Mock theta functions are q-series convergent for 
whose behavior near roots of unity resembles that
of theta functions without being accounted for by a single theta function. Ramanujan
gave 17 examples in his last letter to Hardy (Watson 1936; Ramanujan 1988, pp. 127-131;
Ramanujan 2000, pp. 354-355).
In the formulation of Ramanujan's definition given by Rhoades (2013), a mock theta function 
is of Eulerian form and has exponential singularities at infinitely many roots
of unity. For each root of unity 
, there is a theta function 
such that 
is bounded as 
approaches 
radially. No single theta function has this property at
all roots of unity. Here theta functions are understood
in the modular form sense, which is broader than
Jacobi theta functions.
The asymptotic formulation of Gordon and McIntosh (2000) uses approximations of the form
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(1)
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as 
with 
(Gordon and McIntosh 2000).
In this asymptotic convention, a mock theta function is called strong if, for every root of unity 
there are modular forms 
and real
numbers 
for 
such that
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(2)
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is bounded as 
radially approaches 
(Gordon and McIntosh 2000).
Ramanujan found an additional three mock theta functions in his "lost notebook" which were subsequently rediscovered by Watson (1936). 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
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(3)
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(4)
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(5)
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(6)
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(7)
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(8)
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(9)
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with 
,
,
and 
due to Watson (1936, Dragonette 1952).
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
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(10)
|
where 
is a partition function P and 
is the sequence
1, 0, 
,
4, 
,
4, 
,
8, 
,
0, 
,
8, 
,
... (OEIS A064053) for 
, 1, ....
| function | OEIS | series |
 | A000025 | 1, 1,  ,
3,  ,
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 (1936) proved the fundamental relations connecting Ramanujan's mock theta functions,
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(11)
|
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(12)
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![2rho(q)+omega(q)=3[1/2q^(-3/8)theta_2(0,q^(3/2))]^2product_(r=1)^(infty)(1-q^(2r))^(-1)](/images/equations/MockThetaFunction/Inline82.svg) |
(13)
|
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(14)
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where 
is a Jacobi theta function (Dragonette 1952).
Ramanujan (2000, pp. 354-355) gave 10 mock theta functions of order five, given by
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(15)
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(16)
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(17)
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(18)
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(19)
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(20)
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(21)
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(22)
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(23)
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(24)
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(Andrews 1986). Note that the notation here follows the standard convention 
.
Ramanujan gave seven mock theta functions of order six, given by
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(25)
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(26)
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(27)
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(28)
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(29)
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(30)
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(31)
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(Andrews and Hickerson 1991).
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).
Ramanujan (2000, p. 355) also gave three mock theta functions of order seven, given by
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(32)
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(33)
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(34)
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(Andrews 1986).
Gordon and McIntosh (2000) found eight mock theta functions of order 8,
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(35)
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(36)
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(37)
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(38)
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(39)
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(40)
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(41)
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(42)
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(43)
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(44)
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