49-2 Ml!. II. A. WKIJB ON THE COXVKI.'CKN'CE OF 



13. The formula 



a( r -/3)(-a+l)F(a+l, 0-1, y, x)-(y-a) (- + 1) F (a- 1, 0+1, y, x) 



--)} F(a, & y, r) . . . (25) 



may be easily verified.* 



Let a be a negative integer ; F (, 0, y, .<) becomes a polynomial in .r. Write 

 a = it, +0 = p. We deduce from (25) the resultt 



F(-n, 

 is equal to 



(26). 



Assume that the real parts of y and (/>+! y) are positive. Multiply (26) 

 through by 



and integrate from z to z 1. Making use of (22) and (24), we find after some 

 reduction that, when the real parts of y and (p+l y) are positive, 



iF(0, j>,y,.r).F(l,y, p+l 

 i \ t 



1 F / n f ,, y ,\ 



1 * ( ' P 



! ( n . 



(l> 

 +l, y+n, 



* It may be derived at once from formulae [1], [2], [3], [6], and [7], given by GAUSS, " Disquisitiones 

 gencrales circa seriem inhnittim 



l + ^., + ..., 



l -y 



' Werke,' vol. III., pp. 125-162. 



Reference will be made also to another memoir, " Determinatio seriei nostne per {cquationem dittercntialem 

 secxindi ordinis," GAUSS' ' Wcrke,' vol. III., pp. 207-230. 



t For the methoil of deduction, r/. WHITTAKEK, ' Modern Analysis,' p. 228. 



