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In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. Theta functions are parametrized by points in a tube domain inside a complex Lagrangian Grassmannian,<ref name="Tyurin2002">Template:Cite arXiv</ref> namely the Siegel upper half space.
The most common form of theta function is that occurring in the theory of elliptic functions. With respect to one of the complex variables (conventionally called Template:Mvar), a theta function has a property expressing its behavior with respect to the addition of a period of the associated elliptic functions, making it a quasiperiodic function. In the abstract theory this quasiperiodicity comes from the cohomology class of a line bundle on a complex torus, a condition of descent.
One interpretation of theta functions when dealing with the heat equation is that "a theta function is a special function that describes the evolution of temperature on a segment domain subject to certain boundary conditions".<ref>Template:Cite book</ref>
Throughout this article, <math>(e^{\pi i\tau})^{\alpha}</math> should be interpreted as <math>e^{\alpha \pi i\tau}</math> (in order to resolve issues of choice of branch).<ref group="note">See e.g. https://dlmf.nist.gov/20.1. Note that this is, in general, not equivalent to the usual interpretation <math>(e^z)^\alpha=e^{\alpha \operatorname{Log} e^z}</math> when <math>z</math> is outside the strip <math>-\pi<\operatorname{Im}z\le\pi</math>. Here, <math>\operatorname{Log}</math> denotes the principal branch of the complex logarithm.</ref>
Jacobi theta functionEdit
There are several closely related functions called Jacobi theta functions, and many different and incompatible systems of notation for them. One Jacobi theta function (named after Carl Gustav Jacob Jacobi) is a function defined for two complex variables Template:Mvar and Template:Mvar, where Template:Mvar can be any complex number and Template:Mvar is the half-period ratio, confined to the upper half-plane, which means it has a positive imaginary part. It is given by the formula
- <math>\begin{align}
\vartheta(z; \tau) &= \sum_{n=-\infty}^\infty \exp \left(\pi i n^2 \tau + 2 \pi i n z\right) \\ &= 1 + 2 \sum_{n=1}^\infty q^{n^2} \cos(2\pi n z) \\ &= \sum_{n=-\infty}^\infty q^{n^2}\eta^n \end{align}</math>
where Template:Math is the nome and Template:Math. It is a Jacobi form. The restriction ensures that it is an absolutely convergent series. At fixed Template:Mvar, this is a Fourier series for a 1-periodic entire function of Template:Mvar. Accordingly, the theta function is 1-periodic in Template:Mvar:
- <math>\vartheta(z+1; \tau) = \vartheta(z; \tau).</math>
By completing the square, it is also Template:Mvar-quasiperiodic in Template:Mvar, with
- <math>\vartheta(z+\tau;\tau) = \exp\bigl(-\pi i (\tau + 2 z)\bigr) \vartheta(z;\tau).</math>
Thus, in general,
- <math>\vartheta(z+a+b\tau;\tau) = \exp\left(-\pi i b^2 \tau -2 \pi i b z\right) \vartheta(z;\tau)</math>
for any integers Template:Mvar and Template:Mvar.
For any fixed <math>\tau </math>, the function is an entire function on the complex plane, so by Liouville's theorem, it cannot be doubly periodic in <math>1, \tau </math> unless it is constant, and so the best we can do is to make it periodic in <math>1 </math> and quasi-periodic in <math>\tau </math>. Indeed, since <math display="block">\left|\frac{\vartheta(z+a+b\tau;\tau)}{\vartheta(z;\tau)}\right| = \exp\left(\pi (b^2 \Im(\tau) + 2b \Im(z)) \right) </math>and <math>\Im(\tau)> 0 </math>, the function <math>\vartheta(z, \tau) </math> is unbounded, as required by Liouville's theorem.
It is in fact the most general entire function with 2 quasi-periods, in the following sense:<ref>Template:Cite book</ref>
Auxiliary functionsEdit
The Jacobi theta function defined above is sometimes considered along with three auxiliary theta functions, in which case it is written with a double 0 subscript:
- <math>\vartheta_{00}(z;\tau) = \vartheta(z;\tau)</math>
The auxiliary (or half-period) functions are defined by
- <math>\begin{align}
\vartheta_{01}(z;\tau)& = \vartheta \left(z+\tfrac12;\tau\right)\\[3pt] \vartheta_{10}(z;\tau)& = \exp\left(\tfrac14\pi i \tau + \pi i z\right)\vartheta\left(z + \tfrac12\tau;\tau\right)\\[3pt] \vartheta_{11}(z;\tau)& = \exp\left(\tfrac14\pi i \tau + \pi i\left(z+\tfrac12\right)\right)\vartheta\left(z+\tfrac12\tau + \tfrac12;\tau\right). \end{align}</math>
This notation follows Riemann and Mumford; Jacobi's original formulation was in terms of the nome Template:Math rather than Template:Mvar. In Jacobi's notation the Template:Mvar-functions are written:
- <math>\begin{align}
\theta_1(z;q) &=\theta_1(\pi z,q)= -\vartheta_{11}(z;\tau)\\ \theta_2(z;q) &=\theta_2(\pi z,q)= \vartheta_{10}(z;\tau)\\ \theta_3(z;q) &=\theta_3(\pi z,q)= \vartheta_{00}(z;\tau)\\ \theta_4(z;q) &=\theta_4(\pi z,q)= \vartheta_{01}(z;\tau) \end{align}</math>
The above definitions of the Jacobi theta functions are by no means unique. See Jacobi theta functions (notational variations) for further discussion.
If we set Template:Math in the above theta functions, we obtain four functions of Template:Mvar only, defined on the upper half-plane. These functions are called Theta Nullwert functions, based on the German term for zero value because of the annullation of the left entry in the theta function expression. Alternatively, we obtain four functions of Template:Mvar only, defined on the unit disk <math>|q|<1</math>. They are sometimes called theta constants:<ref group="note"><math>\theta_1(q)=0</math> for all <math>q\in\mathbb{C}</math> with <math>|q|<1</math>.</ref>
- <math>\begin{align}
\vartheta_{11}(0;\tau)&=-\theta_1(q)=-\sum_{n=-\infty}^\infty (-1)^{n-1/2}q^{(n+1/2)^2} \\ \vartheta_{10}(0;\tau)&=\theta_2(q)=\sum_{n=-\infty}^\infty q^{(n+1/2)^2}\\ \vartheta_{00}(0;\tau)&=\theta_3(q)=\sum_{n=-\infty}^\infty q^{n^2}\\ \vartheta_{01}(0;\tau)&=\theta_4(q)=\sum_{n=-\infty}^\infty (-1)^n q^{n^2} \end{align}</math>
with the nome Template:Math. Observe that <math> \theta_1(q)=0 </math>. These can be used to define a variety of modular forms, and to parametrize certain curves; in particular, the Jacobi identity is
- <math>\theta_2(q)^4 + \theta_4(q)^4 = \theta_3(q)^4</math>
or equivalently,
- <math>\vartheta_{01}(0;\tau)^4 + \vartheta_{10}(0;\tau)^4 = \vartheta_{00}(0;\tau)^4</math>
which is the Fermat curve of degree four.
Jacobi identitiesEdit
Jacobi's identities describe how theta functions transform under the modular group, which is generated by Template:Math and Template:Math. Equations for the first transform are easily found since adding one to Template:Mvar in the exponent has the same effect as adding Template:Sfrac to Template:Mvar (Template:Math). For the second, let
- <math>\alpha = (-i \tau)^\frac12 \exp\left(\frac{\pi}{\tau} i z^2 \right).</math>
Then
- <math>\begin{align}
\vartheta_{00}\!\left(\frac{z}{\tau}; \frac{-1}{\tau}\right)& = \alpha\,\vartheta_{00}(z; \tau)\quad& \vartheta_{01}\!\left(\frac{z}{\tau}; \frac{-1}{\tau}\right)& = \alpha\,\vartheta_{10}(z; \tau)\\[3pt] \vartheta_{10}\!\left(\frac{z}{\tau}; \frac{-1}{\tau}\right)& = \alpha\,\vartheta_{01}(z; \tau)\quad& \vartheta_{11}\!\left(\frac{z}{\tau}; \frac{-1}{\tau}\right)& = -i\alpha\,\vartheta_{11}(z; \tau). \end{align}</math>
Theta functions in terms of the nomeEdit
Instead of expressing the Theta functions in terms of Template:Mvar and Template:Mvar, we may express them in terms of arguments Template:Mvar and the nome Template:Mvar, where Template:Math and Template:Math. In this form, the functions become
- <math>\begin{align}
\vartheta_{00}(w, q)& = \sum_{n=-\infty}^\infty \left(w^2\right)^n q^{n^2}\quad& \vartheta_{01}(w, q)& = \sum_{n=-\infty}^\infty (-1)^n \left(w^2\right)^n q^{n^2}\\[3pt] \vartheta_{10}(w, q)& = \sum_{n=-\infty}^\infty \left(w^2\right)^{n+\frac12} q^{\left(n + \frac12\right)^2}\quad& \vartheta_{11}(w, q)& = i \sum_{n=-\infty}^\infty (-1)^n \left(w^2\right)^{n+\frac12} q^{\left(n + \frac12\right)^2}. \end{align}</math>
We see that the theta functions can also be defined in terms of Template:Mvar and Template:Mvar, without a direct reference to the exponential function. These formulas can, therefore, be used to define the Theta functions over other fields where the exponential function might not be everywhere defined, such as fields of [[p-adic number|Template:Mvar-adic numbers]].
Product representationsEdit
The Jacobi triple product (a special case of the Macdonald identities) tells us that for complex numbers Template:Mvar and Template:Mvar with Template:Math and Template:Math we have
- <math>\prod_{m=1}^\infty
\left( 1 - q^{2m}\right) \left( 1 + w^2 q^{2m-1}\right) \left( 1 + w^{-2}q^{2m-1}\right) = \sum_{n=-\infty}^\infty w^{2n}q^{n^2}. </math>
It can be proven by elementary means, as for instance in Hardy and Wright's An Introduction to the Theory of Numbers.
