GAMMA COEFFICIENTS AND SERIES. 
I. THE CorFrFriciEeNts. 
1. The function. 
V(c+y+ /) 
(a +1) r(y+1)" 
will be called a gamma coefficient of codrdinates x, y, , and parameters a,b,” , 
(axby )=(ax+by+ -) 
and a multinomial coefficient when each parameter is unity. We shall use 
Greek letters to denote codrdinates taken from the series 0, 1, 2,3, "7 
At points of discontinuity, the sum of the coérdinates is zero or a negative 
integer. These points are excluded in the following properties. 
2. A gamma coefficient with a negative integral codrdinate is zero. 
3. Zero codrdinates 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. 
Dd. 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 En operate to diminish the n’th codrdinate by 
a unit, then in symbols, *(Note) 
(axby )=(#,\+H#.+..)(axby’ *) 
This may be extended to the n’th repetition of #,+#,+ °° =1, where 
the E’s combine by the laws of numbers. 
6. The above property furnishes an immediate proof of the multinomiat 
theorem. Thus let 
Fn=<X(lalgp'') p*¢'', a+B+ 9 =n 
i. e. the summation extends to every point the sum of whose coérdinates is 
B. 
n, there being a given number of variables p, g, , and corresponding in- 
tegral coérdinates a, 8, °. Applying art. 5 to the coefficients of F'n, we find 
Fn=(p+q+ ')F(n—1),andsince Fl =p+q+ |, therefore Fn =(p+q+ ‘)”. 
7. Zero parameters and corresponding codrdinates may be omitted, if the 
resull be multiplied by the multinomial coefficient of the omitted coérdinates 
and one other, the sum, less 1, of the retained coérdinates, as, 
(OxOybzcw) = (bzew) (Aalylw’), w’ =z+w-1 
8. Equal parameters and their coérdinates may be omitted, except one ta 
* (Note) Read » for 7 throughout this paper. 
