# Limits of Sums for Binomial and Eulerian Numbers and their Associated Distributions

## TL;DR

This paper uses renewal theory to derive new limit results for sums of normalized binomial and Eulerian numbers and explores their related distributions, including binomial and Irwin-Hall distributions.

## Contribution

It introduces a unified probabilistic approach to establish novel limits for sums of normalized binomial and Eulerian numbers and analyzes their associated distributions.

## Key findings

- New limit theorems for normalized binomial sums
- Limit results for Eulerian number sums
- Analysis of associated binomial and Irwin-Hall distributions

## Abstract

We provide a unified, probabilistic approach using renewal theory to derive some novel limits of sums for the normalized binomial coefficients and for the normalized Eulerian numbers. We also investigate some corresponding results for their associated distributions -- the binomial distributions for the binomial coefficients and the Irwin-Hall distributions (uniform B-splines) for the Eulerian numbers.