If we express the theta function in terms of the nome Template:Math (noting some authors instead set Template:Math) and take Template:Math then
- <math>\vartheta(z; \tau) = \sum_{n=-\infty}^\infty \exp(\pi i \tau n^2) \exp(2\pi i z n) = \sum_{n=-\infty}^\infty w^{2n}q^{n^2}. </math>
We therefore obtain a product formula for the theta function in the form
- <math>\vartheta(z; \tau) = \prod_{m=1}^\infty
\big( 1 - \exp(2m \pi i \tau)\big) \Big( 1 + \exp\big((2m-1) \pi i \tau + 2 \pi i z\big)\Big) \Big( 1 + \exp\big((2m-1) \pi i \tau - 2 \pi i z\big)\Big). </math>
In terms of Template:Mvar and Template:Mvar:
- <math>\begin{align}
\vartheta(z; \tau) &= \prod_{m=1}^\infty \left( 1 - q^{2m}\right) \left( 1 + q^{2m-1}w^2\right) \left( 1 + \frac{q^{2m-1}}{w^2}\right) \\ &= \left(q^2;q^2\right)_\infty\,\left(-w^2q;q^2\right)_\infty\,\left(-\frac{q}{w^2};q^2\right)_\infty \\ &= \left(q^2;q^2\right)_\infty\,\theta\left(-w^2q;q^2\right) \end{align}</math>
where Template:Math is the [[q-Pochhammer symbol|Template:Mvar-Pochhammer symbol]] and Template:Math is the [[q-theta function|Template:Mvar-theta function]]. Expanding terms out, the Jacobi triple product can also be written
- <math>\prod_{m=1}^\infty
\left( 1 - q^{2m}\right) \Big( 1 + \left(w^2+w^{-2}\right)q^{2m-1}+q^{4m-2}\Big),</math>
which we may also write as
- <math>\vartheta(z\mid q) = \prod_{m=1}^\infty
\left( 1 - q^{2m}\right) \left( 1 + 2 \cos(2 \pi z)q^{2m-1}+q^{4m-2}\right).</math>
This form is valid in general but clearly is of particular interest when Template:Mvar is real. Similar product formulas for the auxiliary theta functions are
- <math>\begin{align}
\vartheta_{01}(z\mid q) &= \prod_{m=1}^\infty \left( 1 - q^{2m}\right) \left( 1 - 2 \cos(2 \pi z)q^{2m-1}+q^{4m-2}\right),\\[3pt] \vartheta_{10}(z\mid q) &= 2 q^\frac14\cos(\pi z)\prod_{m=1}^\infty \left( 1 - q^{2m}\right) \left( 1 + 2 \cos(2 \pi z)q^{2m}+q^{4m}\right),\\[3pt] \vartheta_{11}(z\mid q) &= -2 q^\frac14\sin(\pi z)\prod_{m=1}^\infty \left( 1 - q^{2m}\right)\left( 1 - 2 \cos(2 \pi z)q^{2m}+q^{4m}\right). \end{align}</math> In particular, <math display="block">\lim_{q\to 0}\frac{\vartheta_{10}(z\mid q)}{2 q^{\frac14}} = \cos(\pi z),\quad \lim_{q\to 0}\frac{-\vartheta_{11}(z\mid q)}{2 q^{-\frac14}} = \sin(\pi z)</math>so we may interpret them as one-parameter deformations of the periodic functions <math>\sin, \cos</math>, again validating the interpretation of the theta function as the most general 2 quasi-period function.
Integral representationsEdit
The Jacobi theta functions have the following integral representations:
- <math>\begin{align}
\vartheta_{00} (z; \tau) &= -i\int_{i-\infty}^{i+\infty} e^{i \pi \tau u^2} \frac{\cos(2\pi uz + \pi u)}{\sin(\pi u)} \mathrm{d}u; \\[6pt] \vartheta_{01} (z; \tau) &= -i\int_{i-\infty}^{i+\infty} e^{i \pi \tau u^2} \frac{\cos (2\pi uz)}{\sin (\pi u)} \mathrm{d}u; \\[6pt] \vartheta_{10} (z; \tau) &= -ie^{i \pi z + \frac14 i \pi\tau} \int_{i-\infty}^{i+\infty} e^{i\pi\tau u^2} \frac{\cos(2\pi uz + \pi u + \pi \tau u)}{\sin (\pi u)} \mathrm{d}u; \\[6pt] \vartheta_{11} (z; \tau) &= e^{i \pi z + \frac14 i \pi \tau} \int_{i-\infty}^{i+\infty} e^{i\pi\tau u^2} \frac{\cos(2\pi uz + \pi \tau u)}{\sin(\pi u)} \mathrm{d}u. \end{align}</math>
The Theta Nullwert function <math> \theta_{3}(q) </math> as this integral identity:
- <math>\theta_{3}(q) = 1 + \frac{4q\sqrt{\ln(1/q)}}{\sqrt{\pi}} \int_{0}^{\infty} \frac{\exp[-\ln(1/q)\,x^2]\{1 - q^2\cos[2\ln(1/q)\,x]\}}{1 - 2q^2\cos[2\ln(1/q)\,x] + q^4} \,\mathrm{d}x </math>
This formula was discussed in the essay Square series generating function transformations by the mathematician Maxie Schmidt from Georgia in Atlanta.
Based on this formula following three eminent examples are given:
- <math>\biggl[\frac{2}{\pi}K\bigl(\frac{1}{2}\sqrt{2}\bigr)\biggr]^{1/2} = \theta_{3}\bigl[\exp(-\pi)\bigr] = 1 + 4\exp(-\pi) \int_{0}^{\infty} \frac{\exp(-\pi x^2)[1 - \exp(-2\pi)\cos(2\pi x)]}{1 - 2\exp(-2\pi)\cos(2\pi x) + \exp(-4\pi)} \,\mathrm{d}x </math>
- <math>\biggl[\frac{2}{\pi}K(\sqrt{2} - 1)\biggr]^{1/2} = \theta_{3}\bigl[\exp(-\sqrt{2}\,\pi)\bigr] = 1 + 4\,\sqrt[4]{2}\exp(-\sqrt{2}\,\pi) \int_{0}^{\infty} \frac{\exp(-\sqrt{2}\,\pi x^2)[1 - \exp(-2\sqrt{2}\,\pi)\cos(2\sqrt{2}\,\pi x)]}{1 - 2\exp(-2\sqrt{2}\,\pi)\cos(2\sqrt{2}\,\pi x) + \exp(-4\sqrt{2}\,\pi)} \,\mathrm{d}x </math>
- <math>\biggl\{\frac{2}{\pi}K\bigl[\sin\bigl(\frac{\pi}{12}\bigr)\bigr]\biggr\}^{1/2} = \theta_{3}\bigl[\exp(-\sqrt{3}\,\pi)\bigr] = 1 + 4\,\sqrt[4]{3}\exp(-\sqrt{3}\,\pi) \int_{0}^{\infty} \frac{\exp(-\sqrt{3}\,\pi x^2)[1 - \exp(-2\sqrt{3}\,\pi)\cos(2\sqrt{3}\,\pi x)]}{1 - 2\exp(-2\sqrt{3}\,\pi)\cos(2\sqrt{3}\,\pi x) + \exp(-4\sqrt{3}\,\pi)} \,\mathrm{d}x </math>
Furthermore, the theta examples <math> \theta_{3}(\tfrac{1}{2}) </math> and <math> \theta_{3}(\tfrac{1}{3}) </math> shall be displayed:
- <math>\theta_{3}\left(\frac{1}{2}\right) = 1+2\sum_{n = 1}^{\infty} \frac{1}{2^{n^2}} = 1 + 2\pi^{-1/2}\sqrt{\ln(2)} \int_{0}^{\infty} \frac{\exp[-\ln(2)\,x^2]\{16 - 4\cos[2\ln(2)\,x]\}}{17 - 8\cos[2\ln(2)\,x]} \,\mathrm{d}x </math>
- <math>\theta_{3}\left(\frac{1}{2}\right) = 2.128936827211877158669\ldots </math>
- <math>\theta_{3}\left(\frac{1}{3}\right) = 1+2\sum_{n = 1}^{\infty} \frac{1}{3^{n^2}} = 1 + \frac{4}{3}\pi^{-1/2}\sqrt{\ln(3)} \int_{0}^{\infty} \frac{\exp[-\ln(3)\,x^2]\{81 - 9\cos[2\ln(3)\,x]\}}{82 - 18\cos[2\ln(3)\,x]} \,\mathrm{d}x </math>
- <math>\theta_{3}\left(\frac{1}{3}\right) = 1.691459681681715341348\ldots </math>
Some interesting relationsEdit
If <math>|q|<1</math> and <math>a>0</math>, then the following theta functions
- <math>\theta_3(a,b;q)=\sum^{\infty}_{n=-\infty}q^{an^2+bn}</math>
- <math>\theta_4(a,b;q)=\sum^{\infty}_{n=-\infty}(-1)^nq^{an^2+bn}</math>
have interesting arithmetical and modular properties. When <math>a,b,p</math> are positive integers, then <ref>Template:Cite arXiv</ref><ref>Template:Cite arXiv</ref>
- <math>\log\left(\frac{\theta_3\left(\frac{p}{2},\frac{p}{2}-a;q\right)}{\theta_3\left(\frac{p}{2},\frac{p}{2}-b;q\right)}\right)=-\sum^{\infty}_{n=1}q^n\left(\sum_{\begin{array}{cc}d|n\\n/d\equiv\pm a(p)\end{array}}\frac{(-1)^d}{d}-\sum_{\begin{array}{cc}d|n\\n/d\equiv\pm b(p)\end{array}}\frac{(-1)^d}{d}\right)</math>
- <math>\log\left(\frac{\theta_4\left(\frac{p}{2},\frac{p}{2}-a;q\right)}{\theta_4\left(\frac{p}{2},\frac{p}{2}-b;q\right)}\right)=-\sum^{\infty}_{n=1}q^n\left(\sum_{\begin{array}{cc}d|n\\n/d\equiv\pm a(p)\end{array}}\frac{1}{d}-\sum_{\begin{array}{cc}d|n\\n/d\equiv\pm b(p)\end{array}}\frac{1}{d}\right)</math>
