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Rogers–Ramanujan type identities

Infinite products associated with Rogers–Ramanujan type identities

This page is a list of infinite products associated with Rogers–Ramanujan type identities. Each listed product links to its associated home page, and can be represented by one or more series representations. The list is sorted by the modulus of a product as determined by the product signature.

Use the search bar on the left to jump to a particular modulus. Click on any product to see the series representations.

For help with notation, see the definition of the \(q\)-Pochhammer symbol and of Ramanujan's theta functions.

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Mod 3

Product representationsProduct signature

Mod 7

Mod 10

Product representationsProduct signature
\(\displaystyle \frac{f(-q^3,-q^7)}{\varphi(-q)} = \frac{1}{(q^{},q^{5},q^{9}; q^{10})_\infty^{2}(q^{2},q^{3},q^{4},q^{6},q^{7},q^{8}; q^{10})_\infty}\) 10:(2,1,1,1,2,1,1,1,2,0)
\(\displaystyle \frac{f(-q^5,-q^5)}{\varphi(-q)} = \frac{1}{(q^{},q^{3},q^{7},q^{9}; q^{10})_\infty^{2}(q^{2},q^{4},q^{6},q^{8}; q^{10})_\infty}\) 10:(2,1,2,1,0,1,2,1,2,0)
\(\displaystyle \frac{H(q^2)}{\chi(-q)} = \frac{f(-q^2,-q^8)}{(q;q)_\infty} = \frac{1}{(q^{},q^{3},q^{4},q^{5},q^{6},q^{7},q^{9}; q^{10})_\infty}\) 10:(1,0,1,1,1,1,1,0,1,0)
\(\displaystyle \frac{G(q^2)}{\chi(-q)} = \frac{f(-q^4,-q^6)}{(q;q)_\infty} = \frac{1}{(q^{},q^{2},q^{3},q^{5},q^{7},q^{8},q^{9}; q^{10})_\infty}\) 10:(1,1,1,0,1,0,1,1,1,0)
\(\displaystyle \frac{f(-q,-q^4)}{(q^2;q^2)_\infty} = \frac{(q,q^5,q^9 ;q^{10})_\infty}{(q^2,q^8 ;q^{10})_\infty}\) 10:(-1,1,0,0,-1,0,0,1,-1,0)
\(\displaystyle \frac{f(-q^2,-q^3)}{(q^2;q^2)_\infty} = \frac{(q^3,q^5,q^7 ;q^{10})_\infty}{(q^4,q^6 ;q^{10})_\infty}\) 10:(0,0,-1,1,-1,1,-1,0,0,0)
\(\displaystyle \frac{f(q,q^4)}{\psi(q)} = \frac{(q^3,q^7;q^{10})_\infty (q^5;q^{10})_\infty^2}{(q^4, q^6 ;q^{10})_\infty} \) 10:(0,0,-1,2,-2,2,-1,0,0,0)
\(\displaystyle \frac{f(q^2,q^3)}{\psi(q)} = \frac{(q,q^9;q^{10})_\infty (q^5;q^{10})_\infty^2}{(q^2, q^8 ;q^{10})^2_\infty}\) 10:(-1,2,0,0,-2,0,0,2,-1,0)
\(\displaystyle \frac{f(-q,-q^9)}{\varphi(-q)} = \frac{1}{(q^{},q^{2},q^{4},q^{6},q^{8},q^{9}; q^{10})_\infty(q^{3},q^{5},q^{7}; q^{10})_\infty^{2}}\) 10:(1,1,2,1,2,1,2,1,1,0)
\(\displaystyle \frac{f\left(1,q^5\right)}{2 \psi (q)} = \frac{(q^{},q^{3},q^{7},q^{9}; q^{10})_\infty}{(q^{2},q^{4},q^{6},q^{8}; q^{10})_\infty} \) 10:(-1,1,-1,1,0,1,-1,1,-1,0)

