Quotients of sequences under the binomial convolution
Year: 2024
Type: article
Abstract: This paper gives expressions for the solution $\{a(n)\}$ of the equation \begin{equation*} \sum_{k=0}^n{n \choose k}a(k)b(n-k)=c(n), \ n=0,1,2,\ldots, \end{equation*} where $b(0)\ne 0$, that is, of the equation $a \circ b = c$ in $a$, where $\circ$ is the binomial convolution. These expressions are classified as recursive, explicit, determinant, exponential generating function and convolutional expressions. These expressions are compared with those under the usual Cauchy convolution. Several special cases and examples of combinatorial nature are also discussed.
Author: Pentti Haukkanen
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W4404066565