Advances and Applications in Discrete Mathematics
Volume 6, Issue 1, Pages 11 - 44
(July 2010)
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A FORMAL CALCULUS ON THE RIORDAN NEAR ALGEBRA
L. Poinsot and G. H. E. Duchamp
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Abstract: The Riordan group is the semi-direct product of a multiplicative group of invertible series and a group, under substitution, of non-units. The Riordan near algebra, as introduced in this paper, is the Cartesian product of the algebra of formal power series and its principal ideal of non-units, equipped with a product that extends the multiplication of the Riordan group. The latter is naturally embedded as a subgroup of units into the former. In this paper, we prove the existence of a formal calculus on the Riordan algebra. This formal calculus plays a role similar to those of holomorphic calculi in the Banach or Fréchet algebras setting, but without the constraint of a radius of convergence. Using this calculus, we define en passant a notion of generalized powers in the Riordan group. |
Keywords and phrases: formal power series, formal substitution, Riordan group, near algebra, generalized powers. |
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