Mod 11

Mod 12

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Product representationsProduct signature
\(\displaystyle \frac{f\left(-q,q^2\right)}{f(-q)} = \frac{\left(q^6;q^{12}\right){}_{\infty }}{\left(q^4;q^{12}\right){}_{\infty } \left(q^8;q^{12}\right){}_{\infty } \left(q^2;q^{12}\right)_{\infty }^2 \left(q^3;q^{12}\right)_{\infty }^2 \left(q^9;q^{12}\right)_{\infty }^2 \left(q^{10};q^{12}\right)_{\infty }^2} \) 12:(0,2,2,1,0,-1,0,1,2,2,0,0)
\(\displaystyle \frac{f(-q,-q^{11})}{(q^4;q^4)_\infty} = \frac{(q^{},q^{11}; q^{12})_\infty}{(q^{4},q^{8}; q^{12})_\infty} \) 12:(-1,0,0,1,0,0,0,1,0,0,-1,0)
\(\displaystyle \frac{f(-q,-q^{11})}{(q;q)_\infty} = \frac{1}{(q^{2},q^{3},q^{4},q^{5},q^{6},q^{7},q^{8},q^{9},q^{10}; q^{12})_\infty} \) 12:(0,1,1,1,1,1,1,1,1,1,0,0)
\(\displaystyle \frac{f(-q^2,-q^{10})}{(q;q)_\infty} = \frac{1}{(q^{},q^{3},q^{4},q^{5},q^{6},q^{7},q^{8},q^{9},q^{11}; q^{12})_\infty} \) 12:(1,0,1,1,1,1,1,1,1,0,1,0)
\(\displaystyle \frac{f(-q^3,-q^9)}{(q;q)_\infty} = \frac{f(q,q^5)}{\varphi(-q^2)} = \frac{1}{(q^{},q^{2},q^{4},q^{5},q^{6},q^{7},q^{8},q^{10},q^{11}; q^{12})_\infty} \) 12:(1,1,0,1,1,1,1,1,0,1,1,0)
\(\displaystyle \frac{f(-q^5,-q^7)}{(q;q)_\infty} = \frac{1}{(q^{},q^{2},q^{3},q^{4},q^{6},q^{8},q^{9},q^{10},q^{11}; q^{12})_\infty} \) 12:(1,1,1,1,0,1,0,1,1,1,1,0)
\(\displaystyle \frac{f(-q^6,-q^6)}{(q;q)_\infty} = \frac{f(q^2,q^4)}{\psi(-q)} =\frac{(q^{6}; q^{12})_\infty}{(q^{},q^{2},q^{3},q^{4},q^{5},q^{7},q^{8},q^{9},q^{10},q^{11}; q^{12})_\infty} \) 12:(1,1,1,1,1,-1,1,1,1,1,1,0)
\(\displaystyle \frac{f(-q,-q^5)}{\psi(-q)} = \frac{(q^6;q^{12})_\infty}{(q^3,q^4,q^8,q^9 ;q^{12})_\infty} \) 12:(0,0,1,1,0,-1,0,1,1,0,0,0)
\(\displaystyle \frac{f(-q^3,-q^3)}{\psi(-q)} = \frac{(q^3, q^6, q^9;q^{12})_\infty}{(q,q^4,q^5,q^7,q^8,q^{11} ;q^{12})_\infty}\) 12:(1,0,-1,1,1,-1,1,1,-1,0,1,0)
\(\displaystyle \frac{f(-q^3,-q^6)}{\psi(-q)} = \frac{Q(q^{12},-q^3)}{\psi(-q)} = \frac{(q^3,q^9,q^{12};q^{12})_\infty (q^6,q^{18};q^{24})_\infty}{\psi(-q)}=\frac{(q^3;q^3)_\infty (-q;q^2)_\infty}{ (q^2;q^2)_\infty } = \frac{(q^{6}; q^{12})_\infty}{(q^{},q^{4},q^{5},q^{7},q^{8},q^{11}; q^{12})_\infty} \) 12:(1,0,0,1,1,-1,1,1,0,0,1,0)
