214 PROCEEDINGS OF THE AMERICAN ACADEMY 



for r = 0, 1, 2, p, .... v. Since yj, 1 ' (x) = € /*jf (#), either of these 



two functions may be denoted simply by^ (x). We then have 



/o w +/; 11 w +/? («> + • • ■ ./? w +/? w = l. i 



for /j = 1, 2, and r = 0, 1, 2, .... fi, . ...v, 



fT (0) . ff (6) = 0, 

 for p, q = 1, 2, and r, s = 0, 1, 2, /x, v, but rfs; and 



*-/o (*) = /o ( tr - * ) 

 = /o(-0) 

 = /oW> 



*. f? (?) = f? (*. 0) 

 =f?{-0) 



= /?(*), 



for /• = 1, 2, 3 . . . . n, v. We also have 



e = o ./ («) + *, V^I/i 1 ' - Ai V^/f * + .... 



+ K V~ifl ] (0) - h v a/^T/I 2 ' (0) ; 



therefore, if f{0) denotes any polynomial in powers of 6 or conver- 

 gent power series in 9, 



/(*) =/o./ o 0) +ffrV=i)f?W +f(-hV~i)f?(e) 

 + .... +f(k.V=i)f»(Q +f(-KV~i)f; } (0). 



Thus 

 ^ = e „ = e oy Q ((9) + ^v=iy« ((9) + ,-^v^iyc-, @) + . . . . 



From the relations given above between the functions with the same 

 subscript it is evident that this matrix is orthogonal. 

 Let now 



