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

Andrews, George E.

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

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

Author: George E. Andrews
Journal: J. Amer. Math. Soc. 3 (1990), 653-669
MSC: Primary 05A10; Secondary 05A30, 11B65
DOI: https://doi.org/10.1090/S0894-0347-1990-1040390-4
MathSciNet review: 1040390
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Abstract: The trinomial coefficients are defined centrally by $\Sigma _{j = - m}^\infty {(_j^m)_2}{x^j} = {(1 + x + {x^{ - 1}})^m}$. Euler observed that for $- 1 \leq m \leq 7$, $3{(_{ \;0}^{m + 1})_2} - {(_{ \;0}^{m + 2})_2} = {F_m}({F_m} + 1)$, where ${F_m}$ is the $m$th Fibonacci number. The assertion is false for $m > 7$. We prove general identities—one of which reduces to Euler’s assertion for $m \leq 7$. 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.

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