Generalizations of mock theta functions and radial limits. (English) Zbl 1515.11020
Summary: In the last letter to Hardy, S. Ramanujan [Collected Papers. Cambridge: Cambridge University Press (1927; JFM 53.0030.02); Reprint New York: Chelsea (1962; Zbl 0639.01023)] introduced seventeen functions defined by \(q\)-series convergent for \(|q|<1\) with a complex variable \(q\), and called these functions “mock theta functions”. Subsequently, mock theta functions were widely studied in the literature. In the survey of B. Gordon and R. J. McIntosh [A survey of classical mock theta functions, Partitions, \(q\)-series, and modular forms, Dev. Math. 23, 95–144 (2012; Zbl 1246.33006)], they showed that the odd (resp. even) order mock theta functions are related to the function \(g_3(x,q)\) (resp. \(g_2(x,q))\). These two functions are usually called “universal mock theta functions”. D. R. Hickerson and E. T. Mortenson [Proc. Lond. Math. Soc. (3) 109, No. 2, 382–422 (2014; Zbl 1367.11047)] expressed all the classical mock theta functions and the two universal mock theta functions in terms of Appell-Lerch sums. In this paper, based on some \(q\)-series identities, we find four functions, and express them in terms of Appell-Lerch sums. For example,
\[
1+(xq^{-1}-x^{-1}q)\sum\limits_{n=0}^{\infty}\frac{(-1;q)_{2n}q^n}{(xq^{-1},x^{-1}q;q^2)_{n+1}}=2m(x,q^2,q).
\]
Then we establish some identities related to these functions and the universal mock theta function \(g_2(x,q)\). These relations imply that all the classical mock theta functions can be expressed in terms of these four functions. Furthermore, by means of \(q\)-series identities and some properties of Appell-Lerch sums, we derive four radial limit results related to these functions.
MSC:
| 11B65 | Binomial coefficients; factorials; \(q\)-identities |
| 11F27 | Theta series; Weil representation; theta correspondences |
| 33D15 | Basic hypergeometric functions in one variable, \({}_r\phi_s\) |
References:
| [1] | Andrews, George E., The fifth and seventh order mock theta functions, Trans. Amer. Math. Soc., 113-134 (1986) · Zbl 0593.10018 · doi:10.2307/2000275 |
| [2] | Andrews, George E., Ramanujan’s lost notebook. Part I, xiv+437 pp. (2005), Springer, New York · Zbl 1075.11001 |
| [3] | Andrews, George E., Ramanujan’s lost notebook. Part V, xii+430 pp. (2018), Springer, Cham · Zbl 1416.11001 · doi:10.1007/978-3-319-77834-1 |
| [4] | Bajpai, J., Bilateral series and Ramanujan’s radial limits, Proc. Amer. Math. Soc., 479-492 (2015) · Zbl 1316.11035 · doi:10.1090/S0002-9939-2014-12249-0 |
| [5] | Berkovich, Alexander, Some observations on Dyson’s new symmetries of partitions, J. Combin. Theory Ser. A, 61-93 (2002) · Zbl 1016.05003 · doi:10.1006/jcta.2002.3281 |
| [6] | Bringmann, Kathrin, Partial theta functions and mock modular forms as \(q\)-hypergeometric series, Ramanujan J., 295-310 (2012) · Zbl 1283.11077 · doi:10.1007/s11139-012-9370-1 |
| [7] | Bringmann, Kathrin, Radial limits of mock theta functions, Res. Math. Sci., Art. 17, 18 pp. (2015) · Zbl 1379.11048 · doi:10.1186/s40687-015-0035-8 |
| [8] | Carroll, Greg, Universal mock theta functions as quantum Jacobi forms, Res. Math. Sci., Paper No. 6, 15 pp. (2019) · Zbl 1470.11097 · doi:10.1007/s40687-018-0169-6 |
| [9] | Dyson, F. J., Some guesses in the theory of partitions, Eureka, 10-15 (1944) |
| [10] | Folsom, Amanda, Mock theta functions and quantum modular forms, Forum Math. Pi, e2, 27 pp. (2013) · Zbl 1294.11083 · doi:10.1017/fmp.2013.3 |
| [11] | Folsom, Amanda, Ramanujan 125. Ramanujan’s radial limits, Contemp. Math., 91-102 (2014), Amer. Math. Soc., Providence, RI · Zbl 1359.11064 · doi:10.1090/conm/627/12534 |
| [12] | Garvan, F. G., Universal mock theta functions and two-variable Hecke-Rogers identities, Ramanujan J., 267-296 (2015) · Zbl 1380.11091 · doi:10.1007/s11139-014-9624-1 |
| [13] | Gasper, George, Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, xxvi+428 pp. (2004), Cambridge University Press, Cambridge · Zbl 1129.33005 · doi:10.1017/CBO9780511526251 |
