Mr Orr, On the product J m (a?) J n (x). 95 



We may suppose that neither m nor n is negative and will also 

 suppose, unless otherwise stated, that none of the constants m, n, 

 ni ± n, is zero or an integer. The four convergent series are then 

 interminable ; none of them contains any terms of the form 0/0, 

 nor can the same power of x occur in any two. We therefore 

 have 



n,m+n 



i m {x) i n o) * = 2»+»n(m)n(») 



p (m + n + 1 m + n + 2 n ,\ 



Fl - , — ; m + n + 1, m + 1, n+ 1 ; x>\ 



(9), J 



and three other analogous equations. 



The case in which m = 1/2 is of great interest, and, therefore, 

 although the relations obtainable are well known, it may be 

 pardonable to investigate them from this point of view. In this 

 case it is seen that of the four convergent series (4), one consists 

 of the terms of odd order, and another of the terms of even order 

 of the series 



aP+*F(n+l/2; 2n + 1 ; 2x) (10), 



a third of the terms of odd order, and the fourth of the terms of 

 even order of the series obtained from (10) by changing the sign 

 of n ; the series last mentioned satisfies the same linear differential 

 equation of the second order as (10). It is also seen that of the 

 two divergent series (5) consists of the terms of odd, and (6) is a 

 multiple of those of even order in the series 



6 (l. + n l _ n . Z±\ = i _ (*+*)(*"*) + (ii) 



which is associated f with (10) and its analogue. Thus by addition 

 and subtraction relations (9) become on division by as* 



e*.I n = (ttx/2)1 (I, + I_ h ) I n = gJ^ F{\ + n ; 1 + 2n; 2x) 



(12) 



*- . I n = (ttx/2)1 (I. h - 1.) /. = 2^- F(± + n;l + 2n;- 2x) 



(13), 



and two analogous equations obtainable by changing the sign of n. 



* Schlafi, Math. Ann., in., gives the equivalent formula 



IT 



2 /"2 

 J m [x)J n {x) = - J m+n (2x cos 4>) cos (m-n)cp.d<p 

 v J o 

 m and n being integral. 



t See the paper referred to above, Art. 3. 



8—2 



