The paper characterizes the reproducing kernel Hilbert spaces with orthonormal bases of the form {(an,0+an,1z+…+an,JzJ)zn, n [ges ] 0}. The primary focus is on the tridiagonal case where J = 1, and on how it compares with the diagonal case where J = 0. The question of when multiplication by z is a bounded operator is investigated, and aspects of this operator are discussed. In the diagonal case, Mz is a weighted unilateral shift. It is shown that in the tridiagonal case, this need not be so, and an example is given in which the commutant of Mz on a tridiagonal space is strikingly different from that on any diagonal space.