Mock Theta Function

In his last letter to Hardy, Ramanujan defined 17 Jacobi theta function-like functions with which he called "mock theta functions" (Watson 1936ab, Ramanujan 1988, pp. 127-131; Ramanujan 2000, pp. 354-355). These functions are q-series with exponential singularities such that the arguments terminate for some power . In particular, if is not a Jacobi theta function, then it is a mock theta function if, for each root of unity , there is an approximation of the form

(1)

as with (Gordon and McIntosh 2000).

If, in addition, for every root of unity there are modular forms and real numbers and such that

(2)

is bounded as radially approaches , then is said to be a strong mock theta function (Gordon and McIntosh 2000).

Ramanujan found an additional three mock theta functions in his "lost notebook" which were subsequently rediscovered by Watson (1936ab). The first formula on page 15 of Ramanujan's lost notebook relates the functions which Watson calls and (equivalent to the third equation on page 63 of Watson's 1936 paper), and the last formula on page 31 of the lost notebook relates what Watson calls and (equivalent to the fourth equation on page 63 of Watson's paper). The orders of these and Ramanujan's original 17 functions were all 3, 5, or 7.

Ramanujan's "lost notebook" also contained several mock theta functions of orders 6 and 10, which, however, were not explicitly identified as mock theta functions by Ramanujan. Their properties have now been investigated in detail (Andrews and Hickerson 1991, Choi 1999).

Unfortunately, while known identities make it clear that mock theta functions of "order" are related to the number , no formal definition for the order of a mock theta function is known. As a result, the term "order" must be regarded merely as a convenient label when applied to mock theta functions (Andrews and Hickerson 1991).

The complete list of mock theta functions of order 3 are

			(3)
			(4)
			(5)
			(6)
			(7)
			(8)
			(9)

with , , and due to Watson (1936ab; Dragonette 1952). Note that the series for does not converge, but the series of even and odd partial sums do converge, so is commonly taken as the average of these two values (Andrews and Hickerson 1991).

The following table summarizes the first few terms of these series. in particular is considered by Dragonette (1952), who showed that the coefficients of the series for satisfy

(10)

where is a partition function P and is the sequence 1, 0, , 4, , 4, , 8, , 8, , ... (OEIS A064053) for , 1, ....

function	OEIS	series
	A000025	1, 1, , 3, , , 7, , 6, ...
	A053250	1, 1, 0, , 1, 1, , , 0, 2, ...
	A053251	0, 1, 1, 1, 2, 2, 2, 3, 3, 4, 5, 5, 6, ...
	A053252	1, 1, 1, 0, 0, 0, 1, 1, 0, 0, , 0, ...
	A053253	1, 2, 3, 4, 6, 8, 10, 14, 18, 22, 29, ...
	A053254	1, , 2, , 2, , 4, , 5, ...
	A053255	1, , 0, 1, 0, , 1, , 0, 1, ...

Watson (1936ab) proved the fundamental relations connecting Ramanujan's mock theta functions,

	(11)
	(12)
	(13)
	(14)

where is a Jacobi theta function (Dragonette 1952).

Ramanujan (2000, pp. 354-355) gave 10 mock theta functions of order five, given by

			(15)
			(16)
			(17)
			(18)
			(19)
			(20)
			(21)
			(22)
			(23)
			(24)

(Andrews 1986). Note that the notation here follows the standard convention .

Ramanujan gave seven mock theta functions of order six, given by

			(25)
			(26)
			(27)
			(28)
			(29)
			(30)
			(31)

(Andrews and Hickerson 1991).

Ramanujan (2000, p. 355) also gave three mock theta functions of order seven, given by

			(32)
			(33)
			(34)

(Andrews 1986).

Gordon and McIntosh (2000) found eight mock theta functions of order 8,

			(35)
			(36)
			(37)
			(38)
			(39)
			(40)
			(41)
			(42)
			(43)
			(44)

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References

Andrews, G. E. "The Fifth and Seventh Order Mock Theta Functions." Trans. Amer. Soc. 293, 113-134, 1986.Andrews, G. E. "Mock Theta Functions." Proc. Sympos. Pure Math. 49, 283-298, 1989.Andrews, G. E. and Berndt, B. Ramanujan's Lost Notebook, Part I. New York: Springer, 2005.Andrews, G. E. and Hickerson, D. "Ramanujan's 'Lost' Notebook VII: The Sixth Order Mock Theta Functions." Adv. Math. 89, 60-105, 1991.Bellman, R. E. A Brief Introduction to Theta Functions. New York: Holt, Rinehart, and Winston, p. 51, 1961.Berndt, B. C. and Rankin, R. A. Ramanujan: Letters and Commentary. Providence, RI: Amer. Math. Soc., pp. 220-224, 1995.Choi, Y.-S. "Tenth Order Mock Theta Functions in Ramanujan's Lost Notebook." Invent. Math. 136, 497-569, 1999.Dragonette, L. A. "Some Asymptotic Formulae for the Mock Theta Series of Ramanujan." Trans. Amer. Math. Soc. 73, 474-500, 1952.Gordon, B. and McIntosh, R. J. "Some Eighth Order Mock Theta Functions." J. London Math. Soc. 62, 321-335, 2000.Gordon, B. and McIntosh, R. J. "Modular Transformations of Ramanujan's Fifth and Seventh Order Mock Theta Functions." Ramanujan J. 7, 193-222, 2003.Ramanujan, S. The Lost Notebook and Other Unpublished Manuscripts. New Delhi, India: Narosa, 1988.Ramanujan, S. Collected Papers of Srinivasa Ramanujan (Ed. G. H. Hardy, P. V. S. Aiyar, and B. M. Wilson). Providence, RI: Amer. Math. Soc., 2000.Selberg, A. "Über die Mock-Thetafunktionen siebenter Ordnung." Arch. Math. og Naturvidenskab 41, 3-15, 1938.Sloane, N. J. A. Sequences A000025/M0433, A053250, A053251, A053252, A053253, A053254, A053255, and A064053 in "The On-Line Encyclopedia of Integer Sequences."Watson, G. N. "The Final Problem: An Account of the Mock Theta Functions." J. London Math. Soc. 11, 55-80, 1936a.Watson, G. N. "The Mock Theta Function (I)." J. London Math. Soc. 11, 55-80, 1936b.Watson, G. N. "The Mock Theta Function (II)." Proc. London Math. Soc. 42, 274-304, 1937.

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Mock Theta Function

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Weisstein, Eric W. "Mock Theta Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MockThetaFunction.html

Mock Theta Function

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