Hiromasa Nakayama
logic1.implies(
logic1.or(set1.in(arith1.unary_minus(b), setname1.N),
logic1.or(set1.in(arith1.unary_minus(c), setname1.N),
set1.in(arith1.unary_minus(d), setname1.N))),
hypergeo1.hypergeometric_pFq(list1.list(a, b, c, d),
list1.list(k - b, k - c, k - d), 1)
=
((hypergeo0.gamma(k - b) * hypergeo0.gamma(k - c) *
hypergeo0.gamma(k - d) * hypergeo0.gamma(k - b - c - d))
/
(hypergeo0.gamma(k - c - d) * hypergeo0.gamma(k - b - c) *
hypergeo0.gamma(k - b - d) * hypergeo0.gamma(k))
*
hypergeo1.hypergeometric_pFq(
list1.list(b, c, d, k / 2 - (a / 2), 1 / 2 + (k / 2) - (a / 2)),
list1.list(k - a, k / 2, 1 / 2 + (k / 2), b + c + d - k + 1),
1)));
Transformation of a nearly-poised series 4_F_3(1)
@@ /. {a->1/2,b->1/3,c->-3,d->1/4,k->1}