Geodesic
MOCK THETA DOUBLE SUMS
Glasgow mathematical journal, Tome 59 (2017) no. 2, pp. 323-348

Voir la notice de l'article provenant de la source Cambridge University Press

We prove a general result on Bailey pairs and show that two Bailey pairs of Bringmann and Kane are special cases. We also show how to use a change of base formula to pass from the pairs of Bringmann and Kane to pairs used by Andrews in his study of Ramanujan's seventh order mock theta functions. We derive several more Bailey pairs of a similar type and use these to construct a number of new q-hypergeometric double sums which are mock theta functions. Finally, we prove identities between some of these mock theta double sums and classical mock theta functions. 
LOVEJOY, JEREMY; OSBURN, ROBERT. MOCK THETA DOUBLE SUMS. Glasgow mathematical journal, Tome 59 (2017) no. 2, pp. 323-348. doi: 10.1017/S0017089516000197
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[1] 1. Andrews, G.E., Multiple series Rogers-Ramanujan identities, Pacific J. Math. 114 (1984), 267–283. Google Scholar | DOI

[2] 2. Andrews, G.E., q-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra, in Regional Conference Series in Mathematics, vol. 66 (American Mathematical Society, Providence, RI, 1986). Google Scholar

[3] 3. Andrews, G.E., The fifth and seventh order mock theta functions, Trans. Amer. Math. Soc. 293 (1986), 113–134. Google Scholar

[4] 4. Andrews, G.E., Dyson, F. and Hickerson, D., Partitions and indefinite quadratic forms, Invent. Math. 91 (1988), no. 3, 391–407. Google Scholar

[5] 5. Andrews, G.E. and Hickerson, D., Ramanujan's “lost” notebook. VII. The sixth order mock theta functions, Adv. Math. 89 (1991), no. 1, 60–105. Google Scholar

[6] 6. Andrews, G.E., Bailey pairs with free parameters, mock theta functions and tubular partitions, Ann. Combin. 18 (2014), 563–578. Google Scholar

[7] 7. Berkovich, A. and Warnaar, S.O., Positivity preserving transformations for q-binomial coefficients, Trans. Amer. Math. Soc. 357 (2005), 2291–2351. Google Scholar | DOI

[8] 8. Bressoud, D., Ismail, M.E.H. and Stanton, D., Change of base in Bailey pairs, Ramanujan J. 4 (2000), 435–453. Google Scholar

[9] 9. Bringmann, K. and Kane, B., Multiplicative q-hypergeometric series arising from real quadratic fields, Trans. Amer. Math. Soc. 363 (4) (2011), 2191–2209. Google Scholar | DOI

[10] 10. Gasper, G. and Rahman, M., Basic Hypergeometric Series, 2nd edition. Encyclopedia of Mathematics and its Applications, (Cambridge University Press, Cambridge, 2004). Google Scholar

[11] 11. Gordon, B. and Mcintosh, R.J., A survey of classical mock theta functions, in Partitions, q-series, and Modular Forms, Developments in Mathematics, vol. 23 (Springer, New York, 2012), 95–144. Google Scholar | DOI

[12] 12. Hickerson, D. and Mortenson, E., Hecke-type double sums, Appell-Lerch sums, and mock theta functions, I, Proc. London Math. Soc. 109 (2014), 382–422. Google Scholar

[13] 13. Lovejoy, J. and Osburn, R., q-hypergeometric double sums as mock theta functions, Pacific J. Math. 264 (2013), no. 1, 151–162. Google Scholar

[14] 14. Lovejoy, J. and Osburn, R., Real quadratic double sums, Indag. Math. 26 (4) (2015), 697–712. Google Scholar | DOI

[15] 15. Ono, K., Unearthing the visions of a master: harmonic Maass forms and number theory, in Proceedings of the 2008 Harvard-MIT Current Developments in Mathematics Conference (International Press, Somerville, MA, 2009), 347–454. Google Scholar

[16] 16. Slater, L.J., A new proof of Rogers's transformtions of infinite series, Proc. London Math. Soc. 53 (2) (1951), 460–475. Google Scholar | DOI

[17] 17. Warnaar, S. O., Algebraic combinatorics and applications, in 50 years of Bailey's lemma, (Gößweinstein, 1999) (Springer, Berlin, 2001), 333–347. Google Scholar

[18] 18. Zagier, D., Ramanujan's mock theta functions and their applications (after Zwegers and Ono-Bringmann), Astérisque 326 (2009), 143–164. Google Scholar

[19] 19. Zwegers, S., Mock Theta Functions, PhD Thesis (Universiteit Utrecht, 2002). Google Scholar

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