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I suspect that the function

$$F(q) = \sum_{n \geq 0} (2n + 1) \, q^\binom{n+1}{2}$$

may be some kind of modular form. It looks like a weighted theta function, but is not exactly an harmonic theta series.

Plotting this function in the complex unit disc suggests a modular behaviour. There is a real zero around $-0.3$, and other zeros inside the disc.

The formula above has an hidden symmetry, because summation over all negative integers would give the opposite function.

Is $F(q)$ indeed a modular form ?

And if so

Does it appear in the literature ?

I am also interested in the expansion around $q=1$, where there is a pole. But this would be another question.

EDIT:

The answer below by Somos is interesting, but it only tells us that another resembling series (with alternating signs) is a modular form. It is not possible to pass from this modular form to $F(q)$ by just changing $q$ to $-q$.

EDIT:

Trying to follow the suggested track of Eichler integrals for modular forms of half-integer weights and quantum modular forms, one auxiliary question is now whether the following is a modular form:

$$\sum_{n \geq 0} q^{(2n+1)^2/8}.$$

Here the sum can also be taken over $\mathbb{Z}$, so this is rather similar to a theta function...

EDIT: ... and in fact is equal to the difference

$$\sum_{n} q^{n^2/8} - \sum_{n} q^{n^2/2}.$$

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    $\begingroup$ According to Section 4.3 of this survey people.smp.uq.edu.au/OleWarnaar/pubs/Partial-theta.pdf this function may be related to quantum modular forms. $\endgroup$ – François Brunault Jan 6 at 14:04
  • $\begingroup$ Thanks for the comment and the keyword "partial theta function". Looking at Warnaar's article, I did not quite see something similar to $F(q)$. $\endgroup$ – F. C. Jan 7 at 9:04
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    $\begingroup$ See the articles of Brigmann-Rolen-Zwegers "On the Fourier coefficients of negative index meromorphic Jacobi forms" (Section 2.2) and Zagier "Vassiliev invariants and a strange identity related to the Dedekind eta-function". There is a partial theta function similar to your $F(q)$ whose limiting values (when $q$ tends to roots of unity) has quantum modular properties. So maybe it would be interesting to show that (up to the factor $q^{1/8}$) your $F(q)$ is a quantum modular form of weight $3/2$. $\endgroup$ – François Brunault Jan 7 at 9:55
  • $\begingroup$ Merci! This seems to be a good direction to go, even more because my context also involves the derivative w.r.t. n of the exponent of q, like in the references you give. $\endgroup$ – F. C. Jan 7 at 21:05
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The OEIS sequence A198954 has your q-series as its generating function. It is noted in the entry that

Note that the g.f. theta_1'(0, q^(1/2)) / (2 * q^(1/8)) = 1 - 3*q + 5*q^3 - 7*q^6 + 9*q^10 + ... which is the same as this sequence except the signs alternate.

Thus, this is the derivative at $0$ of a theta function, except the signs there alternate. This is similar to what Rogers and Ramanujan called a "false theta function". The sequence with alternating signs, OEIS squence A010816 suitably scaled to $q - 3q^9 + 5q^{25} - 7q^{49} + \dots = \eta(8z)^3$ is a modular form.

There is a comment in the entry

The partition function of a heteronuclear diatomic molecule is Sum_{J>=0} (2*J + 1) * exp( - J * (J + 1) * hbar^2 / (2 * I * k * T)) where I is the moment of inertia, hbar is reduced Planck's constant, k is Boltzmann's constant, and T is temperature. The degeneracy for the J-th energy level is 2*J + 1.

The reference this comes from is listed as

G. H. Wannier, Statistical Physics, Dover Publications, 1987, see p. 215 equ. (11.13).

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  • $\begingroup$ Good catch. I had no idea that such lacunary sequences were also in the oeis.. $\endgroup$ – F. C. Jan 4 at 18:23
  • $\begingroup$ Can we deduce something about the case of positive coefficients ? $\endgroup$ – F. C. Jan 4 at 20:51
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Not an answer, just a long comment. I had often wondered whether the superficially similar series

$G(q)=-\displaystyle\sum_{n\geq 1} n q^{n^2}$

has any modular properties. My motivation was that the formal "Borcherds transform" (excuse the vulgarity) of this series, up to a power of $q$, would be the MacMahon function

$M(q)=\displaystyle\prod_{n\geq 1} (1-q^n)^{-n}$,

in the same way as the Borcherds transform of the standard $\theta$-function is the Dedekind $\eta$-function (essentially just delete the linear $n$ in both formulas above).

On the other hand, the MacMahon function is claimed to have some kind of exotic modularity properties in an old physics paper of Cardy (Operator content and modular properties of higher dimensional conformal field theories, Nucl. Phys B366). Unfortunately I could never make head or tail of the discussion there.

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Regarding the function $G(q) = \sum_{n \geq 0} q^{(2n+1)^2/8}$ we have \begin{equation*} G(q) = \frac12 \sum_{n \in \mathbb{Z}} q^{n^2/8} - \frac12 \sum_{n \in \mathbb{Z}} q^{n^2/2} = \frac12 \vartheta\bigl(\frac{\tau}{4}\bigr) - \frac12 \vartheta(\tau) \end{equation*} where $\vartheta(\tau)$ is the Jacobi theta function $\vartheta_{00}(0;\tau)$. It is known that $\vartheta$ is modular for $\Gamma := \Gamma(2) \cup \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \Gamma(2)$ (see for example [1], Annexe E). In general if $f(\tau)$ is a modular form for $\Gamma(N)$ and $\delta \geq 1$ is any integer, then $f(\delta\tau)$ and $f(\tau/\delta)$ are modular forms for $\Gamma(N\delta)$. So $G(q)$ would be a modular form of weight $1/2$ for $\Gamma(8)$, except that the transformation $\tau \to \tau/4$ changes the multiplier system involved in the definition of an half-integral modular form, so one might need to add some level to get the true level of $G$.

Regarding the definition of the multiplier system for half-integral weight modular forms, Shimura in [2] takes $(\gamma,\tau) \mapsto \theta(\gamma \tau)/\theta(\tau)$ with $\theta(\tau)=\vartheta(2\tau)$, which is defined for $\gamma \in \Gamma_0(4)$. The matrix $\begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix}$ conjugates $\Gamma_0(4)$ to $\Gamma(2)$, so here one may simply choose the multiplier $(\gamma,\tau) \mapsto \vartheta(\gamma \tau)/\vartheta(\tau)$ defined on any subgroup of $\Gamma$. This multiplier is computed explicitly in Martin-Royer, see [1, Proposition 166].

[1] Martin, François; Royer, Emmanuel. Formes modulaires et périodes. Formes modulaires et transcendance, 1--117, Sémin. Congr., 12, Soc. Math. France, Paris, 2005.

[2] Shimura, Goro. On modular forms of half integral weight. Ann. of Math. (2) 97 (1973), 440--481.

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