2019/07/29 by E. Burlachenko, Burlachenko, E.
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1907.12272
openalex publication_date 2019/07/29 · openalex created_date 2019/08/13 · openalex updated_date 2026/07/28
Let ( g( x ),xg( x ) ) be a Riordan matrix from the Bell subgroup. We denote ( g( x ),xg( x ) )φ=( g( φ)( x ),xg( φ)( x ) ), where a matrix power is defined in the standard way. The polynomials cn( x ) such that g( φ)( x )=∑\nolimitsn=0∞ cn( φ)xn 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 ( g( x ),xg( x ) ) is a pseudo-involution, then there exists a numerical sequence (B-sequence) with the generating function B( x ) such that g( x )=1+xg( x )B( x2g( x ) ). The matrix whose B-sequence has the generating function φB( x ) will be denoted by ( g[ φ]( x ),xg[ φ]( x ) ). The polynomials un( x ) such that g[ φ]( x )=∑\nolimitsn=0∞ un( φ)xn 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( x )=( 1-x )-1 (RNA matrix), B( x )=1+x, B( x )=C( x ), where C( x ) is the Catalan series, are considered in detail.