\( \displaystyle \frac{f\left(-q^5,-q^7\right)}{f\left(-q^4\right)} = \frac{\left(q^5;q^{12}\right){}_{\infty } \left(q^7;q^{12}\right){}_{\infty }}{\left(q^4;q^{12}\right){}_{\infty } \left(q^8;q^{12}\right){}_{\infty }} \) 12:(0,0,0,1,-1,0,-1,1,0,0,0,0)
\(\displaystyle \frac{f(q,q^5)}{f(-q^2)} = \frac{1}{(q^{},q^{4},q^{5},q^{7},q^{8},q^{11}; q^{12})_\infty} \) 12:(1,0,0,1,1,0,1,1,0,0,1,0)
\(\displaystyle \frac{Q(q^{12},q^3)}{\varphi(-q^2)} = \frac{1}{\left(q^3;q^{12}\right){}_{\infty } \left(q^4;q^{12}\right){}_{\infty } \left(q^8;q^{12}\right){}_{\infty } \left(q^9;q^{12}\right){}_{\infty } \left(q^2;q^{12}\right)_{\infty }^2 \left(q^{10};q^{12}\right)_{\infty }^2} \) 12:(0,2,1,1,0,0,0,1,1,2,0,0)
\(\displaystyle \frac{Q\left(q^{12},q^5\right)}{\psi (-q)} = \frac{(q^{2},q^{10}; q^{12})_\infty}{(q^{},q^{3},q^{4},q^{8},q^{9},q^{11}; q^{12})_\infty(q^{5},q^{7}; q^{12})_\infty^{2}} \) 12:(1,-1,1,1,2,0,2,1,1,-1,1,0)
\(\displaystyle \frac{Q\left(q^{12},q^4\right)}{\psi (-q)} = \frac{f(-q^{12},-q^{24})}{\psi(-q)} = \frac{1}{(q^{},q^{3},q^{4},q^{5},q^{7},q^{8},q^{9},q^{11}; q^{12})_\infty} \) 12:(1,0,1,1,1,0,1,1,1,0,1,0)
\(\displaystyle \frac{Q\left(q^{12},q\right)}{\varphi \left(-q^2\right)} = \frac{1}{(q^{},q^{2},q^{4},q^{8},q^{10},q^{11}; q^{12})_\infty(q^{6}; q^{12})_\infty^{2}} \) 12:(1,1,0,1,0,2,0,1,0,1,1,0)
\(\displaystyle \frac{Q\left(q^{12},q^5\right)}{\varphi \left(-q^2\right)} = \frac{1}{(q^{2},q^{4},q^{5},q^{7},q^{8},q^{10}; q^{12})_\infty(q^{6}; q^{12})_\infty^{2}} \) 12:(0,1,0,1,1,2,1,1,0,1,0,0)
\(\displaystyle \frac{f\left(-q^5,-q^7\right)}{\psi \left(-q^3\right)} = \frac{\left(q^5;q^{12}\right){}_{\infty } \left(q^7;q^{12}\right){}_{\infty }}{\left(q^3;q^{12}\right){}_{\infty } \left(q^9;q^{12}\right){}_{\infty }} \) 12:(0,0,1,0,-1,0,-1,0,1,0,0,0)
\(\displaystyle \frac{f\left(-q,-q^{11}\right)}{\psi \left(-q^3\right)} = \frac{(q^{},q^{11}; q^{12})_\infty}{(q^{3},q^{9}; q^{12})_\infty} \) 12:(-1,0,1,0,0,0,0,0,1,0,-1,0)
\(\displaystyle \frac{Q\left(q^6,-q\right)}{\varphi (-q)} = \frac{1}{(q^{},q^{2},q^{5},q^{7},q^{10},q^{11}; q^{12})_\infty(q^{3},q^{9}; q^{12})_\infty^{2}} \) 12:(1,1,2,0,1,0,1,0,2,1,1,0)
\(\displaystyle \frac{Q\left(q^6,-q^2\right)}{\varphi (-q)} =\frac{(q^{2},q^{10}; q^{12})_\infty}{(q^{},q^{3},q^{5},q^{7},q^{9},q^{11}; q^{12})_\infty^{2}} \) 12:(2,-1,2,0,2,0,2,0,2,-1,2,0)

