In mathematics, the Rogers–Ramanujan identities are two identities related to basic hypergeometric series and integer partitions. The identities were first discovered and proved by Leonard James Rogers (1894), and were subsequently rediscovered (without a proof) by Srinivasa Ramanujan some time before 1913. Ramanujan had no proof, but rediscovered Rogers's paper in 1917, and they then published a joint new proof (Rogers & Ramanujan 1919). Issai Schur (1917) independently rediscovered and proved the identities.
Definition The Rogers–Ramanujan identities are
G ( q ) = ∑ n = 0 ∞ q n 2 ( q ; q ) n = 1 ( q ; q 5 ) ∞ ( q 4 ; q 5 ) ∞ = 1 + q + q 2 + q 3 + 2 q 4 + 2 q 5 + 3 q 6 + ⋯ {\displaystyle G(q)=\sum _{n=0}^{\infty }{\frac {q^{n^{2}}}{(q;q)_{n}}}={\frac {1}{(q;q^{5})_{\infty }(q^{4};q^{5})_{\infty }}}=1+q+q^{2}+q^{3}+2q^{4}+2q^{5}+3q^{6}+\cdots } (sequence A003114 in the OEIS) and
H ( q ) = ∑ n = 0 ∞ q n 2 + n ( q ; q ) n = 1 ( q 2 ; q 5 ) ∞ ( q 3 ; q 5 ) ∞ = 1 + q 2 + q 3 + q 4 + q 5 + 2 q 6 + ⋯ {\displaystyle H(q)=\sum _{n=0}^{\infty }{\frac {q^{n^{2}+n}}{(q;q)_{n}}}={\frac {1}{(q^{2};q^{5})_{\infty }(q^{3};q^{5})_{\infty }}}=1+q^{2}+q^{3}+q^{4}+q^{5}+2q^{6}+\cdots } (sequence A003106 in the OEIS). Here, ( a ; q ) n {\displaystyle (a;q)_{n}} denotes the q-Pochhammer symbol.
Combinatorial interpretation Consider the following:
q n 2 ( q ; q ) n {\displaystyle {\frac {q^{n^{2}}}{(q;q)_{n}}}} is the generating function for partitions with exactly n {\displaystyle n} parts such that adjacent parts have difference at least 2.
1 ( q ; q 5 ) ∞ ( q 4 ; q 5 ) ∞ {\displaystyle {\frac {1}{(q;q^{5})_{\infty }(q^{4};q^{5})_{\infty }}}} is the generating function for partitions such that each part is congruent to either 1 or 4 modulo 5.
q n 2 + n ( q ; q ) n {\displaystyle {\frac {q^{n^{2}+n}}{(q;q)_{n}}}} is the generating function for partitions with exactly n {\displaystyle n} parts such that adjacent parts have difference at least 2 and such that the smallest part is at least 2.
1 ( q 2 ; q 5 ) ∞ ( q 3 ; q 5 ) ∞ {\displaystyle {\frac {1}{(q^{2};q^{5})_{\infty }(q^{3};q^{5})_{\infty }}}} is the generating function for partitions such that each part is congruent to either 2 or 3 modulo 5. The Rogers–Ramanujan identities could be now interpreted in the following way. Let n {\displaystyle n} be a non-negative integer.
The number of partitions of n {\displaystyle n} such that the adjacent parts differ by at least 2 is the same as the number of partitions of n {\displaystyle n} such that each part is congruent to either 1 or 4 modulo 5. The number of partitions of n {\displaystyle n} such that the adjacent parts differ by at least 2 and such that the smallest part is at least 2 is the same as the number of partitions of n {\displaystyle n} such that each part is congruent to either 2 or 3 modulo 5. Alternatively,
The number of partitions of n {\displaystyle n} such that with k {\displaystyle k} parts the smallest part is at least k {\displaystyle k} is the same as the number of partitions of n {\displaystyle n} such that each part is congruent to either 1 or 4 modulo 5. The number of partitions of n {\displaystyle n} such that with k {\displaystyle k} parts the smallest part is at least k + 1 {\displaystyle k+1} is the same as the number of partitions of n {\displaystyle n} such that each part is congruent to either 2 or 3 modulo 5.
