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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Representation of numbers by cascades
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by C. C. Chen and D. E. Daykin PDF
Proc. Amer. Math. Soc. 59 (1976), 394-398 Request permission

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, ${\alpha ^n}C = \alpha ({\alpha ^{n - 1}}C)$.
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Additional Information
  • © Copyright 1976 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 59 (1976), 394-398
  • MSC: Primary 05A17
  • DOI: https://doi.org/10.1090/S0002-9939-1976-0414385-X
  • MathSciNet review: 0414385