Also if <math>q=e^{\pi i z}</math>, <math>Im(z)>0</math>, the functions with :
- <math>\vartheta_{+}(z)=\theta_{+}(a,p;z)=q^{p/8+a^2/(2p)-a/2}\theta_3\left(\frac{p}{2},\frac{p}{2}-a;q\right)</math>
and
- <math>\vartheta_{-}(z)=\theta_{-}(a,p;z)=q^{p/8+a^2/(2p)-a/2}\theta_4\left(\frac{p}{2},\frac{p}{2}-a;q\right)</math>
are modular forms with weight <math>1/2</math> in <math>\Gamma(2p)</math> i.e. If <math>a_1,b_1,c_1,d_1</math> are integers such that <math>a_1,d_1\equiv 1(2p)</math>, <math>b_1,c_1\equiv0(2p)</math> and <math>a_1d_1-b_1c_1=1</math> there exists <math>\epsilon_{\pm}=\epsilon_{\pm}(a_1,b_1,c_1,d_1)</math>, <math>(\epsilon_{\pm})^{24}=1</math>, such that for all complex numbers <math>z</math> with <math>Im(z)>0</math>, we have
- <math>\vartheta_{\pm}\left(\frac{a_1z+b_1}{c_1z+d_1}\right)=\epsilon_{\pm}\sqrt{c_1z+d_1}\vartheta_{\pm}(z)</math>
Explicit valuesEdit
Lemniscatic valuesEdit
Proper credit for most of these results goes to Ramanujan. See Ramanujan's lost notebook and a relevant reference at Euler function. The Ramanujan results quoted at Euler function plus a few elementary operations give the results below, so they are either in Ramanujan's lost notebook or follow immediately from it. See also Yi (2004).<ref name="Yi">Template:Cite journal</ref> Define,
- <math>\quad \varphi(q) =\vartheta_{00}(0;\tau) =\theta_3(0;q)=\sum_{n=-\infty}^\infty q^{n^2}</math>
with the nome <math>q =e^{\pi i \tau},</math> <math>\tau = n\sqrt{-1},</math> and Dedekind eta function <math>\eta(\tau).</math> Then for <math>n = 1,2,3,\dots</math>
- <math>\begin{align}
\varphi\left(e^{-\pi} \right) &= \frac{\sqrt[4]{\pi}}{\Gamma\left(\frac34\right)} = \sqrt2\,\eta\left(\sqrt{-1}\right)\\ \varphi\left(e^{-2\pi}\right) &= \frac{\sqrt[4]{\pi}}{\Gamma\left(\frac34\right)} \frac{\sqrt{2+\sqrt2}}{2}\\ \varphi\left(e^{-3\pi}\right) &= \frac{\sqrt[4]{\pi}}{\Gamma\left(\frac34\right)} \frac{\sqrt{1+\sqrt3}}{\sqrt[8]{108}}\\ \varphi\left(e^{-4\pi}\right) &= \frac{\sqrt[4]{\pi}}{\Gamma\left(\frac34\right)} \frac{2+\sqrt[4]{8}}{4}\\ \varphi\left(e^{-5\pi}\right) &= \frac{\sqrt[4]{\pi}}{\Gamma\left(\frac34\right)} \sqrt{\frac{2+\sqrt5}{5}}\\ \varphi\left(e^{-6\pi}\right) &= \frac{\sqrt[4]{\pi}}{\Gamma\left(\frac34\right)} \frac{\sqrt{\sqrt[4]{1}+\sqrt[4]{3}+\sqrt[4]{4}+\sqrt[4]{9}}}{\sqrt[8]{12^3}}\\ \varphi\left(e^{-7\pi}\right) &= \frac{\sqrt[4]{\pi}}{\Gamma\left(\frac34\right)} \frac{\sqrt{\sqrt{13+\sqrt{7}}+\sqrt{7+3\sqrt{7}}}}{\sqrt[8]{14^3}\cdot\sqrt[16]{7}}\\ \varphi\left(e^{-8\pi}\right) &= \frac{\sqrt[4]{\pi}}{\Gamma\left(\frac34\right)} \frac{\sqrt{2+\sqrt{2}}+\sqrt[8]{128}}{4}\\ \varphi\left(e^{-9\pi}\right) &= \frac{\sqrt[4]{\pi}}{\Gamma\left(\frac34\right)} \frac{1+\sqrt[3]{2+2\sqrt{3}}}{3}\\ \varphi\left(e^{-10\pi}\right) &= \frac{\sqrt[4]{\pi}}{\Gamma\left(\frac34\right)} \frac{\sqrt{\sqrt[4]{64}+\sqrt[4]{80}+\sqrt[4]{81}+\sqrt[4]{100}}}{\sqrt[4]{200}}\\ \varphi\left(e^{-11\pi}\right) &= \frac{\sqrt[4]{\pi}}{\Gamma\left(\frac34\right)} \frac{\sqrt{11+\sqrt{11}+(5+3\sqrt{3}+\sqrt{11}+\sqrt{33})\sqrt[3]{-44+33\sqrt{3}}+(-5+3\sqrt{3}-\sqrt{11}+\sqrt{33})\sqrt[3]{44+33\sqrt{3}}}}{\sqrt[8]{52180524}}\\ \varphi\left(e^{-12\pi}\right) &= \frac{\sqrt[4]{\pi}}{\Gamma\left(\frac34\right)} \frac{\sqrt{\sqrt[4]{1}+\sqrt[4]{2}+\sqrt[4]{3}+\sqrt[4]{4}+\sqrt[4]{9}+\sqrt[4]{18}+\sqrt[4]{24}}}{2\sqrt[8]{108}}\\ \varphi\left(e^{-13\pi}\right) &= \frac{\sqrt[4]{\pi}}{\Gamma\left(\frac34\right)} \frac{\sqrt{13+8\sqrt{13}+(11-6\sqrt{3}+\sqrt{13})\sqrt[3]{143+78\sqrt{3}}+(11+6\sqrt{3}+\sqrt{13})\sqrt[3]{143-78\sqrt{3}}}}{\sqrt[4]{19773}}\\ \varphi\left(e^{-14\pi}\right) &= \frac{\sqrt[4]{\pi}}{\Gamma\left(\frac34\right)} \frac{\sqrt{\sqrt{13+\sqrt{7}}+\sqrt{7+3\sqrt{7}}+\sqrt{10+2\sqrt{7}}+\sqrt[8]{28}\sqrt{4+\sqrt{7}}}}{\sqrt[16]{28^7}}\\ \varphi\left(e^{-15\pi}\right) &= \frac{\sqrt[4]{\pi}}{\Gamma\left(\frac34\right)} \frac{\sqrt{7+3\sqrt{3}+\sqrt{5}+\sqrt{15}+\sqrt[4]{60}+\sqrt[4]{1500}}}{\sqrt[8]{12^3}\cdot\sqrt{5}}\\ 2\varphi\left(e^{-16\pi}\right) &= \varphi\left(e^{-4\pi}\right) + \frac{\sqrt[4]{\pi}}{\Gamma\left(\frac34\right)} \frac{\sqrt[4]{1+\sqrt{2}}}{\sqrt[16]{128}}\\ \varphi\left(e^{-17\pi}\right) &= \frac{\sqrt[4]{\pi}}{\Gamma\left(\frac34\right)} \frac{\sqrt{2}(1+\sqrt[4]{17})+\sqrt[8]{17}\sqrt{5+\sqrt{17}}}{\sqrt{17+17\sqrt{17}}}\\ 2\varphi\left(e^{-20\pi}\right) &= \varphi\left(e^{-5\pi}\right) + \frac{\sqrt[4]{\pi}}{\Gamma\left(\frac34\right)} \sqrt{\frac{3+2\sqrt[4]{5}}{5\sqrt2}}\\ 6\varphi\left(e^{-36\pi}\right) &= 3\varphi\left(e^{-9\pi}\right) + 2\varphi\left(e^{-4\pi}\right) - \varphi\left(e^{-\pi}\right) + \frac{\sqrt[4]{\pi}}{\Gamma\left(\frac34\right)} \sqrt[3]{\sqrt[4]{2}+\sqrt[4]{18}+\sqrt[4]{216}} \end{align}</math>
If the reciprocal of the Gelfond constant is raised to the power of the reciprocal of an odd number, then the corresponding <math> \vartheta_{00} </math> values or <math> \phi </math> values can be represented in a simplified way by using the hyperbolic lemniscatic sine:
- <math> \varphi\bigl[\exp(-\tfrac{1}{5}\pi)\bigr] = \sqrt[4]{\pi}\,{\Gamma\left(\tfrac{3}{4}\right)}^{-1} \operatorname{slh}\bigl(\tfrac{1}{5}\sqrt{2}\,\varpi\bigr)\operatorname{slh}\bigl(\tfrac{2}{5}\sqrt{2}\,\varpi\bigr) </math>
- <math> \varphi\bigl[\exp(-\tfrac{1}{7}\pi)\bigr] = \sqrt[4]{\pi}\,{\Gamma\left(\tfrac{3}{4}\right)}^{-1} \operatorname{slh}\bigl(\tfrac{1}{7}\sqrt{2}\,\varpi\bigr)\operatorname{slh}\bigl(\tfrac{2}{7}\sqrt{2}\,\varpi\bigr) \operatorname{slh}\bigl(\tfrac{3}{7}\sqrt{2}\,\varpi\bigr) </math>
- <math> \varphi\bigl[\exp(-\tfrac{1}{9}\pi)\bigr] = \sqrt[4]{\pi}\,{\Gamma\left(\tfrac{3}{4}\right)}^{-1} \operatorname{slh}\bigl(\tfrac{1}{9}\sqrt{2}\,\varpi\bigr)\operatorname{slh}\bigl(\tfrac{2}{9}\sqrt{2}\,\varpi\bigr) \operatorname{slh}\bigl(\tfrac{3}{9}\sqrt{2}\,\varpi\bigr) \operatorname{slh} \bigl(\tfrac{4}{9}\sqrt{2}\,\varpi\bigr) </math>
- <math> \varphi\bigl[\exp(-\tfrac{1}{11}\pi)\bigr] = \sqrt[4]{\pi}\,{\Gamma\left(\tfrac{3}{4}\right)}^{-1} \operatorname{slh}\bigl(\tfrac{1}{11}\sqrt{2}\,\varpi\bigr)\operatorname{slh}\bigl(\tfrac{2}{11}\sqrt{2}\,\varpi\bigr) \operatorname{slh}\bigl(\tfrac{3}{11}\sqrt{2}\,\varpi\bigr) \operatorname{slh} \bigl(\tfrac{4}{11}\sqrt{2}\,\varpi\bigr) \operatorname{slh}\bigl(\tfrac{5}{11}\sqrt{2}\,\varpi\bigr) </math>
With the letter <math> \varpi </math> the Lemniscate constant is represented.
