Sequence characterizations of double Riordan arrays and their compressions

Inspired by the Fibonacci tree shown in the recent book, Catalan Numbers, by Richard Stanley, we present a combinatorial way to construct double Riordan arrays by using ECO technique. The sequence characterizations of double Riordan arrays and subgroups of the double Riordan group are found. As an extension of Pascal-Fibonacci triangle, the compression forms of double Riordan arrays called double quasi-Riordan arrays are defined. The connections of lower triangular matrices and double Riordan arrays as well as their compressions are given. Those results are also extended to the case of high order Riordan arrays, which are defined in the paper. The pairs of Sheffer polynomials and pairs of summation formulas associated with double Riordan arrays are defined and discussed.