Bailey pair tables
A pair \( (\alpha_n, \beta_n) \) is called a Bailey pair relative to \( x/q \) if \[ \beta_n = \sum_{r=0}^n \frac{\alpha_r}{(x;q)_{n+r} (q;q)_{n-r}} \tag{BP} .\] In each case, Eq. (BP) will be evaluated using a specialization of Bailey's sum of a very-well-poised \({}_6\psi_6\) series (Bailey, 1936); see also Gasper and Rahman (2004, p. 140 ff.) \[ \sum_{n=-\infty}^\infty \frac{(1-aq^{2n})(b,c,d,e;q)_n} {(1-a)(aq/b, aq/c, aq/d, aq/e;q)_n } \Big( \frac{a^2q}{bcd} \Big)^n = \frac{(q,q/a,aq,aq/(bc),aq/(bd),aq/(be), aq/(cd), aq/(ce),aq/(de),q)_\infty} {(q/b,q/c,q/d,q/e,aq/b,aq/c,aq/d,aq/e,a^2q/(bcde);q)_\infty}, \tag{BBS}\] where \( |a^2 q / (bcde)| < 1 \) and \(|q|<1\).
Always, \( \alpha_0 = \beta_0 = 1 . \)
Historical note: Group A Bailey pairs are transformations due to Rogers (1917), translated into Bailey pair language by Slater. Group J pairs are due to Slater (1952). Group P pairs are due to McLaughlin and Sills (2008).
Groups A, J, and P
In (BBS), set \( b = q^{-n/3}, c = q^{(1-n)/3}, d = q^{(2-n)/3},\) then replace \( q \) by \( q^3 \). This yields \[ \sum_{r = -N}^{\lfloor n/3 \rfloor} \frac{(1-aq^{6r})(q^{-n};q)_{3r} (e;q^3)_r a^{2r} q^{3nr} }{(1-a)(aq^{n+1};q)_{3r} (aq^3/e;q^3)_r e^r} = \frac{(a,q^3/a, aq^2/e, aq/e; q^3)_\infty}{(q, q^2, q^3/e, a^2/e; q^3)_\infty} \frac{(q,aq;q)_n (a^2/e;q^3)_n}{ (a;q)_{2n} (aq/e;q)_n }, \] where \( a \) is some power of \( q \) so that the series terminates below.
| Label | \( a \) | \( e \) | \(\beta_n\) | \(\alpha_{3r-1}\) | \(\alpha_{3r}\) | \(\alpha_{3r+1}\) | \(x\) |
|---|---|---|---|---|---|---|---|
| A(1) | \(q\) | \(\infty\) | \(1/(q;q)_{2n}\) | \(-q^{6r^2-5r+1}\) | \(q^{6r^2-r} + q^{6r^2+r}\) | \(-q^{6r^2-5r+1}\) | \(q\) |
| A(2) | \(q^2\) | \(\infty\) | \(1/(q^2;q)_{2n}\) | \(q^{6r^2-r}\) | \(q^{6r^2+r}\) | \(-q^{6r^2-5r+1} - q^{6r^2 + 7r + 2}\) | \(q^2\) |
| A(3) | \(q\) | \(\infty\) | \(q^n/(q;q)_{2n}\) | \(-q^{6r^2-2r}\) | \(q^{6r^2-2r} + q^{6r^2+2r}\) | \(-q^{6r^2+2r}\) | \(q\) |
| A(4) | \(q^2\) | \(\infty\) | \(q^n/(q^2;q)_{2n}\) | \(q^{6r^2-4r}\) | \(q^{6r^2+4r}\) | \(-q^{6r^2+8r+2} - q^{6r^2 + 4r}\) | \(q^2\) |
