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

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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’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 languageEnglish (US)
Pages (from-to)653-669
Number of pages17
JournalJournal of the American Mathematical Society
Volume3
Issue number3
DOIs
StatePublished - Jul 1990

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

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