[Paper Review] Linear recurrence sequences and their convolutions via Bell polynomials

Daniel Birmajer, Juan B. Gil+1 more|arXiv (Cornell University)|May 29, 2014

Advanced Combinatorial Mathematics2 references3 citations

TL;DR

This paper introduces a novel framework for analyzing linear recurrence sequences and their multifold convolutions using partial Bell polynomials. By expressing sequences via the INVERT transform of recurrence coefficients, it derives universal recurrence relations for convolved sequences that preserve the original recurrence depth, enabling systematic derivation of combinatorial identities and generalizations of Girard-Waring formulas.

ABSTRACT

We recast homogeneous linear recurrence sequences with fixed coefficients in terms of partial Bell polynomials, and use their properties to obtain various combinatorial identities and multifold convolution formulas. Our approach relies on a basis of sequences that can be obtained as the INVERT transform of the coefficients of the given recurrence relation. For such a basis sequence $(y_n)$ with generating function $Y(t)$, and for any positive integer $r$, we give a formula for the convolved sequence generated by $Y(t)^r$ and prove that it satisfies an elegant recurrence relation.

Motivation & Objective

• To unify the representation of linear recurrence sequences with fixed coefficients using partial Bell polynomials.
• To develop a systematic method for computing multifold convolutions of such sequences via generating functions and Bell polynomial identities.
• To derive a universal recurrence relation of the same depth as the original sequence for any r-fold convolution.
• To provide an alternative derivation of the Girard-Waring formulas for power sum symmetric functions using this framework.

Proposed method

• Representing linear recurrence sequences via partial Bell polynomials applied to the INVERT transform of recurrence coefficients.
• Defining a basis of sequences using the generating function $ Y(t) = 1/Q(t) $, where $ Q(t) = 1 - \sum c_n t^n $, and expressing any sequence as a linear combination of shifted basis sequences.
• Using Faà di Bruno’s formula to express the basis sequence $ y_n $ in terms of partial Bell polynomials: $ y_n = \sum_{k=0}^n \frac{k!}{n!} B_{n,k}(1!c_1, 2!c_2, \dots) $.
• Deriving a universal recurrence for the r-fold convolution sequence $ a_n^{(r)} $, showing it satisfies a recurrence of the same depth as the original sequence.
• Applying properties of Bell polynomials to derive closed-form expressions for convolved sequences and their generating functions.
• Validating the framework through explicit examples, including Fibonacci, Padovan, and Tribonacci sequences and their convolutions.

Experimental results

Research questions

• RQ1 Can linear recurrence sequences with fixed coefficients be systematically represented using partial Bell polynomials and the INVERT transform of their coefficients?
• RQ2 What universal recurrence relation governs the r-fold convolution of a linear recurrence sequence, and does it preserve the depth of the original recurrence?
• RQ3 How can the Girard-Waring formulas for power sum symmetric functions be re-derived using this Bell polynomial framework?
• RQ4 What combinatorial identities emerge from expressing convolved sequences via Bell polynomial expansions?
• RQ5 Can the method be applied to sequences like Fibonacci, Padovan, and Tribonacci to derive new recurrence relations for their convolutions?

Key findings

• The r-fold convolution of a linear recurrence sequence satisfies a recurrence of the same depth as the original sequence, derived via Bell polynomial identities.
• For any sequence $ (a_n) $ satisfying a linear recurrence of order $ d $, its r-fold convolution $ a_n^{(r)} $ satisfies a recurrence of the form $ n a_n^{(r)} = \sum_{j=1}^d (n + j(r-1)) c_j a_{n-j}^{(r)} $, where $ c_j $ are the recurrence coefficients.
• The generating function of the r-fold convolution is $ Y(t)^r $, and its coefficients are expressed via Bell polynomials: $ y_n^{(r)} = \sum_{k=1}^n \binom{k+r-1}{k} \frac{k!}{n!} B_{n,k}(1!c_1, 2!c_2, \dots) $.
• The framework provides a unified derivation of the Girard-Waring formulas for power sum symmetric functions by expressing elementary symmetric functions in terms of Bell polynomials.
• Explicit recurrence relations are derived for convolutions of Fibonacci, Padovan, and Tribonacci sequences, matching known OEIS sequences such as A001629, A001628, A001872, and A073778.
• The method enables the derivation of new combinatorial identities by isolating coefficients in Bell polynomial expansions, particularly for sequences with sparse or structured coefficients.