Note that the following modular identities hold:
- <math>\begin{align}
2\varphi\left(q^4\right) &= \varphi(q)+\sqrt{2\varphi^2\left(q^2\right)-\varphi^2(q)}\\ 3\varphi\left(q^9\right) &= \varphi(q)+\sqrt[3]{9\frac{\varphi^4\left(q^3\right)}{\varphi(q)}-\varphi^3(q)}\\ \sqrt{5}\varphi\left(q^{25}\right) &= \varphi\left(q^5\right)\cot\left(\frac{1}{2}\arctan\left(\frac{2}{\sqrt{5}}\frac{\varphi(q)\varphi\left(q^5\right)}{\varphi^2(q)-\varphi^2\left(q^5\right)}\frac{1+s(q)-s^2(q)}{s(q)}\right)\right) \end{align}</math>
where <math>s(q)=s\left(e^{\pi i\tau}\right)=-R\left(-e^{-\pi i/(5\tau)}\right)</math> is the Rogers–Ramanujan continued fraction:
- <math>\begin{align}
s(q) &= \sqrt[5]{\tan\left(\frac{1}{2}\arctan\left(\frac{5}{2}\frac{\varphi^2\left(q^5\right)}{\varphi^2(q)}-\frac{1}{2}\right)\right)\cot^2\left(\frac{1}{2}\operatorname{arccot}\left(\frac{5}{2}\frac{\varphi^2\left(q^5\right)}{\varphi^2(q)}-\frac{1}{2}\right)\right)}\\ &= \cfrac{e^{-\pi i/(25\tau)}}{1-\cfrac{e^{-\pi i/(5\tau)}}{1+\cfrac{e^{-2\pi i/(5\tau)}}{1-\ddots}}} \end{align}</math>
Equianharmonic valuesEdit
The mathematician Bruce Berndt found out further values<ref>Template:Cite journal</ref> of the theta function:
- <math>\begin{array}{lll}
\varphi\left(\exp( -\sqrt{3}\,\pi)\right) &=& \pi^{-1}{\Gamma\left(\tfrac{4}{3}\right)}^{3/2}2^{-2/3}3^{13/8} \\ \varphi\left(\exp(-2\sqrt{3}\,\pi)\right) &=& \pi^{-1}{\Gamma\left(\tfrac{4}{3}\right)}^{3/2}2^{-2/3}3^{13/8}\cos(\tfrac{1}{24}\pi) \\ \varphi\left(\exp(-3\sqrt{3}\,\pi)\right) &=& \pi^{-1}{\Gamma\left(\tfrac{4}{3}\right)}^{3/2}2^{-2/3}3^{7/8}(\sqrt[3]{2}+1) \\ \varphi\left(\exp(-4\sqrt{3}\,\pi)\right) &=& \pi^{-1}{\Gamma\left(\tfrac{4}{3}\right)}^{3/2}2^{-5/3}3^{13/8}\Bigl(1+\sqrt{\cos(\tfrac{1}{12}\pi)}\Bigr) \\ \varphi\left(\exp(-5\sqrt{3}\,\pi)\right) &=& \pi^{-1}{\Gamma\left(\tfrac{4}{3}\right)}^{3/2}2^{-2/3}3^{5/8}\sin(\tfrac{1}{5}\pi)(\tfrac{2}{5}\sqrt[3]{100}+\tfrac{2}{5}\sqrt[3]{10}+\tfrac{3}{5}\sqrt{5}+1) \end{array}</math>
Further valuesEdit
Many values of the theta function<ref>Template:Cite journal</ref> and especially of the shown phi function can be represented in terms of the gamma function:
- <math>\begin{array}{lll}
\varphi\left(\exp( -\sqrt{2}\,\pi)\right) &=& \pi^{-1/2}\Gamma\left(\tfrac{9}{8}\right){\Gamma\left(\tfrac{5}{4}\right)}^{-1/2}2^{7/8} \\ \varphi\left(\exp(-2\sqrt{2}\,\pi)\right) &=& \pi^{-1/2}\Gamma\left(\tfrac{9}{8}\right){\Gamma\left(\tfrac{5}{4}\right)}^{-1/2}2^{1/8}\Bigl(1+\sqrt{\sqrt{2}-1}\Bigr) \\ \varphi\left(\exp(-3\sqrt{2}\,\pi)\right) &=& \pi^{-1/2}\Gamma\left(\tfrac{9}{8}\right){\Gamma\left(\tfrac{5}{4}\right)}^{-1/2}2^{3/8}3^{-1/2}(\sqrt{3}+1)\sqrt{\tan(\tfrac{5}{24}\pi)} \\ \varphi\left(\exp(-4\sqrt{2}\,\pi)\right) &=& \pi^{-1/2}\Gamma\left(\tfrac{9}{8}\right){\Gamma\left(\tfrac{5}{4}\right)}^{-1/2}2^{-1/8}\Bigl(1+\sqrt[4]{2\sqrt{2}-2}\Bigr) \\ \varphi\left(\exp(-5\sqrt{2}\,\pi)\right) &=& \pi^{-1/2}\Gamma\left(\tfrac{9}{8}\right){\Gamma\left(\tfrac{5}{4}\right)}^{-1/2} \frac{1}{15}\,2^{3/8} \times \\
&& \times \biggl[\sqrt[3]{5}\,\sqrt{10+2\sqrt{5}}\biggl(\sqrt[3]{5+\sqrt{2}+3\sqrt{3}}+\sqrt[3]{5+\sqrt{2}-3\sqrt{3}}\,\biggr)-\bigl(2-\sqrt{2}\,\bigr)\sqrt{25-10\sqrt{5}}\,\biggr] \\
\varphi\left(\exp( -\sqrt{6}\,\pi)\right) &=& \pi^{-1/2}\Gamma\left(\tfrac{5}{24}\right){\Gamma\left(\tfrac{5}{12}\right)}^{-1/2}2^{-13/24}3^{-1/8}\sqrt{\sin(\tfrac{5}{12}\pi)} \\ \varphi\left(\exp(-\tfrac{1}{2}\sqrt{6}\,\pi)\right) &=& \pi^{-1/2}\Gamma\left(\tfrac{5}{24}\right){\Gamma\left(\tfrac{5}{12}\right)}^{-1/2}2^{5/24}3^{-1/8}\sin(\tfrac{5}{24}\pi) \end{array}</math>
Nome power theoremsEdit
Direct power theoremsEdit
For the transformation of the nome<ref>Andreas Dieckmann: Table of Infinite Products Infinite Sums Infinite Series, Elliptic Theta. Physikalisches Institut Universität Bonn, Abruf am 1. Oktober 2021.</ref> in the theta functions these formulas can be used:
- <math>\theta_{2}(q^2) = \tfrac{1}{2}\sqrt{2[\theta_{3}(q)^2 - \theta_{4}(q)^2]}</math>
- <math>\theta_{3}(q^2) = \tfrac{1}{2}\sqrt{2[\theta_{3}(q)^2 + \theta_{4}(q)^2]}</math>
- <math>\theta_{4}(q^2) = \sqrt{\theta_{4}(q)\theta_{3}(q)}</math>
The squares of the three theta zero-value functions with the square function as the inner function are also formed in the pattern of the Pythagorean triples according to the Jacobi Identity. Furthermore, those transformations are valid:
- <math>\theta_{3}(q^4) = \tfrac{1}{2}\theta_{3}(q) + \tfrac{1}{2}\theta_{4}(q)</math>
These formulas can be used to compute the theta values of the cube of the nome:
- <math>27\,\theta_{3}(q^3)^8 - 18\,\theta_{3}(q^3)^4\theta_{3}(q)^4 - \,\theta_{3}(q)^8 = 8\,\theta_{3}(q^3)^2\theta_{3}(q)^2[2\,\theta_{4}(q)^4 - \theta_{3}(q)^4]</math>
- <math>27\,\theta_{4}(q^3)^8 - 18\,\theta_{4}(q^3)^4\theta_{4}(q)^4 - \,\theta_{4}(q)^8 = 8\,\theta_{4}(q^3)^2\theta_{4}(q)^2[2\,\theta_{3}(q)^4 - \theta_{4}(q)^4]</math>
And the following formulas can be used to compute the theta values of the fifth power of the nome:
- <math>[\theta_{3}(q)^2 - \theta_{3}(q^5)^2][5\,\theta_{3}(q^5)^2 - \theta_{3}(q)^2]^5 = 256\,\theta_{3}(q^5)^2\theta_{3}(q)^2\theta_{4}(q)^4 [\theta_{3}(q)^4 - \theta_{4}(q)^4]</math>
- <math>[\theta_{4}(q^5)^2 - \theta_{4}(q)^2][5\,\theta_{4}(q^5)^2 - \theta_{4}(q)^2]^5 = 256\,\theta_{4}(q^5)^2\theta_{4}(q)^2\theta_{3}(q)^4 [\theta_{3}(q)^4 - \theta_{4}(q)^4]</math>
Transformation at the cube root of the nomeEdit
The formulas for the theta Nullwert function values from the cube root of the elliptic nome are obtained by contrasting the two real solutions of the corresponding quartic equations:
- <math>\biggl[\frac{\theta_{3}(q^{1/3})^2}{\theta_{3}(q)^2} - \frac{3\,\theta_{3}(q^{3})^2}{\theta_{3}(q)^2}\biggr]^2 = 4 - 4\biggl[\frac{2\,\theta_{2}(q)^2 \theta_{4}(q)^2}{\theta_{3}(q)^4}\biggr]^{2/3} </math>
