Abstract
The combinatorial and analytic properties of Dyson’s “favorite” identity are studied in detail. In particular, a q-series analog of the anti-telescoping method is used to provide a new proof of Dyson’s results with mock theta functions popping up in intermediate steps. This leads to the appearance of Chebyshev polynomials of the third and fourth kind in Bailey pairs related to Bailey’s Lemma. The natural relationship with L.J. Rogers’s pioneering work is also presented.
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References
Andrews, G.E.: An analytic proof of the Rogers-Ramanujan-Gordon identities. Amer. J. Math. 88, 844–846 (1966)
Andrews, G.E.: The Theory of Partitions. Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam (1976)
Andrews, G.E.: Multiple series Rogers-Ramanujan type identities. Pacific J. Math. 114(2), 267–283 (1984)
Andrews, G.E.: Parity in partition identities. Ramanujan J. 23(1-3), 45–90 (2010)
Andrews, G.E.: \(q\)-Orthogonal polynomials, Rogers-Ramanujan identites, and mock theta functions. Proc. Steklov Inst. Math. 276(1), 21–32 (2012)
Andrews, G.E.: Differences of partition functions: the anti-telescoping method. In: Farkas, H.M., Gunning, R.C., Knopp, M.I., Taylor, B.A. (eds.) From Fourier Analysis and Number Theory to Radon Transforms and Geometry, Dev. Math., 28, pp. 1–20. Springer, New York (2013)
Andrews, G.E.: \(4\)-Shadows in \(q\)-series and the Kimberling index. (submitted)
Andrews, G.E.: Sequences in partitions, double \(q\)-series and the mock theta function \(\theta _3(q)\). (in preparation)
Andrews, G.E., Askey, R.: Enumeration of partitions: the role of Eulerian series and \(q\)-orthogonal polynomials. In: Aigner, M. (ed.) Higher Combinatorics, pp. 3–26. Reidel, Dordrecht, Holland (1977)
Bailey, W.N.: Identities of the Rogers-Ramanujan type. Proc. London Math. Soc. (2) 50, 1–10 (1948)
Dyson, F.J.: A walk through Ramanujan’s garden. In: Andrews, G.E., Askey, R.A., Berndt, B.C., Ramanathan, K.G., Rankin, R.A. (eds.) Ramanujan Revisited, pp. 7–28. Academic Press, Boston, MA (1988)
Gasper, G., Rahman, M.: Basic Hypergeometric Series. Cambridge University Press, Cambridge (1990)
Gordon, B.: A combinatorial generalization of the Rogers-Ramanujan identities. Amer. J. Math. 83, 393–399 (1961)
Mason, J.C.: Chebyshev polynomials of the second, third and fourth kinds in approximation, indefinite integration, and integral transforms. J. Comput. Appl. Math. 49(1-3), 169–178 (1993)
Rogers, L.J.: On two theorems of combinatory analysis and some allied identities. Proc. London Math. Soc. 16, 315–336 (1917)
Sills, A.V.: Finite Rogers-Ramanujan type identites. Electron. J. Comb. 10, #R13 (2003)
Slater, L.J.: Further identities of the Rogers-Ramanujan types. Proc. London Math. Soc. (2) 54, 147–167 (1952)
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Andrews, G.E. Dyson’s “Favorite” Identity and Chebyshev Polynomials of the Third and Fourth Kind. Ann. Comb. 23, 443–464 (2019). https://doi.org/10.1007/s00026-019-00443-w
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DOI: https://doi.org/10.1007/s00026-019-00443-w
Keywords
- Partitions
- Dyson’s favorite identity
- Bailey pairs
- Bailey’s lemma
- Partitions
- Chebyshev polynomials
- Mock theta functions