Page "Generalized hypergeometric function" Paragraph 8

β < sub > n </ sub > = n !< sup >− 1 </ sup > and β < sub > n + 1 </ sub >/ β < sub > n </ sub > = 1 /( n + 1 ).

So this satisfies the definition with A ( n ) = 1 and B ( n ) = n + 1.