Representation of numbers by cascades

Abstract:

A cascade $C$ is defined as a sum of binomial coefficients \[ C = \left ( {\begin {array}{*{20}{c}} {{a_h}} \\ h \\ \end {array} } \right ) + \left ( {\begin {array}{*{20}{c}} {{a_{h - 1}}} \\ {h - 1} \\ \end {array} } \right ) + \cdots + \left ( {\begin {array}{*{20}{c}} {{a_t}} \\ t \\ \end {array} } \right )\] where ${a_h} > {a_{h - 1}} > \cdots > {a_t}$. In this expression, we assume that $(_h^a) = 0$ whenever $a < h$. Given a cascade $C$ and a sequence $\varepsilon = \langle {\varepsilon _h},{\varepsilon _{h - 1}}, \ldots ,{\varepsilon _t}\rangle$ of signs (i.e. ${\varepsilon _i} = + 1\;{\text {or}}\; - 1$ for each $i$), we define \[ \varepsilon C = {\varepsilon _h}\left ( {\begin {array}{*{20}{c}} {{a_h}} \\ h \\ \end {array} } \right ) + \cdots + {\varepsilon _t}\left ( {\begin {array}{*{20}{c}} {{a_t}} \\ t \\ \end {array} } \right ).\] Also, we put \[ \alpha C = \left ( {\begin {array}{*{20}{c}} {{a_h}} \\ {h + 1} \\ \end {array} } \right ) + \left ( {\begin {array}{*{20}{c}} {{a_{h - 1}}} \\ h \\ \end {array} } \right ) + \cdots + \left ( {\begin {array}{*{20}{c}} {{a_t}} \\ {t + 1} \\ \end {array} } \right ).\] In the paper, we prove that for any sequence $\langle {n_0}, {n_1}, \ldots ,{n_s}\rangle$ of integers, there exist a cascade $C$ and a corresponding sequence $\varepsilon$ of signs such that ${n_i} = \varepsilon {\alpha ^i}C$ for $i = 0,\;1, \ldots ,s$ where ${\alpha ^0}C = C,\;{\alpha ^1}C = \alpha C,\;{\alpha ^2}C = \alpha ({\alpha ^1}C)$, and recursively, ${\alpha ^n}C = \alpha ({\alpha ^{n - 1}}C)$.