Mock theta function

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A mock theta function is a function similar to certain special functions written down by Srinivasa Ramanujan, in his last letter to G. H. Hardy and in his lost notebook. Ramanujan's own definition of mock theta functions is notoriously obscure, but work of Sander Zwegers has recently led to a satisfactory general definition: a mock theta function is the holomorphic part of a weak Maass wave form of weight 1/2. K. Bringmann and K. Ono, University of Wisconsin, (2005) have used Zweger's results and proved the first exact formula for the coefficients of any mock theta function.

1 Definition and properties
2 Examples
3 References

[edit] Definition and properties

Zwegers (2002) showed that the mock theta functions of orders 3, 5, and 7 can be written as the sum of a real analytic modular form of weight 1/2 and a function that is bounded along geodesics ending at cusps.

The real analytic modular form has eigenvalue 3/16 under the hyperbolic Laplacian (the same value as holomorphic modular forms of weight 1/2); however, it increases exponentially fast near cusps, so it does not satisfy the usual growth condition for Maass wave forms. He proved this result by using expressions for the mock theta functions in terms of Hecke's theta functions of indefinite lattices of dimension 2.

Zwegers's result shows that mock theta functions are the "holomorphic parts" of real analytic modular forms of weight 1/2. Kathrin Bringmann and Ken Ono extended the work of Sander P. Zwegers to show that certain q-series arising from the Rogers-Fine basic hypergeometric series are related to holomorphic parts of weight 3/2 harmonic weak Maass forms. They also develop the theory to obtain applications to number theory concerning exact formulas and divisibility properties of coefficients of mock theta functions and Freeman Dyson's partition rank functions. In particular, they solved the longstanding conjecture of George Andrews and Leila Dragonette which dated to the 1960s. Their work on divisibility illustrates how weak Maass forms can be studied using the theory of Galois representations as developed by Jean Pierre Serre and Pierre Deligne.

Mock theta functions have asymptotic expansions at cusps of the modular group, acting on the upper half plane, that resemble those of modular forms of weight 1/2 with poles at the cusps.

Before the work of Zwegers, attempts to give a formal definition, generally followed these lines:

Mock theta functions are power series in q = e^2πiτ that converge in the unit disk.
As functions of τ in the upper half plane, they have an asymptotic expansion at the cusps, similar to that of modular forms of weight 1/2, possibly with poles at cusps.
They cannot be expressed in terms of "ordinary" theta functions.

Moreover mock theta functions often also have the following properties:

They can be written as "pretty" infinite sums, often in several ways.
Under the action of elements of the modular group, they almost transform like modular forms of weight 1/2 (multiplied by suitable powers of q), except that there are "error terms" in the functional equations, usually given as explicit integrals.

Ramanujan associated an order to his mock theta functions. The possible orders of known mock theta functions include

3, 5, 6, 7, 8, 10.

Ramanujan's notion of order corresponds to the conductor of the Nebentypus character of the weight 1/2 harmonic Maass forms which admit Ramanujan's mock theta functions as their holomorphic projections.

[edit] Examples

The following examples are constructed from q-series $(a; q) n$ which are defined as:

$(a;q)_n = \prod_{0\le j<n}(1-aq^j) = (1-a)(1-aq)\cdots(1-aq^{n-1})$

Many of the examples below are treated as basic hypergeometric functions.

[edit] Order 3

Ramanujan mentioned four order-3 mock theta functions in his letter to Hardy, and listed a further three in his lost notebook, which was rediscovered by G. N. Watson. Watson proved the relations between them stated by Ramanujan and also found their transformations under elements of the modular group. The seven known order-3 mock theta functions are

$f(q) = \sum_{n\ge 0} {q^{n^2}\over (-q;q)_n^2} = {2\over \prod_{n>0}(1-q^n)}\sum_{n\in Z}{(-1)^nq^{3n^2/2+n/2}\over 1+q^n}$ , (sequence A000025 in OEIS).

$\phi(q) = \sum_{n\ge 0} {q^{n^2}\over (-q^2;q^2)_n}$ (sequence A053520 in OEIS).

$\psi(q) = \sum_{n\ge 0} {q^{n^2}\over (q;q^2)_n}$ (sequence A053251 in OEIS).

$\chi(q) = \sum_{n\ge 0} {q^{n^2}\over \prod_{1\le i\le n}(1-q^i+q^{2i})}$ (sequence A053252 in OEIS).

$\omega(q) = \sum_{n\ge 0} {q^{2n(n+1)}\over (q;q^2)^2_n}$ (sequence A053253 in OEIS).

$\nu(q) = \sum_{n\ge 0} {q^{n(n+1)}\over (-q;q^2)_n}$ (sequence A053254 in OEIS).

$\rho(q) = \sum_{n\ge 0} {q^{2n(n+1)}\over \prod_{1\le i\le n}(1+q^{2i-1}+q^{4i-2})}$ (sequence A053255 in OEIS).

