Section III., 1914 [185] Trans. R.S.C. 



Two Identities associated with generalized Legendrian coefficients. 



By James Harkness, M.A., F. R.S.C. 



The functions K v , s (a), or K v , s (cos 6), are generalized Legendrian 

 coefficients, denned by 



(1 -2ax-\-x 2 ) -» = 1+ 2 K VtS (a)x* , v * o, 



5=1 



where a = cos d. The expressions for the earlier functions K v , , K v ,\, 

 K v , 2 , etc., are obtained at once by expanding (1— 2ax+x 2 )~ é as 

 a power-series in x and equating coefficients on the two sides of the 

 equation; e.g., putting K v , = 1, they are 

 ' À',, = l, 



K~ v , i = 2v cos 8, 



Kv,2 = 2v(v + l)cos 2 6-v, 



K v ,s = $v{v+1) 0+2) cos :f d-2v(r+l) cos 0, 



etc., 

 when v is put equal to \, these reduce at once, as they should do, to 

 the standard expressions for the zonal harmonics P ( = 1), Pi (cos d), 

 P 2 (cos 0), P 3 (cos 0), etc., in terms of cos 6. 



Since cos" 9, where w=0, 1, 2, 3, ..., is expressible linearly in 



n 



terms of the K's, it is evident that sin 2w and sin 2n — can be represented 



as linear expressions in the K's. We propose to show that the associ- 

 ated formulae form a link of connection between two important 

 results in the theorv of series using Bessel Functions: 



(1) 

 /.-»(* sin fl) / 2 - (v + 2s)T{s + i)T(v) 



r~' n\« ' = 1/ " " 2 —, j j-jT K v ,'2s{<-Ob(f) J v +2s{*)i 



/ z , ( 2.Ysin - ) 



io\ V 2 / w r (v) ^ 



W — V _ Ç = ^1 2 (*+,) J\ + .(*)*,. f (cOS*). 



(2 sin -;,-) " % * =0 



The former result is due to Nielsen (see his Handbuch der Théorie 

 der Cylinderfunktionen, p. 278), the latter to Gegenbauer. Though 

 the two types of infinite series are entirely distinct, there is as Nielsen 

 has pointed out (I.e. p. 281) a resemblance in their appearance which 

 is "in the highest degree remarkable". 



We shall find it convenient to employ the function R v {x) defined 

 by 



2»r(H-l)J, (tf) =x v R v (x), 

 so that 



(3-y ay 



R v (x) = 1 — z j — = h 7—r. ;— : r-p; — 



. l.t'-fl 1.2.3+1. v+2 



