ALGEBRAIC THEORY OF KAPTEYN SERIES 



By James Harkness, M.A., F.R.S.C. 



(Read May Meeting, 1920) 



In 1893 Kapteyn devised a new type of expansion in Bessel Functions, 

 and later it was shown by Nielsen that the Bessel functions employed 

 could be of the most general character, namely those defined by 



(X \''+2« 

 t) 



-/, {X) = S ~ , 



n=o n\ T{v-\-n-\-l) 



where v is an arbitrary parameter, free to take complex as well as real 

 values. 



Employing Nielsen's notation one result arrived at by Kapteyn for 

 the case v = 0, and generalized by Nielsen so as to cover all finite values 

 of u other than negative integers, is that the power series 



f(x) = S a„( -— ) can be expanded in the form 



the coefficients b being expressible in terms of the a's by linear relations 

 of the form 



<n/2 (y+^ -2/?)2 T(p-\-n-p) 



The /-series that arises in this way is called a Kapteyn series of the first 

 kind and its convergence is unconditional within a certain neighbourhood 

 of a; = 0. 



The theorem presupposes the possibility of the expansion 



(y)" = SaL/.+2n((^ + 2w)x). 



(1) 



(See Nielsen's Handbuch der Theorie der Cylinderfunktionen, Chap. 22, 

 p. 300.) 



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