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A study aims to present the method of coefficients, which can be described as one supposes that one has to prove some identity or to evaluate some expression that is, to find a closed or asymptotic formula for the expression. It demonstrates how a simple set of rules taken from G. P. Egorychev can provide a way of solving a number of combinatorial problems that lend themselves to expressions in terms of generating functions.

Using the inverse limit tool, we obtain the Riordan group in its infinite and bi-infinite representations from groups of finite Riordan matrices. We define two different reflections for bi-infinite Riordan matrices. Employing these definitions, we give answers to the problems D ⊥ = D and D ⋄ = D in the Riordan group. So, we describe the self-complementary and self-dual Riordan matrices. [ABSTRACT FROM AUTHOR]