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I'm presently reading Henri Cohen's Introduction to Modular Forms (https://arxiv.org/pdf/1809.10907.pdf) and I'm trying to do exercise 1.5, which partially entails showing that:

$T_2(a)\equiv\sum_{n=-\infty}^\infty \frac1{\cosh(\pi na)}$

is the square of Jacobi's theta function

$T(a)\equiv\sum_{n=-\infty}^\infty e^{-a\pi n^2}$

(that is, $T_2(a)=T(a)^2$).

I have tried expanding both functions as power series, without much luck, in spite of having tried a few useful tricks mostly entailing summation of geometric series. So far I have the following equivalent forms of $T_2(a)$ and $T(a)^2$:

$T_2(a)=-1+4\sum_{n\geq0}\frac{e^{-a\pi n}}{1+e^{-2a\pi n}}=-1+2\coth\frac{a\pi}2+4\sum_{n\geq1}\sum_{m\geq1}e^{-a\pi n}(-1)^me^{-2a\pi nm}$

$T(a)^2=-1+2T(a)+4\sum_{n\geq1}\sum_{m\geq1}e^{-a\pi(n^2+m^2)}$.

For reference, I have some basic experience with Fourier analysis and functional equations. I'm not really sure what I have to do to show these functions are equal. Thank you all in advance for the assistance!

Bethany Morgan
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  • This is a very standard identity in theory of elliptic and theta functions. One way to look at it is via the Fourier series of elliptic function $\operatorname {dn} (u, k) $. Another more elementary proof is given here: https://math.stackexchange.com/a/737894/72031 – Paramanand Singh Jan 25 '21 at 03:02
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    Also note that most identities arising in theory of modular forms can be proved by algebraic manipulation, but do not expect the steps to be obvious or simple. Maybe that's why this technique championed by the likes of Gauss, Jacobi and Ramanujan has been ditched in modern times in preference to modular forms. – Paramanand Singh Jan 25 '21 at 03:06

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