- <math>\biggl[\frac{3\,\theta_{4}(q^{3})^2}{\theta_{4}(q)^2} - \frac{\theta_{4}(q^{1/3})^2}{\theta_{4}(q)^2}\biggr]^2 = 4 + 4\biggl[\frac{2\,\theta_{2}(q)^2 \theta_{3}(q)^2}{\theta_{4}(q)^4}\biggr]^{2/3} </math>
Transformation at the fifth root of the nomeEdit
The Rogers-Ramanujan continued fraction can be defined in terms of the Jacobi theta function in the following way:
- <math>R(q) = \tan\biggl\{\frac{1}{2}\arctan\biggl[\frac{1}{2} - \frac{\theta _{4}(q)^2}{2\,\theta_{4}(q^5)^2}\biggr]\biggr\}^{1/5} \tan\biggl\{\frac{1}{2}\arccot\biggl[\frac{1}{2} - \frac{\theta_{4}(q)^2}{2\,\theta_{4}(q^5)^2}\biggr]\biggr\}^{2/5} </math>
- <math>R(q^2) = \tan\biggl\{\frac{1}{2}\arctan\biggl[\frac{1}{2} - \frac{\theta_{4}(q)^2}{2\,\theta_{4}(q^5)^2}\biggr]\biggr\}^{2/5} \cot\biggl\{\frac{1}{2}\arccot\biggl[\frac{1}{2} - \frac{\theta_{4}(q)^2}{2\,\theta_{4}(q^5)^2}\biggr]\biggr\}^{1/5} </math>
- <math>R(q^2) = \tan\biggl\{\frac{1}{2}\arctan\biggl[\frac{\theta_{3}(q)^2}{2\,\theta_{3}(q^5)^2} - \frac{1}{2}\biggr]\biggr\}^{2/5} \tan\biggl\{\frac{1}{2}\arccot\biggl[\frac{\theta_{3}(q)^2}{2\,\theta_{3}(q^5)^2} - \frac{1}{2}\biggr]\biggr\}^{1/5} </math>
The alternating Rogers-Ramanujan continued fraction function S(q) has the following two identities:
- <math>S(q) = \frac{R(q^4)}{R(q^2)R(q)} = \tan\biggl\{\frac{1}{2}\arctan\biggl [\frac{\theta_{3}(q)^2}{2\,\theta_{3}(q^5)^2} - \frac{1}{2}\biggr]\biggr\}^{1/5} \cot\biggl\{\frac{1}{2}\arccot\biggl[\frac{\theta_{3}(q)^2}{2\,\theta_{3}(q^5)^2} - \frac{1}{2}\biggr]\biggr\}^{2/5}</math>
The theta function values from the fifth root of the nome can be represented as a rational combination of the continued fractions R and S and the theta function values from the fifth power of the nome and the nome itself. The following four equations are valid for all values q between 0 and 1:
- <math>\frac{\theta_{3}(q^{1/5})}{\theta_{3}(q^5)} - 1 = \frac{1}{S(q)}\bigl[S(q)^2 + R(q^2)\bigr]\bigl[1 + R(q^2)S(q)\bigr] </math>
- <math>1 - \frac{\theta_{4}(q^{1/5})}{\theta_{4}(q^5)} = \frac{1}{R(q)}\bigl[R(q^2) + R(q)^2\bigr]\bigl[1 - R(q^2)R(q)\bigr] </math>
- <math>\theta_{3}(q^{1/5})^2 - \theta_{3}(q)^2 = \bigl[\theta_{3}(q)^2 - \theta_{3}(q^5)^2\bigr]\biggl[1+\frac{1}{R(q^2)S(q)}+R(q^2)S(q)+\frac{1}{R(q^2)^2}+R(q^2)^2+\frac{1}{S(q)}-S(q)\biggr] </math>
- <math>\theta_{4}(q)^2 - \theta_{4}(q^{1/5})^2 = \bigl[\theta_{4}(q^5)^2 - \theta_{4}(q)^2\bigr]\biggl[1-\frac{1}{R(q^2)R(q)}-R(q^2)R(q)+\frac{1}{R(q^2)^2}+R(q^2)^2-\frac{1}{R(q)}+R(q)\biggr] </math>
Modulus dependent theoremsEdit
In combination with the elliptic modulus, the following formulas can be displayed:
These are the formulas for the square of the elliptic nome:
- <math>\theta_{4}[q(k)] = \theta_{4}[q(k)^2]\sqrt[8]{1 - k^2}</math>
- <math>\theta_{4}[q(k)^2] = \theta_{3}[q(k)]\sqrt[8]{1 - k^2}</math>
- <math>\theta_{3}[q(k)^2] = \theta_{3}[q(k)]\cos[\tfrac{1}{2}\arcsin(k)]</math>
And this is an efficient formula for the cube of the nome:
- <math> \theta_{4}\biggl\langle q\bigl\{\tan\bigl[\tfrac{1}{2}\arctan(t^3)\bigr]\bigr\}^3 \biggr\rangle =
\theta_{4}\biggl\langle q\bigl\{\tan\bigl[\tfrac{1}{2}\arctan(t^3)\bigr]\bigr\} \biggr\rangle \,3^{-1/2} \bigl(\sqrt{2\sqrt{t^4 - t^2 + 1} - t^2 + 2} + \sqrt{t^2 + 1}\,\bigr)^{1/2} </math>
For all real values <math> t \in \R </math> the now mentioned formula is valid.
And for this formula two examples shall be given:
First calculation example with the value <math> t = 1 </math> inserted:
<math> \theta_{4}\biggl\langle q\bigl\{\tan\bigl[\tfrac{1}{2}\arctan(1)\bigr]\bigr\}^3 \biggr\rangle = \theta_{4}\biggl\langle q\bigl\{\tan\bigl[\tfrac{1}{2}\arctan(1)\bigr]\bigr\} \biggr\rangle \,3^{-1/2} \bigl(\sqrt{3} + \sqrt{2}\,\bigr)^{1/2} </math>
<math> \theta_{4}\bigl[\exp(-3\sqrt{2}\,\pi)\bigr] = \theta_{4}\bigl[\exp(-\sqrt{2}\,\pi)\bigr] \,3^{-1/2} \bigl(\sqrt{3} + \sqrt{2}\,\bigr)^{1/2} </math>
Second calculation example with the value <math> t = \Phi^{-2} </math> inserted:
<math> \theta_{4}\biggl\langle q\bigl\{\tan\bigl[\tfrac{1}{2}\arctan(\Phi^{-6})\bigr]\bigr\}^3 \biggr\rangle = \theta_{4}\biggl\langle q\bigl\{\tan\bigl[\tfrac{1}{2}\arctan(\Phi^{-6})\bigr]\bigr\} \biggr\rangle \,3^{-1/2} \bigl(\sqrt{2\sqrt{\Phi^{-8} - \Phi^{-4} + 1} - \Phi^{-4} + 2} + \sqrt{\Phi^{-4} + 1}\,\bigr)^{1/2} </math>
<math> \theta_{4}\bigl[\exp(-3\sqrt{10}\,\pi)\bigr] = \theta_{4}\bigl[\exp(-\sqrt{10}\,\pi)\bigr] \,3^{-1/2} \bigl(\sqrt{2\sqrt{\Phi^{-8} - \Phi^{-4} + 1} - \Phi^{-4} + 2} + \sqrt{\Phi^{-4} + 1}\,\bigr)^{1/2} </math>
The constant <math> \Phi </math> represents the Golden ratio number <math> \Phi = \tfrac{1}{2}(\sqrt{5} + 1)</math> exactly.
Some series identitiesEdit
Sums with theta function in the resultEdit
The infinite sum<ref>Landau (1899) zitiert nach Borwein, Page 94, Exercise 3.</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> of the reciprocals of Fibonacci numbers with odd indices has the identity:
- <math>\sum_{n=1}^\infty \frac{1}{F_{2n-1}} = \frac{\sqrt{5}}{2}\,\sum_{n=1}^\infty \frac{2(\Phi^{-2})^{n - 1/2}}{1 + (\Phi^{-2})^{2n - 1}} = \frac{\sqrt{5}}{4} \sum_{a=-\infty}^\infty \frac{2(\Phi^{-2})^{a - 1/2}}{1 + (\Phi^{-2})^{2a - 1}} =</math>
- <math>= \frac{\sqrt{5}}{4}\,\theta_{2}(\Phi^{-2})^2 = \frac{\sqrt{5}}{8}\bigl[\theta_{3}(\Phi^{-1})^2 - \theta_{4}(\Phi^{-1})^2\bigr]</math>
By not using the theta function expression, following identity between two sums can be formulated:
- <math>\sum_{n=1}^\infty \frac{1}{F_{2n-1}} = \frac{\sqrt{5}}{4}\,\biggl[ \sum_{n=1}^\infty 2 \,\Phi^{- (2n - 1)^2 /2} \biggr]^2 </math>
- <math>\sum_{n=1}^\infty \frac{1}{F_{2n-1}} = 1.82451515740692456814215840626732817332\ldots </math>
Also in this case <math> \Phi = \tfrac{1}{2}(\sqrt{5} + 1)</math> is Golden ratio number again.