Mod 14

Product representationsProduct signature
\(\displaystyle \frac{f(-q^3,-q^4)}{f(-q^2)} = \frac{(q^3,q^4,q^7;q^7)_\infty}{(q^2;q^2)_\infty} = \frac{(q^3, q^7, q^{11};q^{14})_\infty}{(q^2, q^6, q^8, q^{12}; q^{14})_\infty}\) 14:(0,1,-1,0,0,1,-1,1,0,0,-1,1,0,0)
\(\displaystyle \frac{f(-q,-q^6)}{(q^2;q^2)_\infty} = \frac{(q^{},q^{7},q^{13}; q^{14})_\infty}{(q^{2},q^{4},q^{10},q^{12}; q^{14})_\infty}\) 14:(-1,1,0,1,0,0,-1,0,0,1,0,1,-1,0)
\(\displaystyle \frac{f(-q^2,-q^5)}{(q^2;q^2)_\infty} = \frac{(q^{5},q^{7},q^{9}; q^{14})_\infty}{(q^{4},q^{6},q^{8},q^{10}; q^{14})_\infty}\) 14:(0,0,0,1,-1,1,-1,1,-1,1,0,0,0,0)
\(\displaystyle \frac{f\left(-q^2,-q^{12}\right)}{f(-q)} = \frac{1}{(q^{},q^{3},q^{4},q^{5},q^{6},q^{7},q^{8},q^{9},q^{10},q^{11},q^{13}; q^{14})_\infty}\) 14:(1,0,1,1,1,1,1,1,1,1,1,0,1,0)
\(\displaystyle \frac{f\left(-q^4,-q^{10}\right)}{f(-q)} = \frac{1}{(q^{},q^{2},q^{3},q^{5},q^{6},q^{7},q^{8},q^{9},q^{11},q^{12},q^{13}; q^{14})_\infty} \) 14:(1,1,1,0,1,1,1,1,1,0,1,1,1,0)
\(\displaystyle \frac{f\left(-q^6,-q^8\right)}{f(-q)} = \frac{1}{(q^{},q^{2},q^{3},q^{4},q^{5},q^{7},q^{9},q^{10},q^{11},q^{12},q^{13}; q^{14})_\infty} \) 14:(1,1,1,1,1,0,1,0,1,1,1,1,1,0)
\(\displaystyle \frac{Q(q^7,-q^2)}{\varphi(-q)} = \frac{\left(q^3;q^{14}\right){}_{\infty } \left(q^{11};q^{14}\right){}_{\infty } f\left(-q^2,-q^5\right)}{\varphi (-q)} = \frac{1}{(q^{},q^{13}; q^{14})_\infty^{2}(q^{3},q^{4},q^{5},q^{6},q^{7},q^{8},q^{9},q^{10},q^{11}; q^{14})_\infty} \) 14:(2,0,1,1,1,1,1,1,1,1,1,0,2,0)
\(\displaystyle \frac{Q(q^7,-q)}{\varphi(-q)} = \frac{\left(q^5;q^{14}\right){}_{\infty } \left(q^9;q^{14}\right){}_{\infty } f\left(-q,-q^6\right)}{\varphi (-q)} = \frac{1}{(q^{},q^{2},q^{4},q^{5},q^{7},q^{9},q^{10},q^{12},q^{13}; q^{14})_\infty(q^{3},q^{11}; q^{14})_\infty^{2}} \) 14:(1,1,2,1,1,0,1,0,1,1,2,1,1,0)
\(\displaystyle \frac{Q(q^7,-q^3)}{\varphi(-q)} = \frac{\left(q;q^{14}\right){}_{\infty } \left(q^{13};q^{14}\right){}_{\infty } f\left(-q^3,-q^4\right)}{\varphi (-q)} = \frac{1}{(q^{},q^{2},q^{3},q^{6},q^{7},q^{8},q^{11},q^{12},q^{13}; q^{14})_\infty(q^{5},q^{9}; q^{14})_\infty^{2}} \) 14:(1,1,1,0,2,1,1,1,2,0,1,1,1,0)

