Jacobi's Triple Product Identity and the Quintuple Product Identity

Wenchang Chu

Bollettino dell'Unione Matematica Italiana (2007)

  • Volume: 10-B, Issue: 3, page 867-874
  • ISSN: 0392-4033

Abstract

top
The simplest proof of Jacobi's triple product identity originally due to Cauchy (1843) and Gauss (1866) is reviewed. In the same spirit, we prove by means of induction principle and finite difference method, a finite form of the quintuple product identity. Similarly, the induction principle will be used to give a new proof of another algebraic identity due to Guo and Zeng (2005), which can be considered as another finite form of the quintuple product identity.

How to cite

top

Chu, Wenchang. "Jacobi's Triple Product Identity and the Quintuple Product Identity." Bollettino dell'Unione Matematica Italiana 10-B.3 (2007): 867-874. <http://eudml.org/doc/290428>.

@article{Chu2007,
abstract = {The simplest proof of Jacobi's triple product identity originally due to Cauchy (1843) and Gauss (1866) is reviewed. In the same spirit, we prove by means of induction principle and finite difference method, a finite form of the quintuple product identity. Similarly, the induction principle will be used to give a new proof of another algebraic identity due to Guo and Zeng (2005), which can be considered as another finite form of the quintuple product identity.},
author = {Chu, Wenchang},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {10},
number = {3},
pages = {867-874},
publisher = {Unione Matematica Italiana},
title = {Jacobi's Triple Product Identity and the Quintuple Product Identity},
url = {http://eudml.org/doc/290428},
volume = {10-B},
year = {2007},
}

TY - JOUR
AU - Chu, Wenchang
TI - Jacobi's Triple Product Identity and the Quintuple Product Identity
JO - Bollettino dell'Unione Matematica Italiana
DA - 2007/10//
PB - Unione Matematica Italiana
VL - 10-B
IS - 3
SP - 867
EP - 874
AB - The simplest proof of Jacobi's triple product identity originally due to Cauchy (1843) and Gauss (1866) is reviewed. In the same spirit, we prove by means of induction principle and finite difference method, a finite form of the quintuple product identity. Similarly, the induction principle will be used to give a new proof of another algebraic identity due to Guo and Zeng (2005), which can be considered as another finite form of the quintuple product identity.
LA - eng
UR - http://eudml.org/doc/290428
ER -

