A generalization of Kawanaka's identity for Hall-Littlewood polynomials and applications
Masao Ishikawa1
, Frédéric Jouhet2
and Jiang Zeng2
1Faculty of Education Tottori University Department of Mathematics Tottori 680-8551 Japan Tottori 680-8551 Japan
2Université Claude Bernard (Lyon I) 43 Institut Camille Jordan bd du 11 Novembre 1918 69622 Villeurbanne Cedex France bd du 11 Novembre 1918 69622 Villeurbanne Cedex France
2Université Claude Bernard (Lyon I) 43 Institut Camille Jordan bd du 11 Novembre 1918 69622 Villeurbanne Cedex France bd du 11 Novembre 1918 69622 Villeurbanne Cedex France
DOI: 10.1007/s10801-006-8350-1
Abstract
An infinite summation formula of Hall-Littlewood polynomials due to Kawanaka is generalized to a finite summation formula, which implies, in particular, twelve more multiple q-identities of Rogers-Ramanujan type than those previously found by Stembridge and the last two authors.
Pages: 395–412
Keywords: keywords symmetric functions; Hall-Littlewood polynomials; $Q$-series; Rogers-Ramanujan type identities
Full Text: PDF
References
1. G.E. Andrews, The theory of partitions, Encyclopedia of mathematics and its applications, Addison- Wesley, Reading, Massachusetts, 2 (1976).
2. G.E. Andrews, Multiple series Rogers-Ramanujan type identities, Pacific J. Math., 114(2) (1984), 267- 283.
3. D.M. Bressoud, An analytic generalization of the Rogers-Ramanujan identities with interpretation, Quart. J. Math. Oxford Ser. (2), 31 (124) (1980), 385-399.
4. D.M. Bressoud, Analytic and combinatorial generalizations of the Rogers-Ramanujan identities, Mem. Amer. Math. Soc., 24 (227) (1980).
5. G. Gasper and M. Rahman, Basic Hypergeometric Series, 2nd edn., Encyclopedia of Mathematics and its Applications 96, Cambridge University Press, Cambridge, 2004.
6. F. Jouhet and J. Zeng, Some new identities for Schur functions, Adv. Appl. Math., 27 (2001), 493-509.
7. F. Jouhet and J. Zeng, New Identities of Hall-Littlewood Polynomials and Applications, The Ramanujan J. 10 (2005), 89-112.
8. N. Kawanaka, A q-series identity involving Schur functions and related topics, Osaka J. Math., 36 (1999), 157-176.
9. I.G. Macdonald, Symmetric functions and Hall polynomials, Clarendon Press, second edition, Oxford,
1995. Springer J Algebr Comb (2006) 23: 395-412
10. L.J. Slater, A new proof of Rogers's transformations of infinite series, Proc. London Math. Soc., 53(2) (1951), 460-475.
11. L.J. Slater, Further identities of the Rogers-Ramanujan Type, Proc. London Math. Soc., 54(2) (1951-52), 147-167.
12. J.R. Stembridge, Hall-Littlewood functions, plane partitions, and the Rogers-Ramanujan identities, Trans. Amer. Math. Soc., 319(2) (1990), 469-498.
13. J. Zeng, On the q-variations of Sylvester's bijection, The Ramanujan J. 9 (2005), 289-303.
2. G.E. Andrews, Multiple series Rogers-Ramanujan type identities, Pacific J. Math., 114(2) (1984), 267- 283.
3. D.M. Bressoud, An analytic generalization of the Rogers-Ramanujan identities with interpretation, Quart. J. Math. Oxford Ser. (2), 31 (124) (1980), 385-399.
4. D.M. Bressoud, Analytic and combinatorial generalizations of the Rogers-Ramanujan identities, Mem. Amer. Math. Soc., 24 (227) (1980).
5. G. Gasper and M. Rahman, Basic Hypergeometric Series, 2nd edn., Encyclopedia of Mathematics and its Applications 96, Cambridge University Press, Cambridge, 2004.
6. F. Jouhet and J. Zeng, Some new identities for Schur functions, Adv. Appl. Math., 27 (2001), 493-509.
7. F. Jouhet and J. Zeng, New Identities of Hall-Littlewood Polynomials and Applications, The Ramanujan J. 10 (2005), 89-112.
8. N. Kawanaka, A q-series identity involving Schur functions and related topics, Osaka J. Math., 36 (1999), 157-176.
9. I.G. Macdonald, Symmetric functions and Hall polynomials, Clarendon Press, second edition, Oxford,
1995. Springer J Algebr Comb (2006) 23: 395-412
10. L.J. Slater, A new proof of Rogers's transformations of infinite series, Proc. London Math. Soc., 53(2) (1951), 460-475.
11. L.J. Slater, Further identities of the Rogers-Ramanujan Type, Proc. London Math. Soc., 54(2) (1951-52), 147-167.
12. J.R. Stembridge, Hall-Littlewood functions, plane partitions, and the Rogers-Ramanujan identities, Trans. Amer. Math. Soc., 319(2) (1990), 469-498.
13. J. Zeng, On the q-variations of Sylvester's bijection, The Ramanujan J. 9 (2005), 289-303.