19 
answered by the theorem of PincuErLE, at least not in a simple 
manner, as will appear from what follows. In order to investigate 
the question we should commence to deduce the series (2) from 
the given function w(x). Next we should examine whether the func- 
tion g(t) represented by it has the required properties: to be regular 
without the cirele (1,1), and on the circumference of it of finite 
order. For this we should try to transform the above series into 
another according to negative integral powers of t—1 
ve) 
Cn 
OE 
The relation between the Rn of the two series is given 
by the equations 
cn = w (n+ 1) — & ) w(n) + ...+(—1)"w (1) = A"*[w(1)]') . (4) 
and 
ate (Iet +6 er ER deter) 
By means of (4) we must see if the series (3) converges without 
the circle (1,1), and further if the characteristic à of the coefficients 
Cn, defined by 
n= log n 6) 
is not positive infinite; the latter being the condition that g(t) shall 
be -of non-positive infinite order on the circle (1,1). 
But the relation (4) is rather intricate and so it may be very 
difficult, if not impossible, to perform the just mentioned research. 
Suppose this, however, possible, and let 2 differ from + o. Then 
we have to examine whether the given function w(x) is really the 
coefficientfunction of g(t). For there are a great many functions Q(z) 
giving rise to the same generating function g(t), viz. all those 
contained in the Ee 
2 (x) = 0 (2) + F (a) 
where F(x) is a Mheen that vanishes for positive integral del: 
of zv. It is therefore necessary to consult the definition of the coef- 
ficient-function given by PiNcHerLE and to apply it to the obtained 
y (t) in order to see if the original function w (#) is the result. But 
this is again not very easy. If the characteristic 2’ defined by (5) 
is less than —1, then p (4) is finite and continuous along the circum- 
1) This symbol denotes the n-th difference of (x) at x=1, the increase Ax 
of the argument x being equal to unity. 
2% 
