Abstract
This paper takes up again the study of the Jacobi triple and Watson quintuple identities that have been derived combinatorially in several manners in the classical literature. It also contains a proof of the recent Farkas-Kra septuple product identity that makes use only of “smanipulatorics” methods.
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References
B. Adiga, B. C. Berndt, S. Bhargava and G. N. Watson, Ramanujan’s second notebook: Theta-functions and q-series Chap. 16. Amer. Math. Soc., Prodidence 1985 (Amer. Math. Soc. Memoirs, vol. 315).
K. Alladi, The quintuple product identity and shifted partition functions, J. Comput. Appl. Math., vol. 68, 1996, p. 3–13.
G. E. Andrews, A simple proof of Jacobi’s triple product identity, Proc. Amer. Math. Soc., vol. 16, 1965, p. 333–334.
G. E. Andrews, Applications of basic hypergeometric functions, SIAM Rev., vol. 16, 1974, p. 441–484.
G. E. Andrews, The Theory of Partitions. Addison-Wesley, Reading, 1976 (Encyclopedia of Math. and Its Appl., vol. 2).
G. E. Andrews, Generalized Frobenius partitions. Amer. Math. Soc., Providence, 1984 (Amer. Math. Soc. Memoirs, vol. 49), pp. 4–5.
G. E. Andrews, Private communication, 1998.
A. O. L. Atkin and P. Swinnerton-Dyer, Some properties of partitions, Proc. London. Math. Soc., vol. 4, 1954, p. 84–106.
W. N. Bailey, On the simplification of some identities of the Rogers-Ramanujan type, Proc. London Math. Soc., vol. 1, 1951, p. 217–221.
D. Bressoud, Proofs and Confirmations. The story of the alternating sign matrix conjecture. Preprint, Macalester College, Saint Paul, 1997.
L. Carlitz and M. V. Subbarao, A simple proof of the quintuple product identity, Proc. Amer. Math. Soc., vol. 32, 1972, p. 42–44.
M. S. Cheema, Vector partitions and combinatorial identities, Math. Camp., vol. 18, 1964, p. 414–420.
J. A. Ewell, An easy proof of the triple-product identity, Amer. Math. Monthly, vol. 88, 1981, p. 270–272.
H. M. Farkas and I. Kra, On the Quintuple Product Identity, Proc. Amer. Math. Soc., vol. 27, 1999, p. 771–778.
D. Foata and G.-N. Han, Jacobi and Watson identities combinatorially revisited. See home page: http://cartan.u-strasbg.fr/~foata/paper/pub81.html http://cartan.u-strasbg.fr/~guoniu/papers/jacobi.html.
F. G. Garvan, Generalizations of Dyson’s Rank. Ph. D Thesis, Pennsylvania State University, May 1986, 128 p.
F. G. Garvan, Private communication, April 1999.
G. Gasper and M. Rahman, Basic Hypergeometric Series. Cambridge Univ. Press, Cambridge and New York, 1990 (Encyclopedia of Math. and its Appl., vol. 35), p. 134.
B. Gordon, Some identities in combinatorial analysis, Quart. J. Math. (Oxford), vol. 12, 1961, p. 285–290.
H. Gupta, Selected Topics in Number Theory. Abacus Press, Tunbridge Wells, 1980, pp. 213–215.
R. A. Gustafson, Multilateral summation theorems for ordinary and basic hypergeometric series in U(n), SIAM J. Math. Anal., vol. 18, 1987, p. 1576–1596.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. Clarendon Press, Oxford, 1938. Reprinted with corrections, 1985.
M. D. Hirschhorn, A generalization of the quintuple product identity, J. Austral. Math. Soc., vol. A 44, 1988, p. 42–45.
T. T. Joichi and D. Stanton, An involution for Jacobi’s identity, Discrete Math., vol. 73, 1989, p. 261–272.
V. G. Kac, Infinite-dimensional algebras, Dedekind’s η-function, classical Möbius function and the very strange formula, Advances in Math., vol. 30, 1978, p. 85–136.
V. G. Kac, Infinite Dimensional Lie Algebras 2nd. Edition. Cambridge Univ. Press, Cambridge 1985.
J. Lepowsky and S. Milne, Lie algebraic approaches to classical partition identities, Advances in Math., vol. 29, 1978, p. 15–59.
R. P. Lewis, A combinatorial proof of the triple product identity, Amer. Math. Monthly, vol. 91, 1984, p. 420–423.
(Major) P. A. MacMahon, Combinatory Analysis vol. I, II. Cambridge Univ. Press, 1915. Reprinted by Chelsea, New York, 1960.
I. G. Macdonald, Some conjectures for root systems, SIAM J. Math. Anal., vol. 13, 1982, p. 988–1007.
P. K. Menon, On Ramanujan’s continued fraction and related identities, J. London Math. Soc., vol. 40, 1965, p. 49–54.
S. Milne, An elementary proof of the Macdonald identities for A l (1), Adv. in Math., vol. 57, 1985, p. 34–70.
D. B. Sears, Two identities of Bailey, J. London Math. Soc., vol. 27, 1952, p. 510–511.
M. V. Subbarao and M. Vidyasagar, On Watson’s quintuple product identity, Proc. Amer. Math. Soc., vol. 26, 1970, p. 23–27.
C. Sudler, Two enumerative proofs of an identity of Jacobi, Proc. Edinburgh Math. Soc., vol. 15, 1966, p. 67–71.
J. J. Sylvester, A constructive theory of partitions in three acts, an interact and an exodion, Amer. J. Math., vol. 5, 1882, p. 251–330 (or pp. 1–83 of the ColI. Math. Papers of J.J. Sylvester, vol. 4, Cambridge Univ. Press, London and New York, 1912; reprinted by Chelsea, New York, 1974).
G. N. Watson, Theorems stated by Ramanujan. VII: Theorems on continued fractions, J. London Math. Soc., vol. 4, 1929, p. 39–48.
E. M. Wright, An enumerative proof of an identity of Jacobi, J. London Math. Soc., vol. 40, 1965, p. 55–57.
J. Zolnowsky, A direct combinatorial proof of the Jacobi identity, Discrete Math., vol. 9, 1974, p. 293–298.
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Dedicated to George Andrews on the occasion of his sixtieth birthday
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Foata, D., Han, GN. (2001). The Triple, Quintuple and Septuple Product Identities Revisited. In: Foata, D., Han, GN. (eds) The Andrews Festschrift. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-56513-7_15
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DOI: https://doi.org/10.1007/978-3-642-56513-7_15
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