# A de Bruijn identity for discrete random variables

## TL;DR

This paper introduces a new operation called beamsplitter addition for discrete non-negative integer variables, providing a classical proof of a heat equation and de Bruijn identity related to geometric distributions.

## Contribution

It defines a novel scaled convolution operation for discrete variables and derives a classical proof of related heat and de Bruijn identities.

## Key findings

- Beamsplitter addition acts as a scaled convolution for discrete variables.
- Provides a classical proof of the heat equation for geometric distributions.
- Establishes a de Bruijn identity for discrete random variables.

## Abstract

We discuss properties of the "beamsplitter addition" operation, which provides a non-standard scaled convolution of random variables supported on the non-negative integers. We give a simple expression for the action of beamsplitter addition using generating functions. We use this to give a self-contained and purely classical proof of a heat equation and de Bruijn identity, satisfied when one of the variables is geometric.