Application of Cauchy's equation in combinatorics and genetics

Abstract

We are familiar with the combinatorial formula $\left(\begin{array}{cc} n\\ r \end{array}\right) = \frac{n(n-1) \cdots (n - r + 1)}{r !} = $ number of possible ways of choosing $r$ objects out of $n$ objects\,.

In section 1 of this paper we obtain $\left( \begin{array}{cc} n\\ 2\end{array}\right)$ and $\left( \begin{array}{cc} n\\ 3 \end{array}\right)$ by using a functional equation, {\it the additive Cauchy equation}.

In genetics it is important to know the combinatorial function $g_{r}(n)=$ the number of possible ways of picking $r$ objects at a time from $n$ objects {\it allowing repetitions}, since this function describes the number of possibilities from a gene pool. Again we determine $g_2(n)$ and $g_3(n)$ with the help of the additive Cauchy equation in section 2.

Functional equations are used increasingly in diverse fields. The method of finding $\left( \begin{array}{cc} n\\ 2 \end{array}\right), \left( \begin{array}{cc} n\\ 3 \end{array}\right), g_2 (n)$ and $g_3(n)$ (see Snow [6]) is similar to that of finding the well known sum of powers of integers $S_K(n) = 1^K + 2^K + \cdots + n^K$ (Acz\'{e}l [2], Snow [5]).\\