108 HISTORY OF THE THEORY OF NUMBERS. [CHAP, ill 



integral is 



A' x + Blcr 1 + - - - + X' x ar* + - 



1 - A x a~ l - B x ar* - ... - X x ar* 

 Hence we get Va x ; let its expansion be 



Va x = A + A'cr 1 + A"a~ 2 + . 

 Differentiate its logarithm, regarding x as constant and a variable. Thus 



A' + 2A"a-i + , .,, _ 2 , 



-r~~i A, -, , = r + r'a 1 + r"a 2 + , 



A + A'a l + 



where r (m) is the excess of the sum of the (m + l)th powers of the roots of 

 the denominator over the sum of the (m + l)th powers of the roots of the 

 numerator in the fractional function of or 1 giving Va x . Hence 



A' = Ar, 2A" = A'r + Ar', 3A'" = A'Y + AY + Ar", ", 

 which give A '/A, A "/A, - as functions of r, r', . Hence, evidently, 



Z(y, x) = A4>(y) + A'<j>(y - 1) + A"0(y - 2) + - - -, 

 0(y) = TTia 2 ' + n6 tf + -. 



Consider (pp. 817-21) the number (y, x) of ways y is a sum of x equal 

 or distinct positive integers. Those in which 1 is a part furnish the (y 1, 

 x 1) ways y 1 is a sum of x 1 parts; while those in which each 

 part exceeds 1 give, upon subtracting 1 from every part, the (y x, x) 

 ways y x is a sum of x parts. Hence 



(y, x) = (y - x,x) + (y - l,x - 1). 



It has the integral (y, x) = a y Va x if a x = a~ x a x + a" 1 , whence 



or x 



The sum of the mth powers of the roots of a = 1, a 2 = 1, , a x = 1 is 

 the sum d(m) of those of the numbers m, m/2, m/3, -, m/m which are 

 integral and ^ x. Hence 



r<"> = (m + 1), A' = 5(1), A" = J{(2) + 5 2 (1)}, 



A(m} = JW a ( i)a(m - i) - 



mm m 



(8(3) g(l)g(2) /g(2) 5 2 (l)U(l)U(m-3) 

 I 3 3 \2 2/3/m 



= a" 1 + A 'a" 1 

 (2/, x) = <t>(y ~ x) + A'0(y - x - 1) + A"0(y ~ x - 2) + .... 



Take a: = 1. Then A' = A" = = 1, (y, 1) = <f>(y - 1) + <t>(y - 2) + 

 . -. Replace y by y 1. Thus (y - - 1, 1) = <j>(y - 2) + 0(y - 3) + 

 ... Hence (y, 1) (y 1, 1) = <j>(y 1). By the nature of our 



