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

George E. Andrews

doi:10.1090/S0894-0347-1990-1040390-4

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

George E. Andrews

Mathematics

Research output: Contribution to journal › Article › peer-review

57 Scopus citations

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	https://doi.org/10.1090/S0894-0347-1990-1040390-4
State	Published - Jul 1990

All Science Journal Classification (ASJC) codes

General Mathematics
Applied Mathematics

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10.1090/S0894-0347-1990-1040390-4

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abstract = "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{\textquoteright}s assertion for m ≤ 7. Our main object is to analyze q-analogs extending Euler{\textquoteright}s observation. Among other things we are led to finite versions of dissections of the Rogers-Ramanujan identities into even and odd parts.",

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

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