782 Dr. J. W. Nicholson on 



and finally, we derive the remarkable formula 



n ( \ P ( \ X ? m^ i qv r(m + n--r ) r(m-n + r + l) 



T(m — n — r — j) r(m + n -f? 3 + j -) 



■P 2 ,.+i(»> 



where m and n are integers. 



There is nothing in our argument which causes it to fail 

 when n is pushed to a value exceeding m, and the formula 

 is therefore general. 



The only two Gamma functions which can ever become 

 infinite are T(rn — n — r) in the denominator, and T(m + n— r) 

 in the numerator. The former must occur first, and when 

 it does occur, in the first term (when n>rn), the earlier part 

 of the series vanishes, and we obtain a series starting with 

 the harmonic P 2 m+2»-M- 



If m > n, this series so obtained is accompanied by another 

 of a terminating type, — a gap lying between the two, — and 

 this other is of course found to be a multiple of the series 

 representing 



*2m 0*2/1 — ' Q,2m ±2n 



and already discussed. 



It is more convenient, however, not to separate the two 

 cases, but to include both in the general formula expressed 

 in terms of Gamma f unctions. 



The single series obtained when n>m is found to be a 

 multiple of our Series A , found from the differential equa- 

 tion. This series is therefore recognized in terms of the 

 solution it represents. 



The preceding discussion has proceeded on the supposition 

 that the order of both P and Q is even. When this is not 

 the case, the analysis is very similar, and we may leave it to 

 the reader to prove the following theorems, where m and n 

 are integers in all cases :-— 



Q2i»+l(/*) P2»+l(/V 



_ 1 S (a y TQ + n-r + 1) r(m — w + r+1 ) 

 2Zo { } T(m-n-r) ' r(m + n + r + 3) 



T(m—n— r— J) T(m + n + r + ■§■) 



' r(m + ?i — r + f) " r(m — n + r + f) 



. P 2r+ i(». 



