chapter

B.3 Two Basic Chu–Vandermonde Convolutions

Theorem B.4.

Pages 1

Proof: It is sufficient to prove (B.23) and (B.24) for m = 0. We will prove (B.24), from which (B.23) follows.

Let

S(n, a, b) = ∑ k

{ n

k

} {a,−1}k{b,−1}n−kQa(n−k)Rbk.

Using (B.18), { n

k

} = { n− 1 k − 1

} Rn−k +

{ n− 1 k

} Qk,

we split S into two sums. In the sum with {

we substitute k+1 for k. When we take advantage of common factors in the two sums, this gives us

S(n, a, b) = ∑ k

{ n− 1 k

} {a,−1}k{b,−1}n−1−kQa(n−1−k)Rbk

× ({a + k}Rb+n−1−k + {b + n− 1− k}Qa+k).