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