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We dene Riordan matrix functions associated with Riordan arrays and study their algebraic properties. We also give their applications in the construction of new classes of Bernoulli and Euler polynomials and Bernoulli and Euler numbers, referred to as the duals and conjugate Bernoulli and Euler polynomials and dual and conjugate Bernoulli

and Euler numbers, respectively.

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 …