Infinite sum of the reciprocals of the Fibonacci number squares:
- <math>\sum_{n=1}^\infty \frac{1}{F_{n}^2} = \frac{5}{24}\bigl[2\,\theta_{2}(\Phi^{-2})^4 - \theta_{3}(\Phi^{-2})^4 + 1\bigr] = \frac{5}{24}\bigl[\theta_{3}(\Phi^{-2})^4 - 2\,\theta_{4}(\Phi^{-2})^4 + 1\bigr]</math>
Infinite sum of the reciprocals of the Pell numbers with odd indices:
- <math>\sum_{n=1}^\infty \frac{1}{P_{2n-1}} = \frac{1}{\sqrt{2}}\,\theta_{2}\bigl[(\sqrt{2}-1)^2\bigr]^2 = \frac{1}{2\sqrt{2}}\bigl[\theta_{3}(\sqrt{2}-1)^2 - \theta_{4}(\sqrt{2}-1)^2\bigr]</math>
Sums with theta function in the summandEdit
The next two series identities were proved by István Mező:<ref name="Mezo2">Template:Citation</ref>
- <math>\begin{align}
\theta_4^2(q)&=iq^{\frac14}\sum_{k=-\infty}^\infty q^{2k^2-k}\theta_1\left(\frac{2k-1}{2i}\ln q,q\right),\\[6pt] \theta_4^2(q)&=\sum_{k=-\infty}^\infty q^{2k^2}\theta_4\left(\frac{k\ln q}{i},q\right). \end{align}</math>
These relations hold for all Template:Math. Specializing the values of Template:Mvar, we have the next parameter free sums
- <math>\sqrt{\frac{\pi\sqrt{e^\pi}}{2}}\cdot\frac{1}{\Gamma^2\left(\frac34\right)} =i\sum_{k=-\infty}^\infty e^{\pi\left(k-2k^2\right)} \theta_1 \left(\frac{i\pi}{2}(2k-1),e^{-\pi}\right)</math>
- <math>\sqrt{\frac{\pi}{2}}\cdot\frac{1}{\Gamma^2\left(\frac34\right)} =\sum_{k=-\infty}^\infty\frac{\theta_4\left(ik\pi,e^{-\pi}\right)}{e^{2\pi k^2}}</math>
Zeros of the Jacobi theta functionsEdit
All zeros of the Jacobi theta functions are simple zeros and are given by the following:
- <math>\begin{align}
\vartheta(z;\tau) = \vartheta_{00}(z;\tau) &= 0 \quad &\Longleftrightarrow&& \quad z &= m + n \tau + \frac{1}{2} + \frac{\tau}{2} \\[3pt] \vartheta_{11}(z;\tau) &= 0 \quad &\Longleftrightarrow&& \quad z &= m + n \tau \\[3pt] \vartheta_{10}(z;\tau) &= 0 \quad &\Longleftrightarrow&& \quad z &= m + n \tau + \frac{1}{2} \\[3pt] \vartheta_{01}(z;\tau) &= 0 \quad &\Longleftrightarrow&& \quad z &= m + n \tau + \frac{\tau}{2} \end{align}</math> where Template:Mvar, Template:Mvar are arbitrary integers.
Relation to the Riemann zeta functionEdit
The relation
- <math>\vartheta\left(0;-\frac{1}{\tau}\right)=\left(-i\tau\right)^\frac12 \vartheta(0;\tau)</math>
was used by Riemann to prove the functional equation for the Riemann zeta function, by means of the Mellin transform
- <math>\Gamma\left(\frac{s}{2}\right) \pi^{-\frac{s}{2}} \zeta(s) = \frac{1}{2}\int_0^\infty\bigl(\vartheta(0;it)-1\bigr)t^\frac{s}{2}\frac{\mathrm{d}t}{t}</math>
which can be shown to be invariant under substitution of Template:Mvar by Template:Math. The corresponding integral for Template:Math is given in the article on the Hurwitz zeta function.
Relation to the Weierstrass elliptic functionEdit
The theta function was used by Jacobi to construct (in a form adapted to easy calculation) his elliptic functions as the quotients of the above four theta functions, and could have been used by him to construct Weierstrass's elliptic functions also, since
- <math>\wp(z;\tau) = -\big(\log \vartheta_{11}(z;\tau)\big) + c</math>
where the second derivative is with respect to Template:Mvar and the constant Template:Mvar is defined so that the Laurent expansion of Template:Math at Template:Math has zero constant term.
Relation to the q-gamma functionEdit
The fourth theta function – and thus the others too – is intimately connected to the [[q-gamma function|Jackson Template:Mvar-gamma function]] via the relation<ref name = 'Mezo'>Template:Cite journal</ref>
- <math>\left(\Gamma_{q^2}(x)\Gamma_{q^2}(1-x)\right)^{-1}=\frac{q^{2x(1-x)}}{\left(q^{-2};q^{-2}\right)^3_\infty\left(q^2-1\right)} \theta_4\left(\frac{1}{2i}(1-2x)\log q,\frac{1}{q}\right). </math>
Relations to Dedekind eta functionEdit
Let Template:Math be the Dedekind eta function, and the argument of the theta function as the nome Template:Math. Then,
- <math>\begin{align}
\theta_2(q) = \vartheta_{10}(0;\tau) &= \frac{2\eta^2(2\tau)}{\eta(\tau)},\\[3pt] \theta_3(q) = \vartheta_{00}(0;\tau) &= \frac{\eta^5(\tau)}{\eta^2\left(\frac{1}{2}\tau\right)\eta^2(2\tau)} = \frac{\eta^2\left(\frac{1}{2}(\tau+1)\right)}{\eta(\tau+1)}, \\[3pt] \theta_4(q) = \vartheta_{01}(0;\tau) &= \frac{\eta^2\left(\frac{1}{2}\tau\right)}{\eta(\tau)}, \end{align}</math>
and,
- <math>\theta_2(q)\,\theta_3(q)\,\theta_4(q) = 2\eta^3(\tau).</math>
See also the Weber modular functions.
Elliptic modulusEdit
The elliptic modulus is
- <math>k(\tau) = \frac{\vartheta_{10}(0;\tau)^2 }{\vartheta_{00}(0;\tau)^2} </math>
and the complementary elliptic modulus is
- <math>k'(\tau) = \frac{\vartheta_{01}(0;\tau)^2 }{\vartheta_{00}(0;\tau)^2} </math>
Derivatives of theta functionsEdit
These are two identical definitions of the complete elliptic integral of the second kind:
- <math>E(k) = \int_{0}^{\pi/2} \sqrt{1 - k^2\sin(\varphi)^2} d\varphi</math>
- <math>E(k) = \frac{\pi}{2}\sum_{a = 0}^{\infty} \frac{[(2a)!]^2}{(1 - 2a)16^{a}(a!)^4} k^{2a}</math>
The derivatives of the Theta Nullwert functions have these MacLaurin series:
- <math>\theta_{2}'(x) = \frac{\mathrm{d}}{\mathrm{d}x}\,\theta_{2}(x) = \frac{1}{2} x^{-3/4}+\sum_{n = 1}^{\infty} \frac{1}{2}(2n + 1)^2 x^{(2n-1)(2n+3)/4}</math>
- <math>\theta_{3}'(x) = \frac{\mathrm{d}}{\mathrm{d}x}\,\theta_{3}(x) = 2+\sum_{n = 1}^{\infty} 2(n + 1)^2 x^{n(n+2)}</math>
- <math>\theta_{4}'(x) = \frac{\mathrm{d}}{\mathrm{d}x}\,\theta_{4}(x) = -2+\sum_{n = 1}^{\infty} 2(n + 1)^2 (-1)^{n+1} x^{n(n+2)}</math>
The derivatives of theta zero-value functions<ref>{{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web |_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:EllipticAlphaFunction%7CEllipticAlphaFunction.html}} |title = Elliptic Alpha Function |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}</ref> are as follows:
- <math>\theta_{2}'(x) = \frac{\mathrm{d}}{\mathrm{d}x} \,\theta_{2}(x) = \frac{1}{2\pi x} \theta_{2}(x)\theta_{3}(x)^2 E\biggl[\frac{\theta_{2}(x)^2}{\theta_{3}(x)^2}\biggr]</math>
- <math>\theta_{3}'(x) = \frac{\mathrm{d}}{\mathrm{d}x} \,\theta_{3}(x) = \theta_{3}(x)\bigl[\theta_{3}(x)^2 + \theta_{4}(x)^2\bigr]\biggl\{\frac{1}{2\pi x}E\biggl[\frac{\theta_{3}(x)^2 - \theta_{4}(x)^2}{\theta_{3}(x)^2 + \theta_{4}(x)^2}\biggr] - \frac{\theta_{4}(x)^2}{4\,x}\biggr\}</math>
- <math>\theta_{4}'(x) = \frac{\mathrm{d}}{\mathrm{d}x} \,\theta_{4}(x) = \theta_{4}(x)\bigl[\theta_{3}(x)^2 + \theta_{4}(x)^2\bigr]\biggl\{\frac{1}{2\pi x}E\biggl[\frac{\theta_{3}(x)^2 - \theta _{4}(x)^2}{\theta_{3}(x)^2+\theta_{4}(x)^2}\biggr] - \frac{\theta _{3}(x)^ 2}{4\,x}\biggr\}</math>
The two last mentioned formulas are valid for all real numbers of the real definition interval: <math> -1 < x < 1 \,\cap \,x \in \R </math>
And these two last named theta derivative functions are related to each other in this way:
- <math>\vartheta _{4}(x)\biggl[\frac{\mathrm{d}}{\mathrm{d}x} \,\vartheta _{3}(x)\biggr] - \vartheta _{3}(x)\biggl[\frac{\mathrm{d}}{\mathrm{d}x} \,\theta _{4}(x)\biggr] = \frac{1}{4\,x}\,\theta_{3}(x)\,\theta_{4}(x)\bigl[\theta_{3}(x)^4 - \theta_{4}(x)^4\bigr] </math>
The derivatives of the quotients from two of the three theta functions mentioned here always have a rational relationship to those three functions:
- <math>\frac{\mathrm{d}}{\mathrm{d}x} \,\frac{\theta _{2}(x)}{\theta _{3}(x)} = \frac{\theta_{2}(x)\,\theta _{4}(x)^4}{4\,x\,\theta _{3}(x)}</math>
- <math>\frac{\mathrm{d}}{\mathrm{d}x} \,\frac{\theta _{2}(x)}{\theta _{4}(x)} = \frac{\theta_{2}(x)\,\theta _{3}(x)^4}{4\,x\,\theta _{4}(x)}</math>
- <math>\frac{\mathrm{d}}{\mathrm{d}x} \,\frac{\theta _{3}(x)}{\theta _{4}(x)} = \frac{\theta_{3}(x)^5 - \theta _{3}(x)\,\theta _{4}(x)^4}{4\,x\,\theta _{4}(x)}</math>
For the derivation of these derivation formulas see the articles Nome (mathematics) and Modular lambda function!
