Shift operators defined in the Riordan group and their applications

In this paper, we discuss a linear operator T   defined in Riordan group RR by using the upper shift matrix U   and lower shift matrix UTUT, namely for each R∈RR∈R, T:R↦URUTT:R↦URUT. Some isomorphic properties of the operator T and the structures of its range sets for different domains are studied. By using the operator T   and the properties of Bell subgroup of RR, the Riordan type Chu–Vandermonde identities and the Riordan equivalent identities of Format Last Theorem and Beal Conjecture are shown. The applications of the shift operators to the complementary Riordan arrays and to the Riordan involutions and Riordan pseudo-involutions are also presented.