B-expansion of pseudo-involution in the Riordan group

2017/07/04 by E. Burlachenko, Burlachenko, E. · 1 citation
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1707.00900

Abstract

Each numerical sequence ( b₀,b₁,b₂,... ) with the generating function B( x ) defines the pseudo-involution in the Riordan group ( 1,xg( x ) ) such that g( x )=1+xg( x )B( x²g( x ) ). In the present paper we realize a simple idea: express the coefficients of the series g^m( x ) in terms of the coefficients of the series B( x ). Obtained expansion has a bright combinatorial character, sheds light on the connection of the pseudo-involution in the Riordan group with the generalized binomial series, and is also useful for finding the series g( x ) by the given series B( x ). We compare this expansion with the similar expansion for the sequence ( 1,a₁,a₂,... ) with the generating function A( x ) such that g( x )=A( xg( x ) ).

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