Algebraic Theory of Kapteyn Series 57 



(III) g (.+4)"'+^ 



f + 2.1^+3 . . . v + G 

 2'" 1 



j/+3.;^4-4.v+5.i'+6.z/+7 



3m ^m ^ ^_y^, 



41 1.2.4 + -1.1.3 + -2.-1.2 +*+ -4.-3.-2 



■•J/+3 1^+4 v + 5 j'+T 



(V) (^+8)"'+^ ^ 



j;+4.v+5.j^+6.j'+7.i'+8 



4m 3*" 2*" 1 



(13) 



I 1.2.3 + -1.1.2 + -2.-1.1 + -3.-2.-1 I 



On addition the fractions all vanish. The proof consists in showing that 

 the J/ + 1 — residues in I, II are equal and opposite, 

 the v + 2 — residues in I, III are equal and opposite, 

 the v+S- residues in I, IV, as also in II, III are equal and opposite, 

 the J' +4 — residues in I, V, as also in II, IV are equal and opposite, 

 the 1^+5- residues in II, V, as also in III, IV are equal and opposite, 

 etc. This proof rests on elementary considerations and need not be 

 given here. In all cases, where m is even, the fractions cancel out. 



Let us now consider the case where the integral part, as distinguished 

 from the fractions, drops out. To take first a simple case let us examine 

 the numerator of (11). Its apparent order is w + 2 and this will be less 

 than the degree 5 of the denominator if w = 0, 2; but the third formula in 

 the second sub-system of (9) shows that w = 4 also provides a zero result, 

 which at first sight clashes with the fact that the order of the numerator 

 exceeds that of the denominator by 1, and that, therefore, there should 

 be, after division, an integral part of order 1. 



The explanation is that the true order of the numerator is not w + 2 

 but m, for the coefficient of v^^'^ is 1-3+3-1, =0, and that of 



V IS 



(4 + 5)-3(w+l)2-3.5+3(w+l)4+3-w.6-3, =0. 



The inequality w + 2 <5 must accordingly be replaced by m <5, and we 

 see that 4 is also a permissible value for the even number m. This 