G. N. Watson, The final problem: an account of the mock theta functions, J. London. Math. Soc. 11 (1936), 55-80.
Basic Hypergeometric Series and Applications by Nathan J. Fine ISBN 0-8218-1524-5
Zwegers, S. P. Mock θ-functions and real analytic modular forms. q-series with applications to combinatorics, number theory, and physics (Urbana, IL, 2000), 269-277, Contemp. Math., 291, Amer. Math. Soc., Providence, RI, 2001.

[edit] Order 5

Ramanujan wrote down ten mock theta functions of order 5 in his letter to Hardy, and stated some relations between them that were later proved by Watson. In his lost notebook he stated some further identities relating these functions, known as the mock theta conjectures, that were later proved by Hickerson.

$f_0(q) = \sum_{n\ge 0} {q^{n^2}\over (-q;q)_{n}}$ (sequence A053256 in OEIS)

$f_1(q) = \sum_{n\ge 0} {q^{n^2+n}\over (-q;q)_{n}}$ (sequence A053257 in OEIS)

$\phi_0(q) = \sum_{n\ge 0} {q^{n^2}(-q;q^2)_{n}}$ (sequence A053258 in OEIS)

$\phi_1(q) = \sum_{n\ge 0} {q^{(n+1)^2}(-q;q)_{n}}$ (sequence A053259 in OEIS)

$\psi_0(q) = \sum_{n\ge 0} {q^{(n+1)(n+1)/2}(-q;q)_{n}}$ (sequence A053260 in OEIS)

$\psi_1(q) = \sum_{n\ge 0} {q^{n(n+1)/2}(-q;q)_{n}}$ (sequence A053261 in OEIS)

$\chi_0(q) = \sum_{n\ge 0} {q^{n}\over (q^n;q)_{n}} = 2F_0(q)-\phi_0(-q)$ (sequence A053262 in OEIS)

$\chi_1(q) = \sum_{n\ge 0} {q^{n}\over (q^n;q)_{n+1}} = 2F_1(q)+q^{-1}\phi_1(-q)$ (sequence A053263 in OEIS)

$F_0(q) = \sum_{n\ge 0} {q^{2n^2}\over (q;q^2)_{n}}$ (sequence A053264 in OEIS)

$F_1(q) = \sum_{n\ge 0} {q^{2n^2+2n}\over (q;q^2)_{n+1}}$ (sequence A053265 in OEIS)

$\Psi_0(q) = -1 + \sum_{n \ge 0} { q^{5n^2}\over(1-q)(1-q^4)(1-q^6)(1-q^9)...(1-q^{5n+1})}$ (sequence A053266 in OEIS)

$\Psi_1(q) = -1 + \sum_{n \ge 0} { q^{5n^2}\over(1-q^2)(1-q^3)(1-q^7)(1-q^8)...(1-q^{5n+2}) }$ (sequence A053267 in OEIS)

Hickerson, Dean A proof of the mock theta conjectures. Invent. Math. 94 (1988), no. 3, 639--660.
G. E. Andrews, The fifth and seventh order mock theta functions, Trans. Amer. Math. Soc. 293 (1986) no. 1 113-134.
G. N. Watson, Mock theta functions II, Proc. London Math Soc (2) 42 (1937) 274-304.

[edit] Order 6

Ramanujan wrote down seven mock theta functions of order 6 in his lost notebook.

$\phi(q) = \sum_{n\ge 0} {(-1)^nq^{n^2}(q;q^2)_n\over (-q;q)_{2n}}$ (sequence A053268 in OEIS)

$\psi(q) = \sum_{n\ge 0} {(-1)^nq^{(n+1)^2}(q;q^2)_n\over (-q;q)_{2n+1}}$ (sequence A053269 in OEIS)

$\rho(q) = \sum_{n\ge 0} {q^{n(n+1)/2}(-q;q)_n\over (q;q^2)_{n+1}}$ (sequence A053270 in OEIS)

$\sigma(q) = \sum_{n\ge 0} {q^{(n+1)(n+2)/2}(-q;q)_n\over (q;q^2)_{n+1}}$ (sequence A053271 in OEIS)

$\lambda(q) = \sum_{n\ge 0} {(-1)^nq^{n}(q;q^2)_n\over (-q;q)_{n}}$ (sequence A053272 in OEIS)

$2\mu(q) = \sum_{n\ge 0} {(-1)^nq^{n+1}(1+q^n)(q;q^2)_n\over (-q;q)_{n+1}}$ (sequence A053273 in OEIS)

$\gamma(q) = \sum_{n\ge 0} {q^{n^2}(q;q)_n\over (q^3;q^3)_{n}}$ (sequence A053274 in OEIS)

Andrews, G. E. and Hickerson, D. Ramanujan's 'Lost' Notebook VII: The Sixth Order Mock Theta Functions. Adv. Math. 89, 60-105, 1991.

[edit] Order 7

Ramanujan gave three mock theta functions of order 7 in his letter to Hardy.

