Contents

User:Titus III

I. Rogers-Selberg Mod 7 Identities

${\displaystyle A($q)={\frac {(q^{3},q^{4},q^{7};q^{7})_{\infty }}{(q^{2};q^{2})_{\infty }}}={\frac {f(-q^{3},-q^{4})}{f(-q^{2},-q^{4})}}=\prod _{n=1}^{\infty }{\frac {(1-q^{7n})(1-q^{7n-3})(1-q^{7n-4})}{(1-q^{2n})}}}

${\displaystyle B($q)={\frac {(q^{2},q^{5},q^{7};q^{7})_{\infty }}{(q^{2};q^{2})_{\infty }}}={\frac {f(-q^{2},-q^{5})}{f(-q^{2},-q^{4})}}=\prod _{n=1}^{\infty }{\frac {(1-q^{7n})(1-q^{7n-2})(1-q^{7n-5})}{(1-q^{2n})}}}

${\displaystyle C($q)={\frac {(q,q^{6},q^{7};q^{7})_{\infty }}{(q^{2};q^{2})_{\infty }}}\,={\frac {f(-q,-q^{6})}{f(-q^{2},-q^{4})}}\,=\prod _{n=1}^{\infty }{\frac {(1-q^{7n})(1-q^{7n-1})(1-q^{7n-6})}{(1-q^{2n})}}}

II. Bailey Mod 9 Identities

${\displaystyle A($q)={\frac {(q^{4},q^{5},q^{9};q^{9})_{\infty }}{(q^{3};q^{3})_{\infty }}}={\frac {f(-q^{4},-q^{5})}{f(-q^{3},-q^{6})}}=\prod _{n=1}^{\infty }{\frac {(1-q^{9n})(1-q^{9n-4})(1-q^{9n-5})}{(1-q^{3n})}}}

${\displaystyle B($q)={\frac {(q^{2},q^{7},q^{9};q^{9})_{\infty }}{(q^{3};q^{3})_{\infty }}}={\frac {f(-q^{2},-q^{7})}{f(-q^{3},-q^{6})}}=\prod _{n=1}^{\infty }{\frac {(1-q^{9n})(1-q^{9n-2})(1-q^{9n-7})}{(1-q^{3n})}}}

${\displaystyle C($q)={\frac {(q,q^{8},q^{9};q^{9})_{\infty }}{(q^{3};q^{3})_{\infty }}}\,={\frac {f(-q,-q^{8})}{f(-q^{3},-q^{6})}}\,=\prod _{n=1}^{\infty }{\frac {(1-q^{9n})(1-q^{9n-1})(1-q^{9n-8})}{(1-q^{3n})}}}

III. Rogers Mod 14 Identities

${\displaystyle A($q)={\frac {(q^{6},q^{8},q^{14};q^{14})_{\infty }}{(q;q)_{\infty }}}\,={\frac {f(-q^{6},-q^{8})}{f(-q,-q^{2})}}\,=\prod _{n=1}^{\infty }{\frac {(1-q^{14n})(1-q^{14n-6})(1-q^{14n-8})}{(1-q^{n})}}}

${\displaystyle B($q)={\frac {(q^{4},q^{10},q^{14};q^{14})_{\infty }}{(q;q)_{\infty }}}={\frac {f(-q^{4},-q^{10})}{f(-q,-q^{2})}}=\prod _{n=1}^{\infty }{\frac {(1-q^{14n})(1-q^{14n-4})(1-q^{14n-10})}{(1-q^{n})}}}

${\displaystyle C($q)={\frac {(q^{2},q^{12},q^{14};q^{14})_{\infty }}{(q;q)_{\infty }}}={\frac {f(-q^{2},-q^{12})}{f(-q,-q^{2})}}=\prod _{n=1}^{\infty }{\frac {(1-q^{14n})(1-q^{14n-2})(1-q^{14n-12})}{(1-q^{n})}}}

IV. Dyson Mod 27 Identities

${\displaystyle A($q)={\frac {(q^{12},q^{15},q^{27};q^{27})_{\infty }}{(q;q)_{\infty }}}={\frac {f(-q^{12},-q^{15})}{f(-q,-q^{2})}}=\prod _{n=1}^{\infty }{\frac {(1-q^{27n})(1-q^{27n-12})(1-q^{27n-15})}{(1-q^{n})}}}

