 84, pp. 31-42. 