$F_0(q) = \sum_{n\ge 0}{q^{n^2}\over (q^{n+1};q)_n}$ (sequence A053275 in OEIS)
$F_1(q) = \sum_{n\ge 0}{q^{n^2}\over (q^{n};q)_n}$ (sequence A053276 in OEIS)
$F_2(q) = \sum_{n\ge 0}{q^{n(n+1)}\over (q^{n+1};q)_{n+1}}$ (sequence A053277 in OEIS)
Selberg, A. Über die Mock-Thetafunktionen siebenter Ordnung. (On the mock theta functions of seventh order) Arch. Math. og Naturvidenskab 41, 3-15, 1938.
G. E. Andrews, The fifth and seventh order mock theta functions, Trans. Amer. Math. Soc. 293 (1986) no. 1 113-134.
Dean Hickerson, On the seventh order mock theta functions, Inventiones Mathematicae, 94 (1988) 661-677

[edit] Order 8

Gordon and McIntosh found eight mock theta functions of order 8.

$S_0(q) = \sum_{n\ge 0} {q^{n^2} (-q;q^2)_n \over (-q^2;q^2)_n}$

$S_1(q) = \sum_{n\ge 0} {q^{n(n+2)} (-q;q^2)_n \over (-q^2;q^2)_n}$

$T_0(q) = \sum_{n\ge 0} {q^{(n+1)(n+2)} (-q^2;q^2)_n \over (-q;q^2)_{n+1}}$

$T_1(q) = \sum_{n\ge 0} {q^{n(n+1)} (-q^2;q^2)_n \over (-q;q^2)_{n+1}}$

$U_0(q) = \sum_{n\ge 0} {q^{n^2} (-q;q^2)_n \over (-q^4;q^4)_n}$

$U_1(q) = \sum_{n\ge 0} {q^{(n+1)^2} (-q;q^2)_n \over (-q^2;q^4)_{n+1}}$

$V_0(q) = -1+2\sum_{n\ge 0} {q^{n^2} (-q;q^2)_n \over (q;q^2)_n} = -1+2\sum_{n\ge 0} {q^{2n^2} (-q^2;q^4)_n \over (q;q^2)_{2n+1}}$

$V_1(q) = \sum_{n\ge 0} {q^{(n+1)^2} (-q;q^2)_n \over (q;q^2)_{n+1}} = \sum_{n\ge 0} {q^{2n^2+2n+1} (-q^4;q^4)_n \over (q;q^2)_{2n+2}}$

Gordon, B. and McIntosh, R. J. Some Eighth Order Mock Theta Functions. J. London Math. Soc. 62, (2000), 321-335

[edit] Order 10

Ramanujan listed four order-10 mock theta functions in his lost notebook, and stated some relations between them, which were proved by Choi.

$\phi(q)=\sum_{n\ge 0}{q^{n(n+1)/2}\over (q;q^2)_{n+1}}$ (sequence A053281 in OEIS)
$\psi(q)=\sum_{n\ge 0}{q^{(n+1)(n+2)/2}\over (q;q^2)_{n+1}}$ (sequence A053282 in OEIS)
$\Chi(q)=\sum_{n\ge 0}{(-1)^nq^{n^2}\over (-q;q)_{2n}}$ (sequence A053283 in OEIS)
$\chi(q)=\sum_{n\ge 0}{(-1)^nq^{(n+1)^2}\over (-q;q)_{2n+1}}$ (sequence A053284 in OEIS)
Y.-S. Choi, Tenth order mock theta functions in Ramanujan's lost notebook I, II, IV, Invent math 136 (1999) 497-569, Adv. Math. 156 (2000), Trans. Amer. Math. Soc. 354 (2002).

[edit] References

For references on mock theta functions of particular orders, see the examples section above.

Andrews, G. E. Mock Theta Functions. Proc. Sympos. Pure Math. 49, 283-298, 1989.
K. Bringman, K. Ono, Lifting elliptic cusp forms to Maass forms with an application to partitions. Proceedings of the National Academy of Sciences, USA.
K. Bringmann and K. Ono, The f(q) mock theta function conjecture and partition ranks, Inventiones Mathematicae, 165, 2006, pp. 243-266 Reprint
K. Bringmann, K. Ono, Dyson's ranks and Maass forms, Annals of Mathematics,
Ramanujan, Srinivasa The lost notebook and other unpublished papers. Springer-Verlag, Berlin; Narosa Publishing House, New Delhi, 1988. ISBN 3-540-18726-X
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.
Eric W. Weisstein, Mock Theta Function at MathWorld.
Mock Theta Functions by Sander Pieter Zwegers (2002 Utrecht PhD thesis, ISBN 90-393-3155-3)
Zwegers, S. P. Mock θ-functions and real analytic modular forms. q-series with applications to combinatorics, number theory, and physics (Urbana, IL, 2000), 269-277, Contemp. Math., 291, Amer. Math. Soc., Providence, RI, 2001.