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In this paper, we discuss a linear operator T dened in Riordan group R by using the upper shift matrix U and lower shift matrix UT ; namely for each R 2 R, T : R 7! URUT . Some isomorphic properties of the operator T and the structures of its range sets for dierent domains are studied. By using the operator T and the properties of Bell subgroup of R, 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 …

Here we use row sum generating functions and alternating sum generating functions to characterize Riordan arrays and subgroups of the Riordan group. Numerous applications and examples are presented which include the construction of Girard-Waring type identities. We also show the extensions to weighted sum (generating) functions,

called the expected value (generating) functions of Riordan arrays.