  Based on weighted noncrossing partitions of type B, we obtain type B
analogues of Coker's identities on the Narayana polynomials. A parity reversing
involution is given for the alternating sum of Narayana numbers of type B.
Moreover, we find type B analogues of the refinements of Coker's identities due
to Chen, Deutsch and Elizalde. By combinatorial constructions, we provide type
B analogues of three identities of Mansour and Sun also on the Narayana
polynomials.
