m. 



12 Prof. J. W. Nicholson on Zonal 



Let m be even. Then n is even, and writing 2m, 2n for 

 them, 



cos(2meos-V)_iv QO M M i ,,T(n + m + i)T(n — m + i) p ( , 



• • • (29) 



The case m = gives the old expansion for (1 — At 2 ) - *. 

 Integrating between zero and /jl, 



sin (2m cos' 1 fi) 



. . . (30) 

 of which the case m = is Catalan's theorem. 

 We also have 



. cos (2m + 1 . cos -1 //,) A -p . N 



^j--, =A »^W' 



where _ls T 1 J 



An=^p| P 2 , +1 ( / ,)cos{(2m + l)cos-V} A7 j^ 



_ 4n + 3 r(» + m + I) T(n - m + j) 

 2 r(n + ™ + 2)r(n-m + l)' 



and therefore 



cos (2m -f- 1 . cos _1 /x) 



ra 1 (n + m + 2) 1 (w — m + l) 



and 



•sin (2m + 1 . cos _1 /z) 



/ . iw 00 r(w + m~l-4) r(n— m + i) (-d , . -d / N ") 



= m+i Z w -p , ' p . -fr)V 2n + 2 {f J )-Y 2n (lM)[. 



m L [n + m + 2) 1 [n — m + 1) (_ r/ ) 



. . . (31) 



In order to obtain corresponding developments relating 

 to Q functions, we first find the Fourier series by another 



