publication . Preprint . 2007

On the irrationality of Ramanujan's mock theta functions and other q-series at an infinite number of points

Mingarelli, Angelo B.;
Open Access English
  • Published: 24 Dec 2007
Abstract
We show that all of Ramanujan's mock theta functions of order 3, Watson's three additional mock theta functions of order 3, the Rogers-Ramanujan q-series, and 6 mock theta functions of order 5 take on irrational values at the points q=\pm 1/2,\pm 1/3,\pm 1/4,...
Subjects
free text keywords: Mathematics - Number Theory, Mathematics - Classical Analysis and ODEs, 11J72
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Φ(p/q) = −1 +

[1] G.Andrews, On the theorems of Watson and Dragonette for Ramanujan's mock theta functions, Amer. J. Math. 88 (1966), 454-490

[2] G.Cantor, U¨ber die einfachen Zahlensysteme, Zeit. fu¨r Math. und Phys. 14, (1869), 121-128

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

[4] P.H.Diananda and A.Oppenheim, Criteria for irrationality of certain classes of numbers,II, Amer. Math. Monthly 62 (4), (1955), 222-225

[5] G.Gasper and M.Rahman, Basic Hypergeometric Series, Second Edition, Cambridge University Press, Cambridge (2004), xxvi, 428p

[6] J.Hanˇcl and R.Tijdeman, On the irrationality of Cantor series, J. reine angew. Math. (Crelle), 571 (2004), 145-158

[7] G.H.Hardy and E.M.Wright, An Introduction to the Theory of Numbers, Fourth Edition, Oxford University Press, Oxford, London, (1960)

[8] T. Matala-Aho and K. Va¨a¨na¨nen, On diophantine approximations of mock theta functions of third order, Ramanujan J., 4 (2000), 13-28.

[9] T. Matala-Aho, On irrationality measures of P l = 0∞dl/ Ql1(1 + dj + d2j s), J. Number Theory, 128 (2008), 1-16.

[10] A.Oppenheim, Criteria for irrationality of certain classes of numbers, Amer. Math. Monthly 61 (4), (1954), 235-241

[11] G.N.Watson, The final problem: an account of mock theta functions, J. London Math. Soc., 11 (1936), 55-80.

[12] S.P.Zwegers, Mock Theta Functions, Doctoral Dissertation, University of Utrecht, The Netherlands (2002), 96 pp.

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