866 
In our case we find, by ‚putting h>=kS m; 
ul) (a) 7 A | 
ml < be yee eset 
always on the understanding that m == 22-1, By means of the 
formula of STIRLING: 
6 
k! = kke bh y/2ak ek, (0 <0 <1) \ 
we can also write for it here 
yim (wv) <i 48 2 e 
Leh 5/ (1—&\W am 
ri yim) = Avs m 2 i 
m! 18 (lW am 
The last inequality holds for any m, since, for the values of m, not 
equal to an odd power of 2, a (rm) —= 0. The terms of the series 
(1) therefore will be smaller in absolute valne than those of a 
converging geometrical progression, if 
] 
Aes 12. or & : : 
< 2 : oer eT 
and hence that series will converge in the whole domain («) if 
I 
cay 
As §8=a-+-c here, 8 will be greater than the radius of convergence 
| of the function considered, if 
a>l—e 
The preceding two inequalities may be fulfilled for a number of 
values of «, if the number c is chosen greater than 3. In any 
domain (a) satisfying the inequality 
so that 
the series. (1) produces therefore a transmuted for the function (AAE) 
although the corresponding domain (3) has a radius greater than the 
radius of convergence of that function; this being the case, though 
(oP Tk\ EE) (A+B! ah Ie kJ ee \ERt 
eS ae ern reel i! (h+-k—il) l—a 
and since (h + k—i'! > h! (k—i)!, the right hand member is less than 
(h +. k)! wht Ne k av erin (h d- k)! al 
hl (1 —#) oom i eee = h! a —a)k+1 
