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Mathematics. — “Series of Polynomials”. (Ast part). By Prof, 
J. C. KLUYVER. 
(Communicated in the meeting of February 22, 1902.) 
Starting from a given power series representing within a definite 
circle of convergence an analytical function Z’(x), investigators have 
endeavoured to deduce for this function other developments with a 
different region of convergence. In the first place another power 
series will present itself as a new development; however this has 
the drawback, that each coefficient of whatever order is dependent on 
all coefficients without exception of the given power series. We can 
require the xh term in the new development to be completely deter- 
mined by the first x terms of the given power series. 
This demand is fulfilled by the development of F(x) in a series 
of polynomials according to the idea of Mirrac-LEFFLER, and now 
it is known that for each function such like developments can be 
found in infinite variety. 
The purpose of this paper is in the first place to give a simple 
deduction for such like series of polynomials, in the second place to 
treat of a couple of simple examples which can serve as an expla- 
nation of the peculiar conditions of existence of the polynomials. 
1. When the power series 
h= gh 
F(a) = F(0) + = — F(0) 
r=ih! 
must be continued by a series of polynomials we need some auxiliar 
function g(u), of which we suppose the following : 
L. g(u) is holomorphic in «= 0 and uniform for | «| < 4, where 
k is greater than 1. 
2. If g(u) is not everywhere finite within the circle | «| =4, the 
function is for |u| increasing from nought for the first time 
finite in a point a. : 
5. gO) = 0; alt. 
By these three suppositions one is but slightly limited in the 
choice of the function g(u). From any function f(«) holomorphic 
in w=0, a function g(u) can be deduced; we can take 
at) LOI = FO), 
FH) — {0 
35 
Proceedings Royal Acad, Amsterdam, Vol, IV, 
