Linear recurrence sequences and their convolutions via Bell polynomials

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.