### Abstract

The trinomial coefficients are defined centrally by (Equation presented). Euler observed that for −1 ≤ m ≤ 7, (Equation presented), where F_{m} is the mth Fibonacci number. The assertion is false for m > 7. We prove general identities—one of which reduces to Euler’s assertion for m ≤ 7. Our main object is to analyze q-analogs extending Euler’s observation. Among other things we are led to finite versions of dissections of the Rogers-Ramanujan identities into even and odd parts.

Original language | English (US) |
---|---|

Pages (from-to) | 653-669 |

Number of pages | 17 |

Journal | Journal of the American Mathematical Society |

Volume | 3 |

Issue number | 3 |

DOIs | |

State | Published - Jan 1 1990 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Cite this

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**Euler’s “exemplum memorabile inductionis Fallacis” and q-trinomial coefficients.** / Andrews, George E.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Euler’s “exemplum memorabile inductionis Fallacis” and q-trinomial coefficients

AU - Andrews, George E.

PY - 1990/1/1

Y1 - 1990/1/1

N2 - The trinomial coefficients are defined centrally by (Equation presented). Euler observed that for −1 ≤ m ≤ 7, (Equation presented), where Fm is the mth Fibonacci number. The assertion is false for m > 7. We prove general identities—one of which reduces to Euler’s assertion for m ≤ 7. Our main object is to analyze q-analogs extending Euler’s observation. Among other things we are led to finite versions of dissections of the Rogers-Ramanujan identities into even and odd parts.

AB - The trinomial coefficients are defined centrally by (Equation presented). Euler observed that for −1 ≤ m ≤ 7, (Equation presented), where Fm is the mth Fibonacci number. The assertion is false for m > 7. We prove general identities—one of which reduces to Euler’s assertion for m ≤ 7. Our main object is to analyze q-analogs extending Euler’s observation. Among other things we are led to finite versions of dissections of the Rogers-Ramanujan identities into even and odd parts.

UR - http://www.scopus.com/inward/record.url?scp=84968521872&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84968521872&partnerID=8YFLogxK

U2 - 10.1090/S0894-0347-1990-1040390-4

DO - 10.1090/S0894-0347-1990-1040390-4

M3 - Article

VL - 3

SP - 653

EP - 669

JO - Journal of the American Mathematical Society

JF - Journal of the American Mathematical Society

SN - 0894-0347

IS - 3

ER -