Chessin — On Some Relations Between Bessel Functions. 103 



= (a-^ — ^)xJ_,(ax)J_,(^x), 



whenn is not an integral number. 



When a = ^ the formulas just derived assume the form 

 0=0. This is obvious in the case of (12X, (12)3, (13)i, 

 and (13)3. It is, however, quite as evident in the other 

 cases if we take into consideration the relations 



(14) J,^,(x)Sr^(x)-J„{x)K„^,(x) =i, 

 when n is an integral number, and 



(15) Jn(x)J-n+l(x) + J_,{X)J^,(X) = ^ '''' """^ 



rr X 



when n is not an integral number.* To derive the relations 

 corresponding to (12) and (13) when a = /3 we, therefore, 

 differentiate both sides of the identities (12) and (13) with 

 respect to a and put y8 = a in the results. We then arrive at 

 the formulas : 



( 16)i ^1 x^^ji(ax) — Jn-i(ax)J^^^(ax)2 \ = 2xJI (ax), 



(^^)^ ^ { ^l«^n(«^)/a«^) -Jn-i(<^x)K,^,(ax)'] } 



= 2QcJ„(ax)ir„(ax), 



= 2xK^(ax), 

 when n is an integral number, and 



* See the treatise on Bessel functions by Gray and Mathews, p. 16. 



