ALCUNI TEOREMI SUI COEFFICIENTI DI LEGENPRE il 



P,,,.(cos6)(cos$)--^'(sen5r7?e=|p,,^.(/a)(pr^"'{l-P-^P"f^/^- 



O — I 



= - ^-^ -1 (ay^-^'n-a) ^ (l'I 



_ \ [{it+l){if-l)...{2t + ?>)][l.S...{2f + 2s+l)][l.3...{2r-ì)] ^(1^) 

 ~^j [2.4...2i'][2.4...(2 5+2^ + 2r + 2)] 



[(4^-l)(4^-3)...(2^+l)][1.3...(2^ + 2g-l)][1.3...(2r-l)] 



[2.4...(2f-2)2j[2. 4...(2r+2s+2 0] 

 (-ir[2^+l)(2^- l)...3.1][1.3...(2g+l)][1.3...(2r-l)ì ( 

 [2.4...2^][2.4. .(2r + 25 + 2)] | 



Si avrà del pari : 



T +1 > 



j P,,^, (cos &) (cos ^)-^' (sen^)--^- f7^=r | P,,^. (p.) {u.r^' {l-p.ydiJ, 



_ 2[(4^+l)(4^-l)...(2^+3)][1.3...(2^+2g+l)][2.4...(2r-2)2r] \ (20) 

 ~[2.4...2^][1.3.5...(2r-3j(2'r-l)(2r+l)(2r+3)...(2r+2 5+2^^3)] / 



[(4^_l)(4^_3)...(2f + l)][1.3...(2^+2.?-l)][2.4...(2r-2)2r] 

 ~ [2 . 4 ... (2 ^~ 2)] [1 . 3 . 5 ... (2 r - 1) (2 r +1) ... (2r +25+2^+ 1)] 



+ + 



, [(2^ + l)(2^-l)...3.1][1.3...(2.+ l)][2.4...(2r-2)2r] 

 "^ ^ ' [2.4...2^][1.3...(2r-l)(2r+l).,.(2r + 2s+3)] j 



Moltiplicando la (1) per (sen 6')'""*'' (cos(5)^'*"^' (Z^ ed integrando 

 fra i limiti e tt , avremo applicando la formola (5) agli in- 

 tegrali del secondo membro deirequazione che ne risulta : 



P,, (cos 0) (sen QY+' (cos Oy'^'dO^z j P^, (^.f (1 -p^y (a) -'+• dp. 



" _J /"""^'^'(/ul'-I)^' 



■.'{2t)\J IF^. ' 



\(21) 

 sia r pari o dispari. 