| A(5) | \(q\) | \( 0 \) | \(q^{n^2}/(q;q)_{2n}\) | \(-q^{3r^2-r}\) | \(q^{3r^2-r} + q^{3r^2+r}\) | \(-q^{3r^2+r}\) | \(q\) |
| A(6) | \(q^2\) | \( 0 \) | \(q^{n^2}/(q^2;q)_{2n}\) | \(q^{3r^2+r}\) | \(q^{3r^2-r}\) | \(-q^{3r^2+r} - q^{3r^2 + 5r + 2}\) | \(q^2\) |
| A(7) | \(q\) | \(0 \) | \(q^{n^2-n}/(q;q)_{2n}\) | \(-q^{3r^2-4r+1}\) | \(q^{3r^2-2r} + q^{3r^2+2r}\) | \(-q^{3r^2+4r+1}\) | \(q\) |
| A(8) | \(q^2\) | \(0\) | \(q^{n^2+n}/(q^2;q)_{2n}\) | \(q^{3r^2-2r}\) | \(q^{3r^2+2r}\) | \(-q^{3r^2+4r+1} - q^{3r^2 + 2r}\) | \(q^2\) |
| J(1) | \(1\) | \(1\) | \( (q^3;q^3)_{n-1}/[(q;q)_{2n-1} (q;q)_n]\) | \( 0 \) | \(q^{3r(3r-1)/2} (1+q^{3r})\) | \(0 \) | \(q\) |
| J(2) | \(q \) | \(q^2 \) | \( (q^3;q^3)_{n-1}/[(q;q)_{2n} (q;q)_{n-1}]\) | \( (-1)^{r+1} q^{9r(r-1)/2 + 1 } \) | \( (-1)^r (q^{3r(3r-1)/2} + q^{3r(3r+1)/2} )\) | \( (-1)^{r+1} q^{9r(r+1)/2 + 1} \) | \( q \) |
| J(3) | \(q\) | \(q^2\) | \( q^n (q^3;q^3)_{n-1}/[(q;q)_{2n} (q;q)_{n-1}]\) | \( (-1)^{r+1} q^{3r(r-1)/2 } \) | \( (-1)^n (q^{3r(3r-1)/2} + q^{3r(3r+1)/2} )\) | \( (-1)^{r+1} q^{3r(r+1)/2} \) | \( q \) |
| J(4) | \( q^2 \) | \( q^{5/2} \) | \( (q^{3/2} ;q^3)_{n}/[(q^2;q)_{2n} (q^{1/2};q)_{n}]\) | \( (-1)^{r} q^{9r^2/2 } \) | \( (-1)^r q^{9r^2/2} \) | \( (-1)^{r+1} ( q^{(9r^2+12r+4)/2} - q^{(9r^2 + 6r +1)/2} ) \) | \( q^2 \) |
| J(5) | \( q^2 \) | \( q^{5/2} \) | \( q^n (q^{3/2} ;q^3)_{n}/[(q^2;q)_{2n} (q^{1/2};q)_{n}]\) | \( (-1)^{r} q^{3r(3r-2)/2 } \) | \( (-1)^r (q^{3r(3r+2)/2} )\) | \( (-1)^{r+1} ( q^{(9r^2 + 6r)/2} - q^{(9r^2 + 12r +3)/2} \) | \( q^2 \) |
| J(6) | \( q^3 \) | \( q^{3} \) | \( (q^{3} ;q^3)_{n}/[(q^3;q)_{2n} (q;q)_{n}]\) | \( 0 \) | \( (1-q^{6r+3}) (-1)^n q^{3r(3r+1)/2 } / (1-q^3) \) | \( 0 \) | \( q^4 \) |
| P(1) | \( q \) | \(-q^2 \) | \( (-1 ;q^3)_{n}/[(q;q)_{2n} (-1;q)_{n}]\) | \( -q^{9r^2/2 - 9r/2 +1} \) | \( q^{9r^2/2 - 3r/2} (1+q^{3r}) \) | \( -q^{9r^2/2 + 9r/2 + 1} \) | \( q \) |
| P(2) | \( q \) | \(-q^2 \) | \( q^n (-1 ;q^3)_{n}/[(q;q)_{2n} (-1;q)_{n}]\) | \( -q^{9r^2/2 - 3r/2} \) | \( q^{9r^2/2 - 3r/2} (1+q^{3r}) \) | \( -q^{9r^2/2 + 3r/2 } \) | \( q \) |
