A Generalization of Zwegers' μ-Function According to the q-Hermite-Weber Difference Equation

[1] Andrews G.E., Berndt B.C., Ramanujan’s lost notebook. Part V, Springer, Cham, 2018.

[2] Andrews G.E., Hickerson D., Ramanujan’s “lost” notebook. VII. The sixth order mock theta functions, Adv.

Math. 89 (1991), 60–105.

[3] Beals R., Wong R., Special functions. A graduate text, Cambridge Stud. Adv. Math., Vol. 126, Cambridge

University Press, Cambridge, 2010.

[4] Bradley-Thrush J.G., Properties of the Appell–Lerch function (I), Ramanujan J. 57 (2022), 291–367.

[5] Bringmann K., Folsom A., Ono K., Rolen L., Harmonic Maass forms and mock modular forms: theory and

applications, Amer. Math. Soc. Colloq. Publ., Vol. 64, Amer. Math. Soc., Providence, RI, 2017.

[6] Choi Y.S., The basic bilateral hypergeometric series and the mock theta functions, Ramanujan J. 24 (2011),

345–386.

[7] Garoufalidis S., Wheeler C., Modular q-holonomic modules, arXiv:2203.17029.

[8] Gasper G., Rahman M., Basic hypergeometric series, 2nd ed., Encyclopedia Math. Appl., Vol. 96, Cambridge

University Press, Cambridge, 2004.

A Generalization of Zwegers’ µ-Function

23

[9] Gauss C.F., Summatio quarumdam serierum singularium, Comm. Soc. Reg. Sci. Gottingensis Rec. 1 (1811),

1–40.

[10] Gordon B., McIntosh R.J., A survey of classical mock theta functions, in Partitions, q-Series, and Modular

Forms, Dev. Math., Vol. 23, Springer, New York, 2012, 95–144.

[11] Hickerson D., A proof of the mock theta conjectures, Invent. Math. 94 (1988), 639–660.

[12] Kang S.Y., Mock Jacobi forms in basic hypergeometric series, Compos. Math. 145 (2009), 553–565,

arXiv:0806.1878.

[13] Koekoek R., Lesky P.A., Swarttouw R.F., Hypergeometric orthogonal polynomials and their q-analogues,

Springer Monogr. Math., Springer, Berlin, 2010.

[14] Koelink H.T., Hansen–Lommel orthogonality relations for Jackson’s q-Bessel functions, J. Math. Anal. Appl.

175 (1993), 425–437.

[15] Matsuzaka T., Private communication, 2022.

[16] Ohyama Y., A unified approach to q-special functions of the Laplace type, arXiv:1103.5232.

[17] Ohyama Y., Private communication, 2022.

[18] Ramis J.P., Sauloy J., Zhang C., Local analytic classification of q-difference equations, Ast´erisque 355

(2013), vi+151 pages, arXiv:0903.0853.

[19] Suslov S.K., Some orthogonal very-well-poised 8 ϕ7 -functions that generalize Askey–Wilson polynomials,

Ramanujan J. 5 (2001), 183–218, arXiv:math.CA/9707213.

[20] Weil A., Elliptic functions according to Eisenstein and Kronecker, Classics Math., Springer, Berlin, 1999.

[21] Westerholt-Raum M., H-harmonic Maaß–Jacobi forms of degree 1, Res. Math. Sci. 2 (2015), art. 12, 34 pages.

[22] Zhang C., Une sommation discr`ete pour des ´equations aux q-diff´erences lin´eaires et a

` coefficients analytiques:

th´eorie g´en´erale et exemples, in Differential Equations and the Stokes Phenomenon, World Sci. Publ., River

Edge, NJ, 2002, 309–329.

[23] Zwegers S.P., Mock theta functions, Ph.D. Thesis, Universiteit Utrecht, 2002, available at https://dspace.

library.uu.nl/handle/1874/878.

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