Composition polynomials of RNA matrix and $B$-composition polynomials of Riordan pseudo-involution

Abstract

Let $\left(g\left(x\right),xg\left(x\right)\right)$ be a Riordan matrix from the Bell subgroup. We denote ${\left(g\left(x\right),xg\left(x\right)\right)}^{\phi }=\left(g^{\left(\phi \right)}\left(x\right),x{g}^{\left(\phi \right)}\left(x\right)\right)$, where a matrix power is defined in the standard way. The polynomials $c_n\left(x\right)$ such that $g^{\left(\phi \right)}\left(x\right)={\sum }_{n=0}^{\infty }{c}_n\left(\phi \right)x^n$ will be called composition polynomials. We consider the composition polynomials of the RNA matrix. The construction associated with these polynomials allows the following generalization. If the matrix $\left(g\left(x\right),xg\left(x\right)\right)$ is a pseudo-involution, then there exists a numerical sequence ($B$-sequence) with the generating function $B\left(x\right)$ such that $g\left(x\right)=1+xg\left(x\right)B\left(x^{2}g\left(x\right)\right)$. The matrix whose $B$-sequence has the generating function $\phi B\left(x\right)$ will be denoted by $\left(g^{\left[\phi \right]}\left(x\right),x{g}^{\left[\phi \right]}\left(x\right)\right)$. The polynomials $u_n\left(x\right)$ such that $g^{\left[\phi \right]}\left(x\right)={\sum }_{n=0}^{\infty }{u}_n\left(\phi \right)x^n$ will be called $B$-composition polynomials. Coefficients of these polynomials are expressed in terms of the $B$-sequence. We show that matrices whose rows correspond to the $B$-composition polynomials are connected with exponential Riordan matrices of the Lagrange subgroup in a certain way. The cases $B\left(x\right)={\left(1-x\right)}^{-1}$ (RNA matrix), $B\left(x\right)=1+x$, $B\left(x\right)=C\left(x\right)$, where $C\left(x\right)$ is the Catalan series, are considered in detail.