38 





2{n + s + D! 



+^'' ^ (2n + l){2n + ^){n- sV. 



[105] 



The above formulas are also from Bateman, pp. 363-364. These relations are supplemented 

 by certain additional integrals which will ijow be proved. 



(1) 



lid fi {n + s)' 





s)\ 



[106] 



This may be proved by induction. Call the above integral K^ and assume [106] to hold for 

 K^, . Then making use of [94] 



J _i "+l " 1 - M n- s + 11 J _j 1 - /i J_i 1 - r 



The first term on the right is easily integrated: 



j_;,,,^.jy_i^_|No 



s(2n +1) (n - s)l 



using Equations [102] and [104]. Then 



[2(n - s) + 1] (n + s)\ (n - s)(7i + s)! (71+ s)! 



A'„ = 



s(7i - s + 1)! s(n - s + 1)! s(n - s)! 



so Equation [106] holds for A'„ if it holds for A '„_! • It is easily shown that it holds for n equal 

 to s: 



and 



^.=/>».n^f^ = '--'Jl«>^f^ 



tffi (2s)! 

 2 "s • 0! 



