Abstract
Let \(H_n(t)\) denote the classical Rogers–Szegő polynomial, and let \(\tilde{H}_n(t_1, \ldots ,\) \( t_l)\) denote the homogeneous Rogers–Szegő polynomial in \(l\) variables, with indeterminate \(q\). There is a classical product formula for \(H_k(t)H_n(t)\) as a sum of Rogers–Szegő polynomials with coefficients being polynomials in \(q\). We generalize this to a product formula for the multivariate homogeneous polynomials \(\tilde{H}_n(t_1, \ldots , t_l)\). The coefficients given in the product formula are polynomials in \(q\) which are defined recursively, and we find closed formulas for several interesting cases. We then reinterpret the product formula in terms of symmetric function theory, where these coefficients become structure constants.
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References
Andrews, G.: The Theory of Partitions, Encyclopedia of Mathematics and its Applications, Addison-Wesley, Reading, Mass.-London-Amsterdam, 1976.
Bliem, T., Kousidis, S.: The number of flags in finite vector spaces: asymptotic normality and Mahonian statistics. J. Algebraic Combin. 37(2), 361–380 (2013)
Fine, N.J.: Basic Hypergeometric Series and Applications, Mathematical Surveys and Monographs. American Mathematical Society, Providence (1988)
Hikami, K.: Representations of motifs: new aspect of the Rogers–Szegő polynomials. J. Phys. Soc. Jpn. 64(4), 1047–1050 (1995)
Hikami, K.: Representation of the Yangian invariant motif and the Macdonald polynomial. J. Phys. A 30(7), 2447–2456 (1997)
Hou, Q.-H., Lascoux, A., Mu, Y.-P.: Continued fractions for Rogers–Szegő polynomials. Numer. Algorithms 35(1), 81–90 (2004)
Ismail, M.E.H.: Classical and Quantum Orthogonal Polynomials in One Variable, Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (2005)
Macdonald, I.G.: Symmetric Functions and Hall Polynomials, second edition, with contributions by A. Zelevinsky, Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995.
Macdonald, I.G.: Affine Hecke Algebras and Orthogonal Polynomials, Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (2003)
Rogers, L.J.: On a three-fold symmetry in the elements of Heine’s series. Proc. Lond. Math. Soc. 24, 171–179 (1893)
Rogers, L.J.: On the expansion of some infinite products. Proc. Lond. Math. Soc. 24, 337–352 (1893)
Szegő, G.: Ein Beitrag zur theorie der thetafunktionen, S. B. Preuss. Akad. Wiss. Phys.-Math. Kl., (1926), 242–252.
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The authors thank the anonymous referees for helpful comments which improved this paper.
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The second author was supported in part by NSF Grant DMS-0854849 and in part by a grant from the Simons Foundation.
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Cameron, S., Vinroot, C.R. A product formula for multivariate Rogers–Szegő polynomials. Ramanujan J 35, 479–491 (2014). https://doi.org/10.1007/s11139-014-9571-x
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DOI: https://doi.org/10.1007/s11139-014-9571-x