${\displaystyle B($q)={\frac {(q^{9},q^{18},q^{27};q^{27})_{\infty }}{(q;q)_{\infty }}}={\frac {f(-q^{9},-q^{18})}{f(-q,-q^{2})}}=\prod _{n=1}^{\infty }{\frac {(1-q^{27n})(1-q^{27n-9})(1-q^{27n-18})}{(1-q^{n})}}}

${\displaystyle C($q)={\frac {(q^{6},q^{21},q^{27};q^{27})_{\infty }}{(q;q)_{\infty }}}\,={\frac {f(-q^{6},-q^{21})}{f(-q,-q^{2})}}\,=\prod _{n=1}^{\infty }{\frac {(1-q^{27n})(1-q^{27n-6})(1-q^{27n-21})}{(1-q^{n})}}}

${\displaystyle D($q)={\frac {(q^{3},q^{24},q^{27};q^{27})_{\infty }}{(q;q)_{\infty }}}\,={\frac {f(-q^{3},-q^{24})}{f(-q,-q^{2})}}\,=\prod _{n=1}^{\infty }{\frac {(1-q^{27n})(1-q^{27n-3})(1-q^{27n-24})}{(1-q^{n})}}}

This is similar to the 11/3/2012 update. Let,

${\displaystyle m={\frac {12}{n-1}}}$

be a positive integer for some odd number n. Thus, there are only 4 possibilities: ${\displaystyle n=3,5,7,13}$. Given the standard Ramanujan theta functions ${\displaystyle \phi ($q),\;\psi (q),\;f(-q)} , then,

${\displaystyle \left({\frac {\phi (-q^{2})\phi (-q^{6n})}{\phi (-q^{6})\phi (-q^{2n})}}\right)^{m}\,+\,q^{3}\left({\frac {\psi ($q)\psi (q^{3n})}{\psi (q^{3})\psi (q^{n})}}\right)^{m}-\,q^{3}\left({\frac {\psi (-q)\psi (-q^{3n})}{\psi (-q^{3})\psi (-q^{n})}}\right)^{m}-\,2mq^{2}\left({\frac {f(-q^{2})f(-q^{6n})}{f(-q^{6})f(-q^{2n})}}\right)^{m}=1}

For example, for n = 5, hence m = 3, we have,

${\displaystyle \left({\frac {\phi (-q^{2})\phi (-q^{30})}{\phi (-q^{6})\phi (-q^{10})}}\right)^{3}\,+\,q^{3}\left({\frac {\psi ($q)\psi (q^{15})}{\psi (q^{3})\psi (q^{5})}}\right)^{3}-\,q^{3}\left({\frac {\psi (-q)\psi (-q^{15})}{\psi (-q^{3})\psi (-q^{5})}}\right)^{3}-\,6q^{2}\left({\frac {f(-q^{2})f(-q^{30})}{f(-q^{6})f(-q^{10})}}\right)^{3}=1}

and so on for the other three n.

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Define,

${\displaystyle h_{k}=(-1)^{k-1}q^{k(3k-25)/50}\,{\frac {f(-q^{2k},-q^{25-2k})}{f(-q^{k},-q^{25-k})}},\;\;{\text{for}}\;k=1,...,12.}$

(Note, as usual, that the even-index h_k have a negative sign.) It turns out that appropriate pairsofh_k are roots of a polynomial whose coefficients are in ${\displaystyle j={\frac {\eta (\tau )}{\eta (25\tau )}}}$, analogous to the case for n = 13 discovered by Ramanujan. Since one pair is a constant, ${\displaystyle h_{5}h_{10}=-1}$, then the remaining five are the roots of a quintic (naturally enough) given by,

${\displaystyle 1+(10+10j+4j^{2}+j^{3})x+(5+15j+11j^{2}+5j^{3}+j^{4})x^{2}-2(5+5j+3j^{2}+j^{3})x^{3}+j^{2}x^{4}+x^{5}=0}$

with j as the eta quotient given above. The roots are then,

${\displaystyle [x_{1},\,x_{2},\,x_{3},\,x_{4},\,x_{5}]=[h_{1}h_{7},\,h_{2}h_{11},\,h_{3}h_{4},\,h_{6}h_{8},\,h_{9}h_{12}]}$

(Note: The quintic looks vaguely familiar to me, it seems I've come across it before, or something similar in the course of my research into the Rogers-Ramanujan continued fraction, but I cannot recall precisely in what context.)

Update April 28, 2022: After almost 10 years, I finally found the answer to my "note". This is just the EMMA LEHMER QUINTIC!

