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

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- by George E. Andrews
- J. Amer. Math. Soc.
**3**(1990), 653-669 - DOI: https://doi.org/10.1090/S0894-0347-1990-1040390-4
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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.## References

- George E. Andrews,
*Sieves in the theory of partitions*, Amer. J. Math.**94**(1972), 1214–1230. MR**319883**, DOI 10.2307/2373571 - George E. Andrews,
*The theory of partitions*, Encyclopedia of Mathematics and its Applications, Vol. 2, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. MR**0557013** - George E. Andrews,
*The hard-hexagon model and Rogers-Ramanujan type identities*, Proc. Nat. Acad. Sci. U.S.A.**78**(1981), no. 9, 5290–5292. MR**629656**, DOI 10.1073/pnas.78.9.5290 - George E. Andrews,
*Use and extension of Frobenius’ representation of partitions*, Enumeration and design (Waterloo, Ont., 1982) Academic Press, Toronto, ON, 1984, pp. 51–65. MR**782308** - George E. Andrews and R. J. Baxter,
*Lattice gas generalization of the hard hexagon model. III. $q$-trinomial coefficients*, J. Statist. Phys.**47**(1987), no. 3-4, 297–330. MR**894396**, DOI 10.1007/BF01007513 - David V. Chudnovsky and Richard D. Jenks (eds.),
*Computer algebra*, Lecture Notes in Pure and Applied Mathematics, vol. 113, Marcel Dekker, Inc., New York, 1989. Papers from the International Conference on Computer Algebra as a Tool for Research in Mathematics and Physics held at New York University, New York, April 5–6, 1984. MR**1002975** - George E. Andrews, R. J. Baxter, D. M. Bressoud, W. H. Burge, P. J. Forrester, and G. Viennot,
*Partitions with prescribed hook differences*, European J. Combin.**8**(1987), no. 4, 341–350. MR**930170**, DOI 10.1016/S0195-6698(87)80041-0 - George E. Andrews, R. J. Baxter, and P. J. Forrester,
*Eight-vertex SOS model and generalized Rogers-Ramanujan-type identities*, J. Statist. Phys.**35**(1984), no. 3-4, 193–266. MR**748075**, DOI 10.1007/BF01014383 - David M. Bressoud,
*Extension of the partition sieve*, J. Number Theory**12**(1980), no. 1, 87–100. MR**566873**, DOI 10.1016/0022-314X(80)90077-3 - William H. Burge,
*A correspondence between partitions related to generalizations of the Rogers-Ramanujan identities*, Discrete Math.**34**(1981), no. 1, 9–15. MR**605226**, DOI 10.1016/0012-365X(81)90017-0 - Louis Comtet,
*Advanced combinatorics*, Revised and enlarged edition, D. Reidel Publishing Co., Dordrecht, 1974. The art of finite and infinite expansions. MR**0460128**, DOI 10.1007/978-94-010-2196-8
L. Euler, - L. J. Slater,
*Further identities of the Rogers-Ramanujan type*, Proc. London Math. Soc. (2)**54**(1952), 147–167. MR**49225**, DOI 10.1112/plms/s2-54.2.147
M. Spiegel,

*Observations analyticae*, Novi Commentarii Academiae Scientarum Petropolitanae

**11**(1765), 124-143; also in

*Opera Omnia*, Series 1, vol. 15, Teubner, pp. 50-69. I. Schur,

*Ein Beitrag zur additiven Zahlentheorie und zur theorie der Kettenbrüche*, S.-B. Preuss. Akad. Wiss. Phys.-Math. Kl., 1917, pp. 302-321; reprinted in

*Gesammelte Abhandlungen*, vol. 2, Springer, Berlin, 1973, pp. 117-136.

*Finite differences and difference equations*, McGraw-Hill, New York, 1971. G. N. Watson,

*Proof of certain identities in combinatory analysis*, J. Indian Math. Soc.

**20**(1934), 57-69.

## Bibliographic Information

- © Copyright 1990 American Mathematical Society
- 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