chapter  12
4 Type II(p,p, z)
Theorem 12.12.
Pages 2

The general formula for this family is the following, cf. [Gould 72b, (1.53)]:

Theorem 12.12. For p, n ∈ N, y ∈ C and q, r ∈ N0 satisfying r, q < p and r + q ≤ p, with ρ = e 2πip and z = yp, we have the equivalent identities

( pn + r + q pk + r

) zk =

y−r

p

( 1 + ρjy

)pn+r+q ρ−jr, (12.21)

( n

k

) q∏ j=1

[ n + jp

] k

[ n− jp

] k

[ n + jp

] n−k

[ n− jp

] n−k

zk

=

[ n + jp

] n

[ n− jp

] n

p ( pn+r+q

) yr

( 1 + ρjy

)pn+r+q ρ−jr. (12.22)

Proof: Consider the sum on the right sides of (12.21)–(12.22), and apply the binomial formula to the power:

S = p−1∑ j=0

( 1 + ρjy

)pn+r+q ρ−jr =

( pn + r + q

i

) ρijyiρ−rj

= pn+r+q∑

( pn + r + q

i

) yi

ρ(i−r)j . (12.23)

As we have p−1∑ j=0

ρ(i−r)j =

{ p if i ≡ r (p), 0 if i ≡ r (p),

we can sum only the nonzero terms by changing the summation variable in (12.23) to k, where i = r + kp, to get

S = n∑

( pn + r + q pk + r

) ypk+rp = pyr

( pn + r + q pk + r

) zk. (12.24)

This yields (12.21).