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Here is a generalization of one identity described by Berndt as "...fascinating, but with no direct proof" (Ramanujan's Notebooks III, p.322). Let,

${\displaystyle k={\frac {24}{n-1}}}$

be a positive integer for some odd number n. Of course, there are only 6 possibilities: ${\displaystyle n=3,5,7,9,13,25}$. Given the standard Ramanujan theta functions ${\displaystyle \phi ($q),\;\psi (q),\;f(-q)} , then it is proposed that,

${\displaystyle {\frac {\phi ^{k}(-q^{2n})}{\phi ^{k}(-q^{2})}}+q^{3}\left({\frac {\psi ^{k}(q^{n})}{\psi ^{k}($q)}}-{\frac {\psi ^{k}(-q^{n})}{\psi ^{k}(-q)}}\right)-2kq^{2}\left({\frac {f^{k}(-q^{2n})}{f^{k}(-q^{2})}}\right)=1}

and, as suggested by Michael Somos, by using a Fricke involution,

${\displaystyle {\frac {\phi ^{k}(-q^{2})}{\phi ^{k}(-q^{2n})}}+{\frac {1}{q^{3}}}\left({\frac {\psi ^{k}($q)}{\psi ^{k}(q^{n})}}-{\frac {\psi ^{k}(-q)}{\psi ^{k}(-q^{n})}}\right)-{\frac {2k}{q^{2}}}\left({\frac {f^{k}(-q^{2})}{f^{k}(-q^{2n})}}\right)=n^{k/2}}

Thus, for n = 3, we have,

${\displaystyle {\frac {\phi ^{12}(-q^{6})}{\phi ^{12}(-q^{2})}}+q^{3}\left({\frac {\psi ^{12}(q^{3})}{\psi ^{12}($q)}}-{\frac {\psi ^{12}(-q^{3})}{\psi ^{12}(-q)}}\right)-24q^{2}\left({\frac {f^{12}(-q^{6})}{f^{12}(-q^{2})}}\right)=1}

and,

${\displaystyle {\frac {\phi ^{12}(-q^{2})}{\phi ^{12}(-q^{6})}}+{\frac {1}{q^{3}}}\left({\frac {\psi ^{12}($q)}{\psi ^{12}(q^{3})}}-{\frac {\psi ^{12}(-q)}{\psi ^{12}(-q^{3})}}\right)-{\frac {24}{q^{2}}}\left({\frac {f^{12}(-q^{2})}{f^{12}(-q^{6})}}\right)=3^{6}}

and so on for the other five n, with the case n = 7 given in page 322 which was the inspiration for this generalization. I have no rigorous proof for this "family", but one can easily see via Mathematica that the proposed equality indeed holds true for hundreds of decimal places.

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It seems Ramanujan missed certain aspects of theta quotientsatp = 13.

I. Case p = 7

To illustrate, define the quotients for p = 7 as,

${\displaystyle h_{k}=(-1)^{k-1}q^{k(3k-7)/14}\,{\frac {f(-q^{2k},-q^{7-2k})}{f(-q^{k},-q^{7-k})}}}$

hence,

${\displaystyle h_{1}={\frac {f(-q^{2},-q^{5})}{q^{2/7}f(-q,-q^{6})}}={\frac {1}{q^{2/7}}}\prod _{n=1}^{\infty }{\frac {(1-q^{7n-2})(1-q^{7n-5})}{(1-q^{7n-1})(1-q^{7n-6})}}}$

${\displaystyle h_{2}={\frac {-f(-q^{3},-q^{4})}{q^{1/7}f(-q^{2},-q^{5})}}={\frac {-1}{q^{1/7}}}\prod _{n=1}^{\infty }{\frac {(1-q^{7n-3})(1-q^{7n-4})}{(1-q^{7n-2})(1-q^{7n-5})}}}$

${\displaystyle h_{3}={\frac {q^{3/7}f(-q,-q^{6})}{f(-q^{3},-q^{4})}}={\frac {1}{q^{-3/7}}}\prod _{n=1}^{\infty }{\frac {(1-q^{7n-1})(1-q^{7n-6})}{(1-q^{7n-3})(1-q^{7n-4})}}}$

functions highly analogous to the Rogers–Ramanujan continued fraction. (Kindly note that the even-index ${\displaystyle h_{k}}$isnegative.) Given the Dedekind eta function ${\displaystyle \eta (\tau )}$, let,

