Linear recurrence sequences and their convolutions via Bell polynomials

Daniel Birmajer; Juan B. Gil; Michael D. Weiner

Linear recurrence sequences and their convolutions via Bell polynomials

Combinatorics 2014-12-17 v2 Number Theory

Authors: Daniel Birmajer , Juan B. Gil , Michael D. Weiner

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.

Keywords

integer sequences and recurrences fibonacci numbers special functions and series expansions

Cite

@article{arxiv.1405.7727,
  title  = {Linear recurrence sequences and their convolutions via Bell polynomials},
  author = {Daniel Birmajer and Juan B. Gil and Michael D. Weiner},
  journal= {arXiv preprint arXiv:1405.7727},
  year   = {2014}
}

Comments

12 pages. This is a substantially revised version that emphasizes the aspect of convolutions and incorporates new results in Section 3

Linear recurrence sequences and their convolutions via Bell polynomials

Abstract

Keywords

Cite

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