Chessin — On Some Relations Between Bessel Functions. 103 



(13)3 ^1 ft;[ae7_,,(/3x)J_„+i(ax)— /3J_„(ax)J.„^.i(^x)] 



= (a2_/32)xJ_,(ax)J_„(^cc), 



whenn is not an integral number. 



When a = /8 the formulas just derived assume the form 

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

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

 cases if we take into consideration the relations 



1 



(14) J^^^{x')K,,{x) — J^{x)K,,^^{o:) =-, 



when n is an integral number, and 



2 sin nTT 



(15) Jn(^y-n+l(^) + J-ni^yn-li^) = ^^ , 



when 71 is not an integral number.* To derive the relations 

 corresponding to (12) and (13) when a = ^8 we, therefore, 

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

 respect to a and put /3 = a in the results. We then arrive at 

 the formulas : 



(16)i a^W\yl(''^0 —J^_^{ax)J^^-,(ax)'j\ = 2xJl(ax), 



(16)2 -^ I x'^[j^(ax)K^(ax) —J,^_^(ax)Ii^_^^(ax)'^ I 



= 2xJ^(ax)K„(ax), 



d 



( 16)3 ^ I x2[/lf (ax) — K,,_^{ax) Ii,,+i{ax)'^ 



when n is an integral number, and 



^2 



= 2xlL^(ax), 



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