Mod 15

Mod 16

Product representationsProduct signature
\(\displaystyle \frac{f(q,-q^3)}{\psi(-q)} = \frac{(q^2,q^8,q^{14};q^{16})_\infty}{(q,q^4,q^7,q^9,q^{12}, q^{15}; q^{16})^2_\infty}\) 16:(2,-1,0,2,0,0,2,-1,2,0,0,2,0,-1,2,0)
\(\displaystyle \frac{f(-q,q^3)}{\psi(-q)} = \frac{(q^6, q^8,q^{10};q^{16})_\infty}{(q^3,q^4,q^5,q^{11},q^{12},q^{13};q^{16})^2_\infty}\) 16:(0,0,2,2,2,-1,0,-1,0,-1,2,2,2,0,0,0)
\(\displaystyle \frac{f(q,-q^3)}{\varphi(-q^2)} = \frac{(q^3,q^5,q^8,q^{11},q^{13};q^{16})_\infty}{(q,q^2,q^7,q^9,q^{14},q^{15};q^{16})_\infty (q^4,q^6,q^{10},q^{12} ;q^{16})_\infty^2}\) 16:(1,1,-1,2,-1,2,1,-1,1,2,-1,2,-1,1,1,0)
\(\displaystyle \frac{f(q,q^7)}{(q^2;q^2)_\infty} = \frac{Q(q^8,-q^3)}{f(-q)} = \frac{1}{(q^{},q^{4},q^{6},q^{7},q^{9},q^{10},q^{12},q^{15}; q^{16})_\infty}\) 16:(1,0,0,1,0,1,1,0,1,1,0,1,0,0,1,0)
\(\displaystyle \frac{f(q^3,q^5)}{(q^2;q^2)_\infty} = \frac{Q(q^8,-q)}{f(-q)} = \frac{1}{(q^{2},q^{3},q^{4},q^{5},q^{11},q^{12},q^{13},q^{14}; q^{16})_\infty}\) 16:(0,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0)
\(\displaystyle \frac{f(q,q^7)}{\psi(-q^2)} = \frac{1}{(q^{},q^{6},q^{7},q^{9},q^{10},q^{15}; q^{16})_\infty}\) 16:(1,0,0,0,0,1,1,0,1,1,0,0,0,0,1,0)
\(\displaystyle \frac{f(q^3,q^5)}{\psi(-q^2)} = \frac{1}{(q^{2},q^{3},q^{5},q^{11},q^{13},q^{14}; q^{16})_\infty}\) 16:(0,1,1,0,1,0,0,0,0,0,1,0,1,1,0,0)
\(\displaystyle \frac{f\left(-q^2,-q^{14}\right)}{\psi (-q)} = \frac{(q^{2},q^{14}; q^{16})_\infty}{(q^{},q^{3},q^{4},q^{5},q^{7},q^{8},q^{9},q^{11},q^{12},q^{13},q^{15}; q^{16})_\infty} \) 16:(1,-1,1,1,1,0,1,1,1,0,1,1,1,-1,1,0)
\(\displaystyle \frac{f\left(-q^4,-q^{12}\right)}{\psi (-q)} = \frac{1}{(q^{},q^{3},q^{5},q^{7},q^{8},q^{9},q^{11},q^{13},q^{15}; q^{16})_\infty} \) 16:(1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0)
\( \displaystyle \frac{f\left(-q^6,-q^{10}\right)}{\psi (-q)} = \frac{(q^{6},q^{10}; q^{16})_\infty}{(q^{},q^{3},q^{4},q^{5},q^{7},q^{8},q^{9},q^{11},q^{12},q^{13},q^{15}; q^{16})_\infty} \) 16:(1,0,1,1,1,-1,1,1,1,-1,1,1,1,0,1,0)
\( \displaystyle \frac{f\left(q,q^7\right)}{\psi \left(-q^2\right)} = \frac{1}{(q^{},q^{6},q^{7},q^{9},q^{10},q^{15}; q^{16})_\infty} \) 16:(1,0,0,0,0,1,1,0,1,1,0,0,0,0,1,0)
\( \displaystyle \frac{f\left(q^3,q^5\right)}{\psi \left(-q^2\right)} = \frac{1}{(q^{2},q^{3},q^{5},q^{11},q^{13},q^{14}; q^{16})_\infty} \) 16:(0,1,1,0,1,0,0,0,0,0,1,0,1,1,0,0)