Integrals of theta functionsEdit
For the theta functions these integrals<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> are valid:
- <math>\int_{0}^{1} \theta_{2}(x) \,\mathrm{d}x = \sum _{k = -\infty}^{\infty} \frac{4}{ (2k+1)^2+4} = \pi\tanh(\pi) \approx 3.129881</math>
- <math>\int_{0}^{1} \theta_{3}(x) \,\mathrm{d}x = \sum _{k = -\infty}^{\infty} \frac{1}{ k^2+1} = \pi\coth(\pi) \approx 3.153348</math>
- <math>\int_{0}^{1} \theta_{4}(x) \,\mathrm{d}x = \sum _{k = -\infty}^{\infty} \frac{(-1 )^{k}}{k^2+1} = \pi\,\operatorname{csch}(\pi) \approx 0.272029</math>
The final results now shown are based on the general Cauchy sum formulas.
A solution to the heat equationEdit
The Jacobi theta function is the fundamental solution of the one-dimensional heat equation with spatially periodic boundary conditions.<ref>Template:Cite journal</ref> Taking Template:Math to be real and Template:Math with Template:Mvar real and positive, we can write
- <math>\vartheta (x;it)=1+2\sum_{n=1}^\infty \exp\left(-\pi n^2 t\right) \cos(2\pi nx)</math>
which solves the heat equation
- <math>\frac{\partial}{\partial t} \vartheta(x;it)=\frac{1}{4\pi} \frac{\partial^2}{\partial x^2} \vartheta(x;it).</math>
This theta-function solution is 1-periodic in Template:Mvar, and as Template:Math it approaches the periodic delta function, or Dirac comb, in the sense of distributions
- <math>\lim_{t\to 0} \vartheta(x;it)=\sum_{n=-\infty}^\infty \delta(x-n)</math>.
General solutions of the spatially periodic initial value problem for the heat equation may be obtained by convolving the initial data at Template:Math with the theta function.
Relation to the Heisenberg groupEdit
The Jacobi theta function is invariant under the action of a discrete subgroup of the Heisenberg group. This invariance is presented in the article on the theta representation of the Heisenberg group.
GeneralizationsEdit
If Template:Mvar is a quadratic form in Template:Mvar variables, then the theta function associated with Template:Mvar is
- <math>\theta_F (z)= \sum_{m\in \Z^n} e^{2\pi izF(m)}</math>
with the sum extending over the lattice of integers <math>\mathbb{Z}^n</math>. This theta function is a modular form of weight Template:Math (on an appropriately defined subgroup) of the modular group. In the Fourier expansion,
- <math>\hat{\theta}_F (z) = \sum_{k=0}^\infty R_F(k) e^{2\pi ikz},</math>
the numbers Template:Math are called the representation numbers of the form.
Theta series of a Dirichlet characterEdit
For Template:Mvar a primitive Dirichlet character modulo Template:Mvar and Template:Math then
- <math>\theta_\chi(z) = \frac12\sum_{n=-\infty}^\infty \chi(n) n^\nu e^{2i \pi n^2 z}</math>
is a weight Template:Math modular form of level Template:Math and character
- <math>\chi(d) \left(\frac{-1}{d}\right)^\nu,</math>
which means<ref>Shimura, On modular forms of half integral weight</ref>
- <math>\theta_\chi\left(\frac{az+b}{cz+d}\right) = \chi(d) \left(\frac{-1}{d}\right)^\nu \left(\frac{\theta_1\left(\frac{az+b}{cz+d}\right)}{\theta_1(z)}\right)^{1+2\nu}\theta_\chi(z)</math>
whenever
- <math>a,b,c,d\in \Z^4, ad-bc=1,c \equiv 0 \bmod 4 q^2.</math>
Ramanujan theta functionEdit
Riemann theta functionEdit
Let
- <math>\mathbb{H}_n=\left\{F\in M(n,\Complex) \,\big|\, F=F^\mathsf{T} \,,\, \operatorname{Im} F >0 \right\}</math>
be the set of symmetric square matrices whose imaginary part is positive definite. <math>\mathbb{H}_n</math> is called the Siegel upper half-space and is the multi-dimensional analog of the upper half-plane. The Template:Mvar-dimensional analogue of the modular group is the symplectic group Template:Math; for Template:Math, Template:Math. The Template:Mvar-dimensional analogue of the congruence subgroups is played by
- <math>\ker \big\{\operatorname{Sp}(2n,\Z)\to \operatorname{Sp}(2n,\Z/k\Z) \big\}.</math>
Then, given Template:Math, the Riemann theta function is defined as
- <math>\theta (z,\tau)=\sum_{m\in \Z^n} \exp\left(2\pi i \left(\tfrac12 m^\mathsf{T} \tau m +m^\mathsf{T} z \right)\right). </math>
Here, Template:Math is an Template:Mvar-dimensional complex vector, and the superscript T denotes the transpose. The Jacobi theta function is then a special case, with Template:Math and Template:Math where Template:Math is the upper half-plane. One major application of the Riemann theta function is that it allows one to give explicit formulas for meromorphic functions on compact Riemann surfaces, as well as other auxiliary objects that figure prominently in their function theory, by taking Template:Mvar to be the period matrix with respect to a canonical basis for its first homology group.
The Riemann theta converges absolutely and uniformly on compact subsets of <math>\mathbb{C}^n \times \mathbb{H}_n</math>.
The functional equation is
- <math>\theta (z+a+\tau b, \tau) = \exp\left( 2\pi i \left(-b^\mathsf{T}z-\tfrac12 b^\mathsf{T}\tau b\right)\right) \theta (z,\tau)</math>
which holds for all vectors Template:Math, and for all Template:Math and Template:Math.
Poincaré seriesEdit
The Poincaré series generalizes the theta series to automorphic forms with respect to arbitrary Fuchsian groups.
Derivation of the theta valuesEdit
Identity of the Euler beta functionEdit
In the following, three important theta function values are to be derived as examples:
This is how the Euler beta function is defined in its reduced form:
- <math>\beta(x) = \frac{\Gamma(x)^2}{\Gamma(2x)}</math>
In general, for all natural numbers <math>n \isin \mathbb{N}</math> this formula of the Euler beta function is valid:
- <math>\frac{4^{-1/(n + 2)}}{n + 2}\csc\bigl(\frac{\pi}{n + 2}\bigr)\beta\biggl[\frac{n}{2(n + 2)}\biggr] = \int_{0}^{\infty} \frac{1}{\sqrt{x^{n+2} + 1}} \,\mathrm {d}x</math>
Exemplary elliptic integralsEdit
In the following some Elliptic Integral Singular Values<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> are derived:
|
The ensuing function has the following lemniscatically elliptic antiderivative:
For the value <math>n = 2</math> this identity appears:
This result follows from that equation chain:
|
|
The following function has the following equianharmonic elliptic antiderivative:
For the value <math>n = 4</math> that identity appears:
This result follows from that equation chain:
|
|
And the following function has the following elliptic antiderivative:
For the value <math>n = 6</math> the following identity appears:
This result follows from that equation chain:
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Combination of the integral identities with the nomeEdit
The elliptic nome function has these important values:
- <math>q(\tfrac{1}{2}\sqrt{2}) = \exp(-\pi)</math>
- <math>q[\tfrac{1}{4}(\sqrt{6} - \sqrt{2})] = \exp(-\sqrt{3}\,\pi)</math>
- <math>q(\sqrt{2} - 1) = \exp(-\sqrt{2}\,\pi)</math>
For the proof of the correctness of these nome values, see the article Nome (mathematics)!
