182 PROCEEDINGS OF THE AMERICAN ACADEMY. 



if we write 



(42) (T+ m-1 )<"" p2,„ = Q2.n, (T+m- i)'"" p^+i = Q,„,+i ; 

 and, evidently, formulae (41) hold for any constant multiples of Q2,„ and 

 Q2,n+i, respectively. 



Now (32) shows that 



( T + m) • P2„ = 2 m (m — i) ' P2m-2 

 and, by continued application of the same formula, 



(43) ( T + m)'*=' p2„, = 2' m^'^ (m - i) '*> ■ p,^_.j, 



for ^ i' < m — 1 ; in particular, 



(T+ m)""-'V2„. = 2"'-i m""-i' (m - i)""-^' p2 

 and 



(44) ( T + m)'"" p2„. = 2'» • m ! (m - i)'"" po, 



because, by (34) and (36), 



m! =: w""> = m""-i> • 1, 



(^_f)<'«> = (m-i)"«-^''i, 



(7^+ l)p,^p, = po; 

 so that (43) holds also for k = m. Then, by (42), (38), (43), and (36), 



m 



Q,,_, =. V (- 1)' m'" {r+ m)""-'' p2„, 



(45) m 



= 2". (« - «.-' ■ « :;^/- 1)' (7) 5r(j^, 



for ^ m. 



Similarly, we have, from (32), 



{T+ m + i) ■ p2,„+i = 2 m (m + I) • p2,„_i 



and, by continued application, 



(46) (T+ m + *)<*•' • p2„+i = 2' m<^-' (m + i)'*' * P2™-2*+i 

 for ^ i ^ m. Therefore, by (42), (38), (46), and (36), 



m ^ 



<?o„.+i =]S-^~ l)'m<"(r+ m + i)""-'V2m+i 

 (47) 



2'»+i (m + i)"''+" • m !]^.(- 1)'(7) 



P2,+l 



2.+1 (i + ^)<w» 







for ^ m. 