Mod 18

\( \displaystyle \frac{f\left(-q^3,-q^{15}\right)}{\varphi (-q)} = \frac{1}{(q^{},q^{5},q^{7},q^{9},q^{11},q^{13},q^{17}; q^{18})_\infty^{2}(q^{2},q^{3},q^{4},q^{6},q^{8},q^{10},q^{12},q^{14},q^{15},q^{16}; q^{18})_\infty} \) 18:(2,1,1,1,2,1,2,1,2,1,2,1,2,1,1,1,2,0)
\( \displaystyle \frac{f\left(-q^9,-q^9\right)}{\varphi (-q)} = \frac{1}{(q^{},q^{3},q^{5},q^{7},q^{11},q^{13},q^{15},q^{17}; q^{18})_\infty^{2}(q^{2},q^{4},q^{6},q^{8},q^{10},q^{12},q^{14},q^{16}; q^{18})_\infty} \) 18:(2,1,2,1,2,1,2,1,0,1,2,1,2,1,2,1,2,0)
\( \displaystyle \frac{Q\left(q^{18},q^3\right)}{f\left(-q^2\right)} =\frac{1}{(q^{2},q^{3},q^{4},q^{8},q^{10},q^{14},q^{15},q^{16}; q^{18})_\infty} \) 18:(0,1,1,1,0,0,0,1,0,1,0,0,0,1,1,1,0,0)
\( \displaystyle \frac{Q\left(q^{18},q^5\right)}{f\left(-q^2\right)} = \frac{1}{(q^{2},q^{4},q^{5},q^{6},q^{12},q^{13},q^{14},q^{16}; q^{18})_\infty} \) 18:(0,1,0,1,1,1,0,0,0,0,0,1,1,1,0,1,0,0)
\( \displaystyle \frac{Q\left(q^{18},q^7\right)}{f\left(-q^2\right)} = \frac{1}{(q^{2},q^{6},q^{7},q^{8},q^{10},q^{11},q^{12},q^{16}; q^{18})_\infty} \) 18:(0,1,0,0,0,1,1,1,0,1,1,1,0,0,0,1,0,0)
\( \displaystyle \frac{Q\left(q^9,-q\right)}{f(-q)} = \frac{1}{(q^{2},q^{3},q^{4},q^{5},q^{6},q^{12},q^{13},q^{14},q^{15},q^{16}; q^{18})_\infty} \) 18:(0,1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0)
\( \displaystyle \frac{Q\left(q^9,-q^2\right)}{f(-q)} = \frac{1}{(q^{},q^{3},q^{4},q^{6},q^{8},q^{10},q^{12},q^{14},q^{15},q^{17}; q^{18})_\infty} \) 18:(1,0,1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0)
\( \displaystyle \frac{Q\left(q^9,-q^3\right)}{f(-q)} = \frac{(q^{3},q^{15}; q^{18})_\infty}{(q^{},q^{2},q^{4},q^{5},q^{7},q^{8},q^{10},q^{11},q^{13},q^{14},q^{16},q^{17}; q^{18})_\infty} \) 18:(1,1,-1,1,1,0,1,1,0,1,1,0,1,1,-1,1,1,0)
\( \displaystyle \frac{Q\left(q^9,-q^4\right)}{f(-q)} = \frac{1}{(q^{2},q^{3},q^{6},q^{7},q^{8},q^{10},q^{11},q^{12},q^{15},q^{16}; q^{18})_\infty} \) 18:(0,1,1,0,0,1,1,1,0,1,1,1,0,0,1,1,0,0)

