Closed-form formula for some recursively-defined integro-difference sequence of functions

TL;DR

This paper derives a closed-form solution for a recursively defined sequence of integro-difference functions starting from a characteristic function, providing explicit formulas for each sequence element.

Contribution

The paper introduces a novel closed-form formula for a specific sequence of integro-difference functions defined recursively, expanding understanding of such sequences.

Findings

Explicit closed-form solution for the sequence derived

Sequence starts from characteristic function of an interval

Provides analytical expressions for all sequence elements

Abstract

The main purpose of this paper is to derive the closed form solution the sequence $(g_n)_{n\in \mathbb{N}}$ of integro-difference equations that is defined recursively as follows: \begin{align*} g_1(x) & = \chi_{(-1/2, 1/2)} (x), g_{n+1}(x) & = g_n(x + 1/2)- g_n(x- 1/2) + \int_{x-\frac{1}{2}}^{x + \frac{1}{2}} g_n(s)ds, \, n\in \mathbb{N}, \end{align*} where $ g_1(x)= \chi_{(-1/2, 1/2)} (x) $ is the characteristic function of the unit interval $(-1/2, 1/2) $ has value equal to $ 1 $ on $(-1/2, 1/2) $ and $0$ elsewhere in $ \mathbb{R} $.