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5
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Define the function
$$f(q,z,y) = \sum_{n \ge 0,m,l} c(n,m,l) q^n z^m y^l$$
where $c(n,m,l)$ is defined by
$$ c(n,m,l) =
\begin{cases}
(-1)^{s+l} & \mbox{if } 4n - m^2 + l^2 = 2s(s+1)
\end{cases}
$$
$$
\begin{cases}
0 & otherwise \;\;\;\;\;\;\;\;\;\;\;\;\;\;
\end{cases}
$$
for some integer $s$
and $c(n,m,l) = 0$ unless $4n - m^2 -l^2 \ge 0.$
$f(q,z,1)$ is known to be related to a Mock modular form. I conjecture that
$$f(q,1,-1) = \sum_{n \ge 0} (-1)^n (2n + 1) q^{n(n+1)/2}.$$
Is there an elementary proof of the above conjecture?
Is the function $f(q,z,y)$ a known mathematical object, perhaps related to a Siegel modular form?
Update:
$f(q,z,y)$ is a product of Jacobi theta functions and $\mu(q;z,y),$
where $\mu(q;z,y)$ is a Lerch sum studied by Zweger's Zweger in his thesis. Zweger's thesis also relates mock modular forms to indefinite quadratic forms of signature $(1,n),$ so perhaps it isn't too unsurprising that $f(q,z,y)$ takes a "nice" form.
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4
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Define the function
$$f(q,z,y) = \sum_{n \ge 0,m,l} c(n,m,l) q^n z^m y^l$$
where $c(n,m,l)$ is defined by
$$ c(n,m,l) =
\begin{cases}
(-1)^{s+l} & \mbox{if } 4n - m^2 + l^2 = s(s+12s(s+1)
\end{cases}
$$
$$
\begin{cases}
0 & otherwise \;\;\;\;\;\;\;\;\;\;\;\;\;\;
\end{cases}
$$
for some integer $s$
and $c(n,m,l) = 0$ unless $4n - m^2 -l^2 \ge 0.$
$f(q,z,1)$ is known to be related to a Mock modular form. I conjecture that
$$f(q,1,-1) = \sum_{n \ge 0} (-1)^n (2n + 1) q^{n(n+1)/2}.$$
Is there an elementary proof of the above conjecture?
Is the function $f(q,z,y)$ a known mathematical object, perhaps related to a Siegel modular form?
Update:
$f(q,z,y)$ is a product of Jacobi theta functions and $\mu(q;z,y),$
where $\mu(q;z,y)$ is a Lerch sum studied by Zweger's in his thesis. Zweger's thesis also relates mock modular forms to indefinite quadratic forms of signature $(1,n),$ so perhaps it isn't too unsurprising that $f(q,z,y)$ takes a "nice" form.
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3
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Mock modular forms and (indefinite) quadratic forms
Define the function
$$f(q,z,y) = \sum_{n \ge 0,m,l} c(n,m,l) q^n z^m y^l$$
where $c(n,m,l)$ is defined by
$$ c(n,m,l) =
\begin{cases}
(-1)^{s+l} & \mbox{if } 4n - m^2 + l^2 = s(s+1)
\end{cases}
$$
$$
\begin{cases}
0 & otherwise \;\;\;\;\;\;\;\;\;\;\;\;\;\;
\end{cases}
$$
for some integer $s$
and $c(n,m,l) = 0$ unless $4n - m^2 -l^2 \ge 0.$
$f(q,z,1)$ is known to be related to a Mock modular form. I conjecture that
$$f(q,1,-1) = \sum_{n \ge 0} (-1)^n (2n + 1) q^{n(n+1)/2}.$$
Is there an elementary proof of the above conjecture?
Is the function $f(q,z,y)$ a known mathematical object, perhaps related to a Siegel modular form?
Update:
$f(q,z,y)$ is a product of Jacobi theta functions and $\mu(q;z,y),$
where $\mu(q;z,y)$ is a Lerch sum studied by Zweger's in his thesis. Zweger's thesis also relates mock modular forms to indefinite quadratic forms of signature $(1,n),$ so perhaps it isn't too unsurprising that $f(q,z,y)$ takes a "nice" form.
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2
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Define the function
$$f(q,z,y) = \sum_{n \ge 0,m,l} c(n,m,l) q^n z^m y^l$$
where $c(n,m,l)$ is defined by
$$ c(n,m,l) =
\begin{cases}
(-1)^{s+l} & \mbox{if } 4n - m^2 + l^2 = s(s+1)
\end{cases}
$$
$$
\begin{cases}
0 & otherwise \;\;\;\;\;\;\;\;\;\;\;\;\;\;
\end{cases}
$$
for some integer $s$
and $c(n,m,l) = 0$ unless $4n - m^2 -l^2 \ge 0.$
$f(q,z,1)$ is known to be related to a Mock modular form. I conjecture that
$$f(q,z,-1) $f(q,1,-1) = \sum_{n \ge 0} (-1)^n (2n + 1) q^{n(n+1)/2}.$$
Is the function $f(q,z,y)$ a known mathematical object, perhaps related to a Siegel modular form?
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1
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Mock modular forms and quadratic forms
Define the function
$$f(q,z,y) = \sum_{n \ge 0,m,l} c(n,m,l) q^n z^m y^l$$
where $c(n,m,l)$ is defined by
$$ c(n,m,l) =
\begin{cases}
(-1)^{s+l} & \mbox{if } 4n - m^2 + l^2 = s(s+1)
\end{cases}
$$
$$
\begin{cases}
0 & otherwise \;\;\;\;\;\;\;\;\;\;\;\;\;\;
\end{cases}
$$
for some integer $s$
and $c(n,m,l) = 0$ unless $4n - m^2 -l^2 \ge 0.$
$f(q,z,1)$ is known to be related to a Mock modular form. I conjecture that
$$f(q,z,-1) = \sum_{n \ge 0} (-1)^n (2n + 1) q^{n(n+1)/2}.$$
Is the function $f(q,z,y)$ a known mathematical object, perhaps related to a Siegel modular form?
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