{"paper":{"title":"$B$-expansion of pseudo-involution in the Riordan group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"E. Burlachenko","submitted_at":"2017-07-04T10:39:03Z","abstract_excerpt":"Each numerical sequence $\\left( {{b}_{0}},{{b}_{1}},{{b}_{2}},... \\right)$ with the generating function $B\\left( x \\right)$ defines the pseudo-involution in the Riordan group $\\left( 1,xg\\left( x \\right) \\right)$ such that $g\\left( x \\right)=1+xg\\left( x \\right)B\\left( {{x}^{2}}g\\left( x \\right) \\right)$. In the present paper we realize a simple idea: express the coefficients of the series ${{g}^{m}}\\left( x \\right)$ in terms of the coefficients of the series $B\\left( x \\right)$. Obtained expansion has a bright combinatorial character, sheds light on the connection of the pseudo-involution in "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.00900","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}