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.
- George E. Andrews, Sieves in the theory of partitions, Amer. J. Math. 94 (1972), 1214–1230. MR 319883, DOI https://doi.org/10.2307/2373571
- George E. Andrews, The theory of partitions, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. Encyclopedia of Mathematics and its Applications, Vol. 2. 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 https://doi.org/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 https://doi.org/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 https://doi.org/10.1016/S0195-6698%2887%2980041-0
- 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 https://doi.org/10.1007/BF01014383
- David M. Bressoud, Extension of the partition sieve, J. Number Theory 12 (1980), no. 1, 87–100. MR 566873, DOI https://doi.org/10.1016/0022-314X%2880%2990077-3
- 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 https://doi.org/10.1016/0012-365X%2881%2990017-0
- Louis Comtet, Advanced combinatorics, Revised and enlarged edition, D. Reidel Publishing Co., Dordrecht, 1974. The art of finite and infinite expansions. MR 0460128 L. Euler, 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.
- L. J. Slater, Further identities of the Rogers-Ramanujan type, Proc. London Math. Soc. (2) 54 (1952), 147–167. MR 49225, DOI https://doi.org/10.1112/plms/s2-54.2.147 M. Spiegel, 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.
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© Copyright 1990
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