Application to partitions Since the terms occurring in the identity are generating functions of certain partitions, the identities make statements about partitions (decompositions) of natural numbers. The number sequences resulting from the coefficients of the Maclaurin series of the Rogers–Ramanujan functions G and H are special partition number sequences of level 5:
G ( x ) = 1 ( x ; x 5 ) ∞ ( x 4 ; x 5 ) ∞ = 1 + ∑ n = 1 ∞ P G ( n ) x n {\displaystyle G(x)={\frac {1}{(x;x^{5})_{\infty }(x^{4};x^{5})_{\infty }}}=1+\sum _{n=1}^{\infty }P_{G}(n)x^{n}}
H ( x ) = 1 ( x 2 ; x 5 ) ∞ ( x 3 ; x 5 ) ∞ = 1 + ∑ n = 1 ∞ P H ( n ) x n {\displaystyle H(x)={\frac {1}{(x^{2};x^{5})_{\infty }(x^{3};x^{5})_{\infty }}}=1+\sum _{n=1}^{\infty }P_{H}(n)x^{n}}
The number sequence P G ( n ) {\displaystyle P_{G}(n)} (sequence A003114 in the OEIS)) represents the number of possibilities for the affected natural number n to decompose this number into summands of the patterns 5a + 1 or 5a + 4 with a ∈ N 0 {\displaystyle \mathbb {N} _{0}} . Thus P G ( n ) {\displaystyle P_{G}(n)} gives the number of decays of an integer n in which adjacent parts of the partition differ by at least 2, equal to the number of decays in which each part is equal to 1 or 4 mod 5 is. And the number sequence P H ( n ) {\displaystyle P_{H}(n)} (sequence A003106 in the OEIS)) analogously represents the number of possibilities for the affected natural number n to decompose this number into summands of the patterns 5a + 2 or 5a + 3 with a ∈ N 0 {\displaystyle \mathbb {N} _{0}} . Thus P H ( n ) {\displaystyle P_{H}(n)} gives the number of decays of an integer n in which adjacent parts of the partition differ by at least 2 and in which the smallest part is greater than or equal to 2 is equal the number of decays whose parts are equal to 2 or 3 mod 5. This will be illustrated as examples in the following two tables:
Rogers–Ramanujan continued fractions R and S
Definition of the continued fractions
The following continued fraction R ( q ) {\displaystyle R(q)} is called Rogers–Ramanujan continued fraction, Continuing fraction S ( q ) {\displaystyle S(q)} is called alternating Rogers–Ramanujan continued fraction!
The factor q 1 5 {\displaystyle q^{\frac {1}{5}}} creates a quotient of module functions and it also makes these shown continued fractions modular: This definition applies for the continued fraction mentioned:
R ( q ) = q 1 / 5 ( q ; q 5 ) ∞ ( q 4 ; q 5 ) ∞ ( q 2 ; q 5 ) ∞ ( q 3 ; q 5 ) ∞ {\displaystyle R(q)=q^{1/5}{\frac {(q;q^{5})_{\infty }(q^{4};q^{5})_{\infty }}{(q^{2};q^{5})_{\infty }(q^{3};q^{5})_{\infty }}}}
