269 



Gamma Coefficients and Series. 



I. The Coefficients. 

 1. The function. 



T(x+y+") 



(axby' ) =(ax + by+' ') 



T(x + l)T(y + l)-' 



will be called a gamma coefficient of coordinates x, y, , and parameters a, b, ' , 



and a multinomial coefficient when each parameter is unity. We shall use 



Greek letters to denote coordinates taken from the series 0, 1, 2, 3, 



At points of discontinuity, the sum of the coordinates is zero or a negative 



integer. These points are excluded in the following properties. 



2. A gamma coefficient with a negative integral coordinate is zero. 



3. Zero coordinates and their parameters may be omitted, as (axbycO) = 

 (axby). 



4. The gamma coefficient of a point upon an axis equals the parameter of 

 that axis, as {ax) = a. 



5. ■ The gamma coefficient of any point is the sum of the gamma coefficient 

 of the preceding points (a preceding point being found by diminishing one 

 coordinate by a unit). Let E?7 operate to diminish the n'th coordinate by 

 a unit, then in symbols, *(Note) 



(axby' ')=(E l +E 2 + ..)(axby' ') 

 This may be extended to the n'th repetition of E x + E- 2 + "'= 1 , where 

 the E's combine by the laws of numbers. 



6. The above property furnishes an immediate proof of the multinomial 

 theorem. Thus let 



Fre = S(lal/3' ') pV 3 ' ', «+/3+' ' =n 

 i. e. the summation extends to every point the sum of whose coordinates is 

 n, there being a given number of variables p, q, ' ' , and corresponding in- 

 tegral coordinates a, p, ' ' . Applying art. 5 to the coefficients of Fn, Ave find 

 Fn = (p+q J r )F(?i — l), andsince Fl =p-\-q-\- , therefore Fn = (p + q+ ) ■ 



7. Zero parameters and corresponding coordinates may be omitted, if the 

 result be multiplied by the multinomial coefficient of the omitted coordinates 

 and one other, the sum, less 1, of the retained coordinates, as, 



(OxOybzcw) = (bzcw) (lxlylw'), w' =z-\-w — 1 



8. Equal parameters and their coordinates may be omitted, except one to 



* (Note) Road n for -q throughout this paper. 