On the basis of these integral identities and the above-mentioned Definition and identities to the theta functions in the same section of this article, exemplary theta zero values shall be determined now:
<math>\theta_{3}[q(k)] = \sqrt{2\pi^{-1} K(k)}</math>
- <math>\theta_{3}[\exp(-\pi)] = \theta_{3}[q(\tfrac{1}{2}\sqrt{2})] = \sqrt{2\pi^ {-1}K(\tfrac{1}{2}\sqrt{2})} = 2^{-1/2}\pi^{-1/2}\beta(\tfrac{1}{4} )^{1/2} = 2^{-1/4}\sqrt[4]{\pi}\,{\Gamma\bigl(\tfrac{3}{4}\bigr)}^{-1} </math>
- <math>\theta _{3}[\exp(-\sqrt{3}\,\pi )] = \theta _{3}\bigl\{q\bigl[\tfrac{1}{4}(\sqrt {6} - \sqrt{2})\bigr]\bigr\} = \sqrt{2\pi^{-1}K\bigl[\tfrac{1}{4}(\sqrt{6} - \sqrt {2})\bigr]} = 2^{-1/6}3^{-1/8}\pi^{-1/2}\beta(\tfrac{1}{3})^{1/ 2}</math>
- <math>\theta _{3}[\exp(-\sqrt{2}\,\pi )] = \theta _{3}[q(\sqrt{2} - 1)] = \sqrt{2\pi ^{-1}K(\sqrt{2} - 1)} = 2^{-1/8}\cos(\tfrac{1}{8}\pi)\,\pi^{-1/2} \beta(\tfrac{3}{8})^{1/2}</math>
<math>\theta_{4}[q(k)] = \sqrt[4]{1 - k^2}\,\sqrt{2\pi^{-1} K(k)}</math>
- <math>\theta_{4}[\exp(-\sqrt{2}\,\pi)] = \theta_{4}[q(\sqrt{2} - 1)] = \sqrt[4]{ 2\sqrt{2} - 2}\,\sqrt{2\pi^{-1}K(\sqrt{2} - 1)} = 2^{-1/4}\cos(\tfrac{1} {8}\pi)^{1/2}\,\pi^{-1/2}\beta(\tfrac{3}{8})^{1/2}</math>
Partition sequences and Pochhammer productsEdit
Regular partition number sequenceEdit
The regular partition sequence <math>P(n)</math> itself indicates the number of ways in which a positive integer number <math>n</math> can be split into positive integer summands. For the numbers <math>n = 1</math> to <math>n = 5</math>, the associated partition numbers <math>P</math> with all associated number partitions are listed in the following table:
| n | P(n) | paying partitions |
|---|---|---|
| 0 | 1 | () empty partition/empty sum |
| 1 | 1 | (1) |
| 2 | 2 | (1+1), (2) |
| 3 | 3 | (1+1+1), (1+2), (3) |
| 4 | 5 | (1+1+1+1), (1+1+2), (2+2), (1+3), (4) |
| 5 | 7 | (1+1+1+1+1), (1+1+1+2), (1+2+2), (1+1+3), (2+3), (1+4), (5) |
The generating function of the regular partition number sequence can be represented via Pochhammer product in the following way:
- <math>\sum _{k = 0}^\infty P(k)x^k = \frac{1}{(x;x)_{\infty}} = \theta_{3}(x)^{-1/6}\theta_{4}(x)^{-2/3} \biggl[\frac{\theta_{3}(x)^4 - \theta_{4}(x)^4}{16\,x}\biggr]^{-1/24}</math>
The summandization of the now mentioned Pochhammer product is described by the Pentagonal number theorem in this way:
- <math>(x;x)_{\infty} = 1 + \sum_{n = 1}^{\infty} \bigl[- x^{\text{Fn}(2n-1)} - x^ {\text{Kr}(2n-1)} + x^{\text{Fn}(2n)} + x^{\text{Kr}(2n)}\bigr]</math>
The following basic definitions apply to the pentagonal numbers and the card house numbers:
- <math>\text{Fn}(z) = \tfrac{1}{2}z(3z-1)</math>
- <math>\text{Kr}(z) = \tfrac{1}{2}z(3z+1)</math>
As a further application<ref>Ramanujan's theta-function identities involving Lambert series</ref> one obtains a formula for the third power of the Euler product:
- <math>(x;x)^3 = \prod_{n=1}^\infty (1-x^n)^3 = \sum _{m=0}^\infty (-1)^m(2m +1)x^{m(m+1)/2}</math>
Strict partition number sequenceEdit
And the strict partition sequence <math>Q(n)</math> indicates the number of ways in which such a positive integer number <math>n</math> can be splitted into positive integer summands such that each summand appears at most once<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> and no summand value occurs repeatedly. Exactly the same sequence<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> is also generated if in the partition only odd summands are included, but these odd summands may occur more than once. Both representations for the strict partition number sequence are compared in the following table:
| n | Q(n) | Number partitions without repeated summands | Number partitions with only odd addends |
|---|---|---|---|
| 0 | 1 | () empty partition/empty sum | () empty partition/empty sum |
| 1 | 1 | (1) | (1) |
| 2 | 1 | (2) | (1+1) |
| 3 | 2 | (1+2), (3) | (1+1+1), (3) |
| 4 | 2 | (1+3), (4) | (1+1+1+1), (1+3) |
| 5 | 3 | (2+3), (1+4), (5) | (1+1+1+1+1), (1+1+3), (5) |
| 6 | 4 | (1+2+3), (2+4), (1+5), (6) | (1+1+1+1+1+1), (1+1+1+3), (3+3), (1+5) |
| 7 | 5 | (1+2+4), (3+4), (2+5), (1+6), (7) | (1+1+1+1+1+1+1), (1+1+1+1+3), (1+3+3), (1+1+5), (7) |
| 8 | 6 | (1+3+4), (1+2+5), (3+5), (2+6), (1+7), (8) | (1+1+1+1+1+1+1+1), (1+1+1+1+1+3), (1+1+3+3), (1+1+1+ 5), (3+5), (1+7) |
The generating function of the strict partition number sequence can be represented using Pochhammer's product:
- <math>\sum _{k = 0}^\infty Q(k)x^k = \frac{1}{(x;x^2)_{\infty }} = \theta_{3}(x)^{1/6}\theta_{4}(x)^{-1/3} \biggl[\frac{\theta_{3}(x)^4 - \theta_{4}(x)^4}{16\,x}\biggr]^{1/24}</math>
Overpartition number sequenceEdit
The Maclaurin series for the reciprocal of the function Template:Math has the numbers of over partition sequence as coefficients with a positive sign:<ref>Template:Cite journal</ref>
- <math>\frac{1}{\theta_{4}(x)} = \prod_{n=1}^{\infty} \frac{1 + x^{n}}{1 - x^{n }} = \sum_{k=0}^{\infty} \overline{P}(k)x^{k}</math>
- <math>\frac{1}{\theta_{4}(x)} = 1+2x+4x^2+8x^3+14x^4+24x^5+40x^6+64x^7+100x^ 8+154x^9+232x^{10} + \dots</math>
If, for a given number <math>k</math>, all partitions are set up in such a way that the summand size never increases, and all those summands that do not have a summand of the same size to the left of themselves can be marked for each partition of this type, then it will be the resulting number<ref>Template:Cite journal</ref> of the marked partitions depending on <math>k</math> by the overpartition function <math>\overline{P}(k)</math> .
First example:
- <math>\overline{P}(4) = 14</math>
These 14 possibilities of partition markings exist for the sum 4:
| (4), (4), (3+1), (3+1), (3+1), (3+1), (2+2), (2+2), (2+1+1), (2+1+1), (2+1+1), (2+1+1), (1+1+1+1), (1+1+1+1) |
Second example:
- <math>\overline{P}(5) = 24</math>
These 24 possibilities of partition markings exist for the sum 5:
| (5), (5), (4+1), (4+1), (4+1), (4+1), (3+2), (3+2), (3+2), (3+2), (3+1+1), (3+1+1), (3+1+1), (3+1+1), (2+2+1), (2+2+1), (2+2+1), (2+2+1),
(2+1+1+1), (2+1+1+1), (2+1+1+1), (2+1+1+1), (1+1+1+1+1), (1+1+1+1+1) |
Relations of the partition number sequences to each otherEdit
In the Online Encyclopedia of Integer Sequences (OEIS), the sequence of regular partition numbers <math>P(n)</math> is under the code A000041, the sequence of strict partitions is <math> Q(n)</math> under the code A000009 and the sequence of superpartitions <math>\overline{P}(n)</math> under the code A015128. All parent partitions from index <math>n = 1</math> are even.
The sequence of superpartitions <math>\overline{P}(n)</math> can be written with the regular partition sequence P<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> and the strict partition sequence Q<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> can be generated like this:
- <math>\overline{P}(n) = \sum_{k=0}^{n} P(n - k)Q(k)</math>
In the following table of sequences of numbers, this formula should be used as an example:
| n | P(n) | Q(n) | <math>\overline{P}(n)</math> |
|---|---|---|---|
| 0 | 1 | 1 | 1 = 1*1 |
| 1 | 1 | 1 | 2 = 1 * 1 + 1 * 1 |
| 2 | 2 | 1 | 4 = 2 * 1 + 1 * 1 + 1 * 1 |
| 3 | 3 | 2 | 8 = 3 * 1 + 2 * 1 + 1 * 1 + 1 * 2 |
| 4 | 5 | 2 | 14 = 5 * 1 + 3 * 1 + 2 * 1 + 1 * 2 + 1 * 2 |
| 5 | 7 | 3 | 24 = 7 * 1 + 5 * 1 + 3 * 1 + 2 * 2 + 1 * 2 + 1 * 3 |
Related to this property, the following combination of two series of sums can also be set up via the function Template:Math:
- <math>\theta_{4}(x) = \biggl[\sum_{k = 0}^{\infty} P(k) x^k \biggr]^{-1} \biggl[\sum_{k = 0}^{\infty} Q(k) x^k \biggr]^{-1}</math>
NotesEdit
ReferencesEdit
- Template:Cite book
- Template:Cite book
- Template:Cite book. (for treatment of the Riemann theta)
- Template:Cite book
- Template:Cite book
- Template:Cite book
- Template:Cite book
- Template:Dlmf
- Template:Cite book (history of Jacobi's Template:Mvar functions)
Further readingEdit
Harry Rauch with Hershel M. Farkas: Theta functions with applications to Riemann Surfaces, Williams and Wilkins, Baltimore MD 1974, Template:ISBN.
- Charles Hermite: Sur la résolution de l'Équation du cinquiéme degré Comptes rendus, C. R. Acad. Sci. Paris, Nr. 11, March 1858.
External linksEdit
- {{#invoke:citation/CS1|citation
|CitationClass=web }}
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