R ( q ) = q 1 / 5 ∏ k = 0 ∞ ( 1 − q 5 k + 1 ) ( 1 − q 5 k + 4 ) ( 1 − q 5 k + 2 ) ( 1 − q 5 k + 3 ) = q 1 / 5 H ( q ) G ( q ) {\displaystyle R(q)=q^{1/5}\prod _{k=0}^{\infty }{\frac {(1-q^{5k+1})(1-q^{5k+4})}{(1-q^{5k+2})(1-q^{5k+3})}}=q^{1/5}{\frac {H(q)}{G(q)}}}
This is the definition of the Ramanujan theta function:
f ( a , b ) = ∑ k = − ∞ ∞ a k ( k + 1 ) 2 b k ( k − 1 ) 2 {\displaystyle f(a,b)=\sum _{k=-\infty }^{\infty }a^{\frac {k(k+1)}{2}}b^{\frac {k(k-1)}{2}}}
With this function, the continued fraction R can be created this way:
R ( q ) = q 1 / 5 f ( − q , − q 4 ) f ( − q 2 , − q 3 ) {\displaystyle R(q)=q^{1/5}{\frac {f(-q,-q^{4})}{f(-q^{2},-q^{3})}}} . The connection between the continued fraction and the Rogers–Ramanujan functions was already found by Rogers in 1894 (and later independently by Ramanujan). The continued fraction can also be expressed by the Dedekind eta function:
R ( q ) = tan { 1 2 arccot [ η W ( q 1 / 5 ) 2 η W ( q 5 ) + 1 2 ] } {\displaystyle R(q)=\tan {\biggl \{}{\frac {1}{2}}\operatorname {arccot} {\biggl [}{\frac {\eta _{W}(q^{1/5})}{2\eta _{W}(q^{5})}}+{\frac {1}{2}}{\biggr ]}{\biggr \}}}
The alternating continued fraction S ( q ) {\displaystyle S(q)} has the following identities to the remaining Rogers–Ramanujan functions and to the Ramanujan theta function described above:
S ( q ) = q 1 / 5 H ( − q ) G ( − q ) {\displaystyle S(q)=q^{1/5}{\frac {H(-q)}{G(-q)}}}
S ( q ) = q 1 / 5 f ( q , − q 4 ) f ( − q 2 , q 3 ) {\displaystyle S(q)=q^{1/5}{\frac {f(q,-q^{4})}{f(-q^{2},q^{3})}}}
S ( q ) = R ( q 4 ) R ( q ) R ( q 2 ) {\displaystyle S(q)={\frac {R(q^{4})}{R(q)R(q^{2})}}}
S ( q ) = q 1 / 5 G ( q ) G ( q 2 ) H ( q 4 ) H ( q ) H ( q 2 ) G ( q 4 ) {\displaystyle S(q)=q^{1/5}{\frac {G(q)G(q^{2})H(q^{4})}{H(q)H(q^{2})G(q^{4})}}}
Identities with Jacobi theta functions The following definitions are valid for the Jacobi "Theta-Nullwert" functions:
ϑ 00 ( x ) = 1 + 2 ∑ n = 1 ∞ x n 2 {\displaystyle \vartheta _{00}(x)=1+2\sum _{n=1}^{\infty }x^{n^{2}}}
ϑ 01 ( x ) = 1 − 2 ∑ n = 1 ∞ ( − 1 ) n + 1 x n 2 {\displaystyle \vartheta _{01}(x)=1-2\sum _{n=1}^{\infty }(-1)^{n+1}x^{n^{2}}}
ϑ 10 ( x ) = 2 x 1 / 4 + 2 x 1 / 4 ∑ n = 1 ∞ x n 2 + n {\displaystyle \vartheta _{10}(x)=2x^{1/4}+2x^{1/4}\sum _{n=1}^{\infty }x^{n^{2}+n}}
And the following product definitions are identical to the total definitions mentioned:
