chapter  8
Sums of Type II(2,2,z)
Pages 3

Rather than consider the expression (8.1) with generalized binomial coefficients, it is better to multiply the equation by the integral denominator [n]n. Then (8.1) can be written as

( n

k

) [x]k[y]n−k = [x + y]n. (8.2)

But this expression allows a generalization with arbitrary step size:

Theorem 8.1. For n ∈ N0, x, y ∈ C, and d ∈ C, we have n∑

( n

k

) [x, d]k[y, d]n−k = [x + y, d]n. (8.3)

Proof: Let

S(n) := ∑ k

( n

k

) [x, d]k[y, d]n−k.