${\displaystyle c=\left({\frac {\eta (\tau )}{\eta (7\tau )}}\right)^{4}}$

then Ramanujan found that the 3 roots of the cubic,

${\displaystyle x^{3}-(57+14c+c^{2})x^{2}-(289+126c+19c^{2}+c^{3})x+1=0}$

are,

${\displaystyle x_{1}={h_{1}}^{7},\;\;x_{2}={h_{2}}^{7},\;\;x_{3}={h_{3}}^{7}}$

Note also another use for the eta quotient ${\displaystyle c}$ is,

${\displaystyle {\begin{aligned}&j(\tau )={\frac {(y^{2}+5y+1)^{3}(y^{2}+13y+49)}{y}}\\&j(\tau )-1728={\frac {(y^{4}+14y^{3}+63y^{2}+70y-7)^{2}}{y}}\end{aligned}}}$

where ${\displaystyle y={\frac {7^{2}}{c}}.}$

II. Case p = 13

It turns out that for p = 13, then the 13th power of the analogous theta quotients are the 6 roots of a sextic.

Define,

${\displaystyle h_{k}=(-1)^{k-1}q^{k(-13+3k)/26}\,{\frac {f(-q^{2k},-q^{13-2k})}{f(-q^{k},-q^{13-k})}}}$

hence,

${\displaystyle h_{1}={\frac {f(-q^{2},-q^{11})}{q^{5/13}f(-q,-q^{12})}},\;\;h_{2}={\frac {-f(-q^{4},-q^{9})}{q^{7/13}f(-q^{2},-q^{11})}},\;\;h_{3}={\frac {f(-q^{6},-q^{7})}{q^{6/13}f(-q^{3},-q^{10})}}}$

${\displaystyle h_{4}={\frac {-f(-q^{5},-q^{8})}{q^{2/13}f(-q^{4},-q^{9})}},\;\;h_{5}={\frac {q^{5/13}f(-q^{3},-q^{10})}{f(-q^{5},-q^{8})}},\;\;h_{6}={\frac {-q^{15/13}f(-q,-q^{12})}{f(-q^{6},-q^{7})}}}$

(As before, the even-index ${\displaystyle h_{k}}$ are negative.) Let,

${\displaystyle d=\left({\frac {\eta (\tau )}{\eta (13\tau )}}\right)^{2}}$

Ramanujan discovered that the 3 roots of the cubic,

${\displaystyle d={\frac {y^{3}+y^{2}-4y+1}{y(1-y)}}}$

are,

${\displaystyle y_{1}=h_{1}h_{5},\;\;y_{2}=h_{2}h_{3},\;\;y_{3}=h_{4}h_{6}}$

Note: Incidentally, ${\displaystyle y^{3}+y^{2}-4y+1=0}$ is a well-known equation for ${\displaystyle y=2\cos(2\pi /13)+2\cos(10\pi /13)}$ and similar roots.

However, if a modular equation can be found between, say, ${\displaystyle h_{1},h_{5}}$, then we can eliminate ${\displaystyle h_{5}}$, and have an equation solelyin${\displaystyle h_{1}}$ and ${\displaystyle d}$. After some effort using Mathematica's integer relations algorithm, I found that, let,

${\displaystyle u=h_{1},\,v=h_{5},\,x=h_{1}h_{5}}$

then,

${\displaystyle {\begin{aligned}u^{2}v^{2}(uv-1)^{5}(u^{13}+v^{13})&=1-7x+19x^{2}-27x^{3}+35x^{4}-68x^{5}+100x^{6}-84x^{7}\\&+69x^{8}-86x^{9}+70x^{10}-29x^{11}+11x^{12}-4x^{13}+x^{14}\end{aligned}}}$

(The same relation exists between the other pairs.) Eliminating ${\displaystyle h_{5}}$, one gets the sextic in ${\displaystyle z=h_{1}^{13}}$,

${\displaystyle z^{6}+P_{1}z^{5}+P_{2}z^{4}+P_{3}z^{3}+P_{4}z^{2}+P_{5}z-1=0}$

where the ${\displaystyle P_{i}}$ are polynomials in the eta quotient ${\displaystyle d}$ of degrees 7, 13, 18, 20, 15, respectively. Explicitly, the first one is,

${\displaystyle {\begin{aligned}P_{1}&=-(h_{1}^{13}+h_{2}^{13}+h_{3}^{13}+h_{4}^{13}+h_{5}^{13}+h_{6}^{13})\\&=d^{7}+13d^{6}+91d^{5}+377d^{4}+962d^{3}+1040d^{2}-845d-4083\end{aligned}}}$