chapter

B.9A Basic Transformation of a II(2,2,z) Sum

Theorem B.13.

Pages 2

The form (B.44) is symmetric in Q and R, and reversal of the direction of summation corresponds to the exchange of a with b and x with y.

Proof: It is sufficient to prove (B.44) for m = 0. In the left-hand expression we substitute (B.9) in the form {a}k = (−1)k{−a + k − 1}k(QR)ka−(

Next we apply the Chu-Vandermonde convolution (8.1) to the factorial and get

{−a + k − 1}k = ∑ j

{ k

j

} {n− a− b− 1}j{b− n + k}k−j

×Q(n−a−b−1−j)(k−j)R(b−n+j)j .

Substitution in the left-hand expression yields, after exchanging the order of summation,