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.
All Science Journal Classification (ASJC) codes
- Applied Mathematics