References

top
  1. ALLADI, K., The quintuple product identity and shifted partition functions, J. Comput. Appl. Math., 68 (1996), 3-13. Zbl0867.11071MR1418747DOI10.1016/0377-0427(95)00251-0
  2. ANDREWS, G. E., A simple proof of Jacobi's triple product identity, Proc. Amer. Math., 16 (1965), 333-334. Zbl0132.30901MR171725DOI10.2307/2033875
  3. ANDREWS, G. E., Applications of basic hypergeometric functions, SIAM Review, 16 (1974), 441-484. Zbl0299.33004MR352557DOI10.1137/1016081
  4. ANDREWS, G. E. - ASKEY, R. - ROY, R., Special Functions, Cambridge University Press, Cambridge, 2000. MR1688958DOI10.1017/CBO9781107325937
  5. ATKIN, A. O. L. - SWINNERTON-DYER, P., Some properties of partitions, Proc. London Math. Soc., 4 (1954), 84-106. Zbl0055.03805MR60535DOI10.1112/plms/s3-4.1.84
  6. BAILEY, W. N., Series of hypergeometric type which are infinite in both directions, Quart. J. Math. (Oxford) 7 (1936), 105-115. Zbl62.0410.05
  7. BAILEY, W. N., On the simplification of some identities of the Rogers-Ramanujan type, Proc. London Math. Soc., 1 (1951), 217-221. Zbl0043.06103MR43839DOI10.1112/plms/s3-1.1.217
  8. CARLITZ, L. - SUBBARAO, M. V., A simple proof of the quintuple product identity, Proc. Amer. Math. Society, 32, 1 (1972), 42-44. MR289316DOI10.2307/2038301
  9. CHEN, W. Y. C. - CHU, W. - GU, N. S. S., Finite form of the quintuple product identity, Journal of Combinatorial Theory (Series A), 113, 1 (2006), 185-187. Zbl1145.11036MR2192776DOI10.1016/j.jcta.2005.04.002
  10. CHU, W., Durfee rectangles and the Jacobi triple product identity, Acta Math. Sinica , 9, 1 (1993), 24-26. Zbl0782.05008MR1235637DOI10.1007/BF02559979
  11. CHU, W., Abel's Lemma on summation by parts and Ramanujan's ψ 1 1 -series Identity, Aequationes Mathematicae, 72, 1/2 (2006), 172-176. Zbl1116.33018MR2258814DOI10.1007/s00010-006-2830-1
  12. CHU, W., Abel's Method on summation by parts and Hypergeometric Series, Journal of Difference Equations and Applications, 12, 8 (2006). Zbl1098.33003MR2248785DOI10.1080/10236190600704096
  13. CHU, W., Abel's Lemma on summation by parts and Basic Hypergeometric Series, Advances in Applied Mathematics, 39, 4 (2007), 490-514. Zbl1131.33008MR2356433DOI10.1016/j.aam.2007.02.001
  14. CHU, W. - JIA, C. Z., Abel's Method on summation by parts and Terminating Well- Poised q-Series Identities, Journal of Computational and Applied Mathematics, 207, 2 (2007), 360-370. Zbl1123.33013MR2345255DOI10.1016/j.cam.2006.10.011
  15. COPPER, S., The quintuple product identity, International J. of Number Theory, 2, 1 (2006), 115-161. Zbl1159.33300MR2217798DOI10.1142/S1793042106000401
  16. EVANS, R. J., Theta function identities, J. Math. Anal. Appl., 147, 1 (1990), 97-121. Zbl0707.11033MR1044689DOI10.1016/0022-247X(90)90387-U
  17. EWELL, J. A., An easy proof of the triple product identity, Amer. Math. Month., 88 (1981), 270-272. Zbl0471.40001MR610489DOI10.2307/2320552
  18. FARKAS, H. M. - KRA, IRWIN, On the quintuple product identity, Proc. Amer. Math. Soc., 127, 3 (1999), 771-778. Zbl0932.11029MR1487364DOI10.1090/S0002-9939-99-04791-7
  19. GASPER, G. - RAHMAN, M., Basic Hypergeometric Series (2nd edition), Cambridge University Press, 2004. Zbl1129.33005MR2128719DOI10.1017/CBO9780511526251
  20. GORDON, B., Some identities in combinatorial analysis, Quart. J. Math. Oxford, 12 (1961), 285-290. Zbl0107.25101MR136551DOI10.1093/qmath/12.1.285
  21. HIRSCHHORN, M. D.A generalization of the quintuple product identity, J. Austral. Math. Soc., A44 (1988), 42-45. Zbl0656.05008MR914402
  22. LEWIS, R. P., A combinatorial proof of the triple product identity, Amer. Math. Month., 91 (1984), 420-423. Zbl0551.05016MR759217DOI10.2307/2322993
  23. MORDELL, L. J., An identity in combinatorial analysis, Proc. Glasgow Math. Ass., 5 (1961), 197-200. Zbl0107.25102MR138900
  24. PAULE, P., Short and easy computer proofs of the Rogers-Ramanujan identities and of identities of similar type, Electronic J. of Combinatorics, 1 (1994), R#10. Zbl0814.05009MR1293400
  25. SEARS, D. B., Two identities of Bailey, J. London Math. Soc., 27 (1952), 510-511. MR50067DOI10.1112/jlms/s1-27.4.510
  26. SUBBARAO, M. V. - VIDYASAGAR, M., On Watson's quintuple product identity, Proc. Amer. Math. Society, 26, 1 (1970), 23-27. Zbl0203.30502MR263770DOI10.2307/2036795
  27. WATSON, G. N., Theorems stated by Ramanujan VII: Theorems on continued fractions, J. London Math. Soc., 4 (1929), 39-48. Zbl55.0273.01MR1574903DOI10.1112/jlms/s1-4.1.39
  28. WATSON, G. N., Ramanujan's Vertumung über Zerfallungsanzahlen, J. Reine Angrew. Math., 179 (1938), 97-128. MR1581588DOI10.1515/crll.1938.179.97
  29. WRIGHT, E. M., An enumerative proof of an identity of Jacobi, J. of London Math. Soc., 40 (1965), 55-57. Zbl0125.02503MR169826DOI10.1112/jlms/s1-40.1.55
  30. GUO, V. J. W. - ZENG, J., Short proofs of summation and transformation formulas for basic hypergeometric series, Journal of Mathematical Analysis and Applications, 327, 1 (2007), 310-325. Zbl1106.33017MR2277415DOI10.1016/j.jmaa.2006.04.040

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.