ϑ 00 ( x ) = ∏ n = 1 ∞ ( 1 − x 2 n ) ( 1 + x 2 n − 1 ) 2 {\displaystyle \vartheta _{00}(x)=\prod _{n=1}^{\infty }(1-x^{2n})(1+x^{2n-1})^{2}}
ϑ 01 ( x ) = ∏ n = 1 ∞ ( 1 − x 2 n ) ( 1 − x 2 n − 1 ) 2 {\displaystyle \vartheta _{01}(x)=\prod _{n=1}^{\infty }(1-x^{2n})(1-x^{2n-1})^{2}}
ϑ 10 ( x ) = 2 x 1 / 4 ∏ n = 1 ∞ ( 1 − x 2 n ) ( 1 + x 2 n ) 2 {\displaystyle \vartheta _{10}(x)=2x^{1/4}\prod _{n=1}^{\infty }(1-x^{2n})(1+x^{2n})^{2}}
These three so-called theta zero value functions are linked to each other using the Jacobian identity:
ϑ 00 ( x ) 4 = ϑ 10 ( x ) 4 + ϑ 01 ( x ) 4 {\displaystyle \vartheta _{00}(x)^{4}=\vartheta _{10}(x)^{4}+\vartheta _{01}(x)^{4}}
The mathematicians Edmund Taylor Whittaker and George Neville Watson discovered these definitional identities. The Rogers–Ramanujan continued fraction functions R ( x ) {\displaystyle R(x)} and S ( x ) {\displaystyle S(x)} have these relationships to the theta Nullwert functions:
R ( x ) = tan ⟨ 1 2 arccot { ϑ 01 ( x 1 / 5 ) [ 5 ϑ 01 ( x 5 ) 2 − ϑ 01 ( x ) 2 ] 2 ϑ 01 ( x 5 ) [ ϑ 01 ( x ) 2 − ϑ 01 ( x 1 / 5 ) 2 ] + 1 2 } ⟩ {\displaystyle R(x)=\tan {\biggl \langle }{\frac {1}{2}}\operatorname {arccot} {\biggl \{}{\frac {\vartheta _{01}(x^{1/5})[5\,\vartheta _{01}(x^{5})^{2}-\vartheta _{01}(x)^{2}]}{2\,\vartheta _{01}(x^{5})[\vartheta _{01}(x)^{2}-\vartheta _{01}(x^{1/5})^{2}]}}+{\frac {1}{2}}{\biggr \}}{\biggr \rangle }}
S ( x ) = tan ⟨ 1 2 arccot { ϑ 00 ( x 1 / 5 ) [ 5 ϑ 00 ( x 5 ) 2 − ϑ 00 ( x ) 2 ] 2 ϑ 00 ( x 5 ) [ ϑ 00 ( x 1 / 5 ) 2 − ϑ 00 ( x ) 2 ] − 1 2 } ⟩ {\displaystyle S(x)=\tan {\biggl \langle }{\frac {1}{2}}\operatorname {arccot} {\biggl \{}{\frac {\vartheta _{00}(x^{1/5})[5\,\vartheta _{00}(x^{5})^{2}-\vartheta _{00}(x)^{2}]}{2\,\vartheta _{00}(x^{5})[\vartheta _{00}(x^{1/5})^{2}-\vartheta _{00}(x)^{2}]}}-{\frac {1}{2}}{\biggr \}}{\biggr \rangle }}
The element of the fifth root can also be removed from the elliptic nome of the theta functions and transferred to the external tangent function. In this way, a formula can be created that only requires one of the three main theta functions:
R ( x ) = tan { 1 2 arctan [ 1 2 − ϑ 01 ( x ) 2 2 ϑ 01 ( x 5 ) 2 ] } 1 / 5 tan { 1 2 arccot [ 1 2 − ϑ 01 ( x ) 2 2 ϑ 01 ( x 5 ) 2 ] } 2 / 5 {\displaystyle R(x)=\tan {\biggl \{}{\frac {1}{2}}\arctan {\biggl [}{\frac {1}{2}}-{\frac {\vartheta _{01}(x)^{2}}{2\vartheta _{01}(x^{5})^{2}}}{\biggr ]}{\biggr \}}^{1/5}\tan {\biggl \{}{\frac {1}{2}}\operatorname {arccot} {\biggl [}{\frac {1}{2}}-{\frac {\vartheta _{01}(x)^{2}}{2\vartheta _{01}(x^{5})^{2}}}{\biggr ]}{\biggr \}}^{2/5}}
S ( x ) = tan { 1 2 arctan