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The Overlays of Riordan Arrays

An array (formally, a function 𝐴: ℤ􀬶 → 𝔽) can be thought of as a matrix that extends infinitely in all directions. We can discuss operators 𝑋 and 𝑌 which shift an array right by one column or up by one row, respectively. A polynomial in 𝑋 and 𝑌 that annihilates 𝐴 is called a “template” whereas an “overlay” is an array with finite support that contains the coefficients of a template. Our research looks at new ways in which machinery such as templates and overlays can be applied to well-studied arrays, specifically focusing on Riordan arrays.

Riordan arrays are lower triangular arrays that extend infinitely only to the right and downwards. They were originally motivated by a generalization of Pascal’s triangle and can be described in one of two ways: Either column-wise by their generating functions or row-wise by their associated patterned sequence. This pattern sequence often corresponds to a natural representation as an overlay. Our research focuses on exploring that representation and the requirements for such a representation to exist.