[Paper Review] Motzkin numbers, central trinomial coefficients and hybrid polynomials

Paweł Błasiak, G. Dattoli+3 more|ArXiv.org|Feb 1, 2008

Advanced Combinatorial Mathematics9 references3 citations

TL;DR

This paper introduces a unified framework using hybrid Hermite-Laguerre polynomials to study Motzkin numbers and central trinomial coefficients (CTC), revealing their deep algebraic connections through operational calculus. The key contribution is identifying CTC and Motzkin numbers as special values of hybrid polynomials, enabling new recurrence relations and generalized forms, including p-associated and m-th order CTC sequences with closed-form recurrences.

ABSTRACT

We show that the formalism of hybrid polynomials, interpolating between Hermite and Laguerre polynomials, is very useful in the study of Motzkin numbers and central trinomial coefficients. These sequences are identified as special values of hybrid polynomials, a fact which we use to derive their generalized forms and new identities satisfied by them.

Motivation & Objective

• To establish a unified algebraic framework for Motzkin numbers and central trinomial coefficients using hybrid polynomial formalism.
• To derive new recurrence relations connecting $ c_{n+1} $, $ c_n $, and $ m_{n-1} $ via Hermite-Kampé de Fériét polynomials.
• To generalize CTC and Motzkin numbers into p-associated and m-th order forms, extending their combinatorial and algebraic properties.
• To explore combinatorial interpretations of the resulting integer sequences through parameterized Hermite polynomials and hypergeometric representations.

Proposed method

• Defining hybrid Hermite-Laguerre polynomials $ \Pi_n(x,y) = H_n(y, \widehat{D}_x^{-1}) \mathbf{1} $, where $ \widehat{D}_x^{-1} $ is the inverse derivative operator.
• Using operational rules: $ H_n(x,y) = \exp(y \partial_x^2) \cdot x^n $ and exponential generating function $ \sum_{n=0}^\infty \frac{t^n}{n!} H_n(x,y) = \exp(xt + yt^2) $.
• Expressing central trinomial coefficients as $ c_n = \Pi_n(1,1) $, and Motzkin numbers as $ m_n = \frac{1}{n+1} \binom{n+1}{1}_2 $, linking them to hybrid polynomial evaluations.
• Deriving generalized CTC via $ c_n^p = n! \sum_{k=0}^{[n/2]} \frac{1}{(n-2k)!k!(k+p)!} $, and establishing recurrence $ c_n^{p+1} = \frac{c_{n+2}^p - c_{n+1}^p}{2(n+1)} $.
• Introducing m-th order p-associated CTC as $ {}_m c_n^p = n! \sum_{k=0}^{[n/m]} \frac{1}{(n-mk)!k!(k+p)!} $, with recurrence $ {}_m c_{n+1}^p = {}_m c_n^p + m \frac{n!}{(n-m+1)!} {}_m c_{n-m+1}^p $.
• Connecting results to hypergeometric functions via $ H_n(x,y) = x^n \cdot {}_2F_0(-n/2, (1-n)/2; 4y/x) $, enabling combinatorial interpretation for specific parameter choices.

Experimental results

Research questions

• RQ1 How can Motzkin numbers and central trinomial coefficients be systematically unified under a single polynomial formalism?
• RQ2 What recurrence relations emerge when CTC and Motzkin numbers are expressed as evaluations of hybrid Hermite-Laguerre polynomials?
• RQ3 What are the algebraic and combinatorial properties of generalized p-associated and m-th order central trinomial coefficients?
• RQ4 How do specific parameter choices in Hermite polynomials yield known integer sequences with combinatorial interpretations?
• RQ5 Can the hybrid polynomial framework provide new insights into the combinatorial meaning of CTC and MN beyond standard interpretations?

Key findings

• The central trinomial coefficient $ c_n $ is exactly equal to $ \Pi_n(1,1) $, establishing a direct link between CTC and hybrid Hermite-Laguerre polynomials.
• The Motzkin number $ m_n $ is expressed as $ \frac{1}{n+1} \binom{n+1}{1}_2 $, and is identified as the p-associated CTC with $ p=1 $, i.e., $ c_n^1 $.
• A new recurrence $ c_n^{p+1} = \frac{c_{n+2}^p - c_{n+1}^p}{2(n+1)} $ is derived for p-associated CTC, valid for $ p \geq 0 $, with $ c_n^p $ not necessarily integer for $ p > 1 $.
• The m-th order p-associated CTC $ {}_m c_n^p $ satisfies the recurrence $ {}_m c_{n+1}^p = {}_m c_n^p + m \frac{n!}{(n-m+1)!} {}_m c_{n-m+1}^p $, generalizing the standard CTC structure.
• For $ x=1, y=1/2 $, the Hermite polynomial $ H_n(1,1/2) $ generates the involution numbers (A000085), counting set partitions into singletons and pairs.
• For $ x=y=1/2 $, $ 2^n H_n(1/2,1/2) $ produces sequence A115329, counting colored pair partitions, demonstrating how parameter choices yield new combinatorial interpretations.