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
HTML articles powered by AMS MathViewer

by George E. Andrews

J. Amer. Math. Soc. 3 (1990), 653-669

DOI: https://doi.org/10.1090/S0894-0347-1990-1040390-4

PDF | Request permission

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

Observations analyticae

11

Opera Omnia

Ein Beitrag zur additiven Zahlentheorie und zur theorie der Kettenbrüche

Gesammelte Abhandlungen

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

Finite differences and difference equations

Proof of certain identities in combinatory analysis

20