chapter
Rewriting the Double Sum as Two Single Sums
Pages 10

Now it is time to change variables in (14.34). We write i = j + k and change the summation indices to j and k and get

∑ j

[x + j]m−n[x− 1− j]n−1 ∑ k

( m + 1− n 2j + k + 1

)( n

k

) (x− j − k). (14.35)

The inner sum can be written as the difference of two sums

∑ k

( m + 1− n 2j + k + 1

)( n

k

) (x− j)−

∑ k

( m + 1− n 2j + k + 1

)( n

k

) k. (14.36)

Both are Chu-Vandermonde sums, cf. (8.1), i.e.,

∑ k

( m + 1− n 2j + k + 1

)( n

k

) (x− j) = (x− j)

∑ k

( m + 1− n

m− 2j − n− k )(

n

k

)

= (x− j) (

m + 1 m− 2j − n

) ,

∑ k

( m + 1− n 2j + k + 1

)( n

k

) k = n

∑ k

( m + 1− n

m− 2j − n− k )(

n− 1 k − 1

)

= n (

m

m− 2j − n− 1 ) .