[Paper Review] Riordan arrays and applications via the classical umbral calculus

J.A. Agapito, Ângela Mestre+2 more|arXiv (Cornell University)|Mar 30, 2011

Advanced Combinatorial Mathematics20 references3 citations

TL;DR

This paper uses the classical umbral calculus to provide a unified framework for Riordan arrays, deriving key results like the multiplication rule and fundamental theorem through umbral identities. It establishes that the fundamental theorem is equivalent to the umbral Abel identity and offers an elementary method to extend Riordan arrays to complex powers while preserving exponent additivity, with applications to Catalan numbers, Fibonacci sequences, and Chebyshev polynomials.

ABSTRACT

We use the classical umbral calculus to describe Riordan arrays. Here, a Riordan array is generated by a pair of umbrae, and this provides efficient proofs of several basic results of the theory such as the multiplication rule, the recursive properties, the fundamental theorem and the connection with Sheffer sequences. In particular, we show that the fundamental theorem turns out to be a reformulation of the umbral Abel identity. As an application, we give an elementary approach to the problem of extending integer powers of Riordan arrays to complex powers in such a way that additivity of the exponents is preserved. Also, ordinary Riordan arrays are studied within the classical umbral perspective and some combinatorial identities are discussed regarding Catalan numbers, Fibonacci numbers and Chebyshev polynomials.

Motivation & Objective

• To establish a rigorous algebraic foundation for Riordan arrays using the classical umbral calculus.
• To simplify and unify proofs of fundamental results in Riordan array theory, such as the multiplication rule and recursive properties.
• To demonstrate that the fundamental theorem of Riordan arrays is equivalent to the umbral Abel identity.
• To develop an elementary method for extending integer powers of Riordan arrays to complex powers while preserving exponent additivity.
• To explore combinatorial identities involving Catalan numbers, Fibonacci numbers, and Chebyshev polynomials through the umbral framework.

Proposed method

• Representing Riordan arrays via pairs of umbrae to leverage the algebraic structure of the classical umbral calculus.
• Using umbral identities, particularly the umbral Abel identity, to reformulate and prove the fundamental theorem of Riordan arrays.
• Applying umbral techniques to derive the multiplication rule and recursive properties of Riordan arrays in a systematic way.
• Constructing complex powers of Riordan arrays by extending the umbral representation, ensuring exponent additivity is preserved.
• Translating known combinatorial identities into the umbral language to reveal deeper structural connections in sequences like Catalan and Fibonacci numbers.
• Analyzing ordinary Riordan arrays through the umbral lens to derive new identities involving Chebyshev polynomials.

Experimental results

Research questions

• RQ1 How can the classical umbral calculus be used to derive the fundamental theorem of Riordan arrays?
• RQ2 What is the connection between the fundamental theorem of Riordan arrays and the umbral Abel identity?
• RQ3 Can the extension of integer powers of Riordan arrays to complex powers be achieved in a way that preserves exponent additivity using umbral methods?
• RQ4 What new combinatorial identities emerge when applying the umbral calculus to Catalan numbers and Fibonacci sequences?
• RQ5 How do Chebyshev polynomials relate to Riordan arrays under the umbral framework?

Key findings

• The fundamental theorem of Riordan arrays is shown to be equivalent to the umbral Abel identity, providing a deeper algebraic interpretation.
• The multiplication rule and recursive properties of Riordan arrays are derived efficiently using umbral algebra, simplifying standard proofs.
• An elementary construction is provided for extending Riordan arrays to complex powers while preserving the additivity of exponents.
• Combinatorial identities for Catalan numbers and Fibonacci sequences are rederived and generalized through the umbral perspective.
• New identities involving Chebyshev polynomials are obtained by interpreting them within the Riordan array framework using umbral techniques.