Mod 20

Product representationsProduct signature
\(\displaystyle \frac{f(-q,-q^4)}{\psi(-q)} = \frac{(q^6,q^{10},q^{14};q^{20})_\infty}{(q^3,q^7,q^8,q^{12},q^{13},q^{17} ;q^{20})_\infty}\) 20:(0,0,1,0,0,-1,1,1,0,-1,0,1,1,-1,0,0,1,0,0,0)
\(\displaystyle \frac{f(-q^2,-q^3)}{\psi(-q)} = \frac{(q^2,q^{10},q^{18};q^{20})_\infty} {(q,q^4,q^9,q^{11},q^{16},q^{19} ;q^{20})_\infty}\) 20:(1,-1,0,1,0,0,0,0,1,-1,1,0,0,0,0,1,0,-1,1,0)
\(\displaystyle \frac{f(-q,-q^4)}{\varphi(-q^2)} = \frac{(q,q^5,q^9,q^{11},q^{15},q^{19};q^{20})_\infty} {(q^6, q^8,q^{10},q^{12},q^{14};q^{20})_\infty(q^2,q^{18} ;q^{20})_\infty^2}\) 20:(-1,2,0,0,-1,1,0,1,-1,1,-1,1,0,1,-1,0,0,2,-1,0)
\({\color{red} \displaystyle \frac{f(-q^2,-q^3)}{\varphi(-q^2)} = \frac{(q^3,q^5,q^7,q^{13},q^{15},q^{17};q^{20})_\infty} {(q^2,q^4,q^{10},q^{16},q^{18} ;q^{20})_\infty(q^6,q^{14} ;q^{20})_\infty^2} }\) 20:(0,1,-1,1,-1,2,-1,0,0,1,0,0,-1,2,-1,1,-1,1,0,0)
\(\displaystyle \frac{G(q^4)}{\chi(-q)} = \frac{f\left(-q^8,-q^{12}\right)}{\psi (-q)} = \frac{Q(q^{10},-q^2)}{f(-q)} = \frac{1}{(q^{},q^{3},q^{4},q^{5},q^{7},q^{9},q^{11},q^{13},q^{15},q^{16},q^{17},q^{19}; q^{20})_\infty} \) 20:(1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0)
\(\displaystyle \frac{H(-q)}{\chi(-q)} = \frac{Q\left(q^{10},-q^3\right)}{f(-q)} = \frac{1}{(q^{},q^{2},q^{5},q^{6},q^{8},q^{9},q^{11},q^{12},q^{14},q^{15},q^{18},q^{19}; q^{20})_\infty} \) 20:(1,1,0,0,1,1,0,1,1,0,1,1,0,1,1,0,0,1,1,0)
\(\displaystyle \frac{Q\left(q^{10},-q^3\right)}{\psi (-q)} = \frac{(q^{10}; q^{20})_\infty}{(q^{},q^{5},q^{8},q^{9},q^{11},q^{12},q^{15},q^{19}; q^{20})_\infty} \) 20:(1,0,0,0,1,0,0,1,1,-1,1,1,0,0,1,0,0,0,1,0)
\(\displaystyle \frac{H(q^4)}{\chi(-q)} = \frac{f(-q^4,-q^{16})}{\psi(-q)} = \frac{Q\left(q^{10},-q^4\right)}{f(-q)} = \frac{1}{(q^{},q^{3},q^{5},q^{7},q^{8},q^{9},q^{11},q^{12},q^{13},q^{15},q^{17},q^{19}; q^{20})_\infty} \) 20:(1,0,1,0,1,0,1,1,1,0,1,1,1,0,1,0,1,0,1,0)
\(\displaystyle \frac{G(-q)}{\chi(-q)} = \frac{Q\left(q^{10},-q\right)}{f(-q)} = \frac{1}{(q^{2},q^{3},q^{4},q^{5},q^{6},q^{7},q^{13},q^{14},q^{15},q^{16},q^{17},q^{18}; q^{20})_\infty} \) 20:(0,1,1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,1,0,0)
\(\displaystyle \frac{Q\left(q^{10},-q\right)}{\psi (-q)} = \frac{(q^{10}; q^{20})_\infty}{(q^{3},q^{4},q^{5},q^{7},q^{13},q^{15},q^{16},q^{17}; q^{20})_\infty} \) 20:(0,0,1,1,1,0,1,0,0,-1,0,0,1,0,1,1,1,0,0,0)