| [14] | Gordon, Basil, Partitions, \(q\)-series, and modular forms. A survey of classical mock theta functions, Dev. Math., 95-144 (2012), Springer, New York · Zbl 1246.33006 · doi:10.1007/978-1-4614-0028-8\_9 |
| [15] | Gu, Nancy S. S., Families of multisums as mock theta functions, Adv. in Appl. Math., 98-124 (2016) · Zbl 1401.11092 · doi:10.1016/j.aam.2016.04.003 |
| [16] | Hickerson, Dean, A proof of the mock theta conjectures, Invent. Math., 639-660 (1988) · Zbl 0661.10059 · doi:10.1007/BF01394279 |
| [17] | Hickerson, Dean R., Hecke-type double sums, Appell-Lerch sums, and mock theta functions, I, Proc. Lond. Math. Soc. (3), 382-422 (2014) · Zbl 1367.11047 · doi:10.1112/plms/pdu007 |
| [18] | Jang, Min-Joo, Radial limits of the universal mock theta function \(g_3\), Proc. Amer. Math. Soc., 925-935 (2017) · Zbl 1421.11033 · doi:10.1090/proc/13065 |
| [19] | Kang, Soon-Yi, Mock Jacobi forms in basic hypergeometric series, Compos. Math., 553-565 (2009) · Zbl 1233.11049 · doi:10.1112/S0010437X09004060 |
| [20] | M. Lerch, Pozn\'amky \(k\) theorii funkc\'i elliptick\'ych, Rozpravy Cesk\'e Akademie C\'isare Frantiska Josefa pro vedy, slovesnost a umen\'i v praze, 24 (1892), 465-480. |
| [21] | M. Lerch, Nov\'a analogie rady theta a nekter\'e zvl\'astn\'i hypergeometrick\'e rady Heineovy, Rozpravy, 3 (1893), 1-10. |
| [22] | Liu, Zhi-Guo, Some operator identities and \(q\)-series transformation formulas, Discrete Math., 119-139 (2003) · Zbl 1021.05010 · doi:10.1016/S0012-365X(02)00626-X |
| [23] | Lovejoy, Jeremy, Rank and conjugation for the Frobenius representation of an overpartition, Ann. Comb., 321-334 (2005) · Zbl 1114.11081 · doi:10.1007/s00026-005-0260-8 |
| [24] | Lovejoy, Jeremy, Bailey pairs and indefinite quadratic forms, J. Math. Anal. Appl., 1002-1013 (2014) · Zbl 1320.33025 · doi:10.1016/j.jmaa.2013.09.009 |
| [25] | Lovejoy, Jeremy, \(M_2\)-rank differences for partitions without repeated odd parts, J. Th\'{e}or. Nombres Bordeaux, 313-334 (2009) · Zbl 1270.11103 |
| [26] | Lovejoy, Jeremy, The Bailey chain and mock theta functions, Adv. Math., 442-458 (2013) · Zbl 1290.33019 · doi:10.1016/j.aim.2013.02.005 |
| [27] | Lovejoy, Jeremy, \(q\)-hypergeometric double sums as mock theta functions, Pacific J. Math., 151-162 (2013) · Zbl 1303.33017 · doi:10.2140/pjm.2013.264.151 |
| [28] | McIntosh, Richard J., The \(H\) and \(K\) family of mock theta functions, Canad. J. Math., 935-960 (2012) · Zbl 1264.33022 · doi:10.4153/CJM-2011-066-0 |
| [29] | McIntosh, Richard J., Some identities for Appell-Lerch sums and a universal mock theta function, Ramanujan J., 767-779 (2018) · Zbl 1426.11044 · doi:10.1007/s11139-016-9869-y |
| [30] | Mortenson, Eric T., On the dual nature of partial theta functions and Appell-Lerch sums, Adv. Math., 236-260 (2014) · Zbl 1367.11048 · doi:10.1016/j.aim.2014.07.018 |
| [31] | Mortenson, Eric, Ramanujan’s radial limits and mixed mock modular bilateral \(q\)-hypergeometric series, Proc. Edinb. Math. Soc. (2), 787-799 (2016) · Zbl 1407.11037 · doi:10.1017/S0013091515000425 |
| [32] | S. Ramanujan, Collected Papers, Cambridge Univ. Press, 1927; Reprinted, Chelsea, New York, 1962. · JFM 53.0030.02 |
| [33] | Ramanujan, Srinivasa, The lost notebook and other unpublished papers, xxviii+419 pp. (1988), Springer-Verlag, Berlin; Narosa Publishing House, New Delhi · Zbl 0639.01023 |
| [34] | Watson, G. N., The Final Problem : An Account of the Mock Theta Functions, J. London Math. Soc., 55-80 (1936) · Zbl 0013.11502 · doi:10.1112/jlms/s1-11.1.55 |
| [35] | Watson, G. N., The Mock Theta Functions (2), Proc. London Math. Soc. (2), 274-304 (1936) · Zbl 0015.30402 · doi:10.1112/plms/s2-42.1.274 |
| [36] | Zudilin, Wadim, On three theorems of Folsom, Ono and Rhoades, Proc. Amer. Math. Soc., 1471-1476 (2015) · Zbl 1311.11026 · doi:10.1090/S0002-9939-2014-12364-1 |
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