chapter  14
Sums of Types III–IV
Pages 12

The classical formulas of type III are the Hagen-Rothe formula and the Jensen formula. The Hagen-Rothe formula is a generalization of the ChuVandermonde identity:

x

x + kz

( x + kz

k

) y

y + (n− k)z ( y + (n− k)z

n− k )

(14.1)

= x + y

x + y + nz

( x + y + nz

n

) . (14.2)

It was established in 1793 by H. A. Rothe [Rothe 93] and quoted by J. G. Hagen [Hagen 91] in 1891. It can be found in H. W. Gould [Gould 72b, (3.142) or (3.146)], J. Kaucky´ [Kaucky´ 75, (6.4)], J. Riordan [Riordan 68, p. 169] or in [Andersen 89] and [Andersen and Larsen 94].