Chessin — On Some Relations Between Bessel Functions. 107 



(27) JniayJ'-ni^) + J-n+l{<^)J'n+M 



= 2 i xJn {ax)J_,{ 



*^0 



Further, if n denote a positive, but not integral, number, 

 formulas (20)^ and (21)^ yield the relations 



(28) a^„(^)e/„+,(«)— ««(«K«+i(^) 



= (a2_^2) xJn{ax)J^{/3x)dx, 





(29) Jl{a)—J„_^(a)J„+^(a) = 2\ x4(ax)dx, 



I I xJI^ 



in form identical with (22) and (24) ; while n being a nega- 

 tive, but not integral, number, we derive from (20)3 and 

 (21)3 the formulas 



(30) aJ_„(^)J_„+i(a) _^./_„(a)./_„+i(^) 



xJ_n{ax)J_n(^x)dx, 



= (a2 — /32) j ; 



a)=2rxe/I„i 



(31) J_„(a) — J_„_i(a) J_„+^(a)=2 I xJ_„(ax)dx, 



which are, practically, identical with (28) and (29), since 

 — n in (30) and (31) is a positive, but not integral, number. 



The integration between the limits and 1 in formulas 

 (20), and (21), when n<0, or in (20)3 and (2I3) when ?i>0, 

 is only possible if | n \ <1. Hence, formulas (28) and (29) 

 are still valid if — l<?i<0 ; and formulas (30) and (31) hold 

 good also when 0<>i<l. 



Finally, in the relations (18)3 and (19)3 the integration 

 may be taken between the limits and 1 when?i = 0. We 

 then obtain the formulas 