| P(3) | \( q^2 \) | \(-q \) | \( (-q^3 ;q^3)_{n}/[(q^2;q)_{2n} (-q;q)_{n}]\) | \( -q^{9r^2/2 - 3r/2} \) | \( q^{9r^2/2 + 3r/2} \) | \( -2 q^{9r^2/2 + 9r/2 + 1} \) | \( q^2 \) |
| P(4) | \( q^2 \) | \(-q^{5/2} \) | \( (-q^{3/2} ;q^3)_{n}/[(q^2;q)_{2n} (-q^{1/2};q)_{n}]\) | \( q^{9r^2/2} \) | \( q^{9r^2/2} \) | \( - q^{9r^2/2 + 3r + 1/2} (1+q^{3r}) \) | \( q^2 \) |
| P(5) | \( q^2 \) | \(-q^{5/2} \) | \( q^n (-q^{3/2} ;q^3)_{n}/[(q^2;q)_{2n} (-q^{1/2};q)_{n}]\) | \( q^{9r^2/2 - 3r} \) | \( q^{9r^2/2 + 3r} \) | \( - q^{9r^2/2 } (q^{6r+3/2} + q^{3r}) \) | \( q^2 \) |
| P(6) | \( q \) | \(-q^2 \) | \( (1-q) (-1 ;q^3)_{n}/[(q;q)_{2n} (-1;q)_{n}]\) | \( -q^{9r^2/2 - 9r/2 +1} (1-q^{6r-1}) \) | \( q^{9r^2/2 - 3r/2} (1+q^{6r+1}) \) | \( 0 \) | \( q^2 \) |
| P(7) | \( q \) | \(q^2 \) | \( (q^3 ;q^3)_{n-1}/[(q^2;q)_{2n-1} (q;q)_{n-1}]\) | \( (-1)^{r+1} q^{9r^2/2 - 9r/2+1} (1-q^{6r-1})/(1-q) \) | \( (-1)^r q^{9r^2/2 - 3r/2} (1-q^{6r+1})/(1-q) \) | \( (-1)^{r+1} q^{9r^2/2 + 3r/2+1} (1-q^{6r+3})/(1-q)\) | \( q^2 \) |
Groups B, E (partial), F, and H
In (BBS), set \( b = q^{-n} \) and \(e = a\). This yields \[ \sum_{r = 0}^{n} \frac{(1-aq^{2r}) (-1)^r a^r q^{r(r+1)/2} (a,c,d;q)_r}{(1-a) (aq;q)_{n+r} (q,aq/c, aq/d;q)_r (cd)^r} = \frac{(aq/[cd];q)_n}{(q,aq/c,aq/d;q)_n}. \]
Historical note: Group B and F Bailey pairs are transformations due to Rogers (1917), translated into Bailey pair language by Slater. Group H pairs are due to Slater (1951, 1952). Group E is named by Slater to correspond to Rogers' transformations, but it makes more sense to put part of the E group here and the rest below.
| Label | \( a \) | \( c \) | \( d\) | \(\beta_n\) | \(\alpha_{r}\) | \(x\) |
|---|---|---|---|---|---|---|
| B(1) | \(q\) | \(\infty\) | \(\infty\) | \(1/(q;q)_{n}\) | \( (-1)^r q^{r(3r-1)/2} (1+q^r) \) | \(q\) | B(2) | \(q^n/(q;q)_{n}\) | \( (-1)^n q^{3r(r-1)/2} (1+q^{3r}) \) | \(q\) | B(3) | \(q\) | \(\infty\) | \(\infty\) | \(1/(q;q)_{n}\) | \( (-1)^r q^{r(3r+1)/2} (1-q^{2r+1})/(1-q) \) | \(q^2\) | B(4) | \(q^n/(q;q)_{n}\) | \( \{ (-1)^r q^{(3r+2)(r-1)/2} (1-q^r) + q^{(3r+5)/2}(1-q^{r+1})\}/(1-q) \) | \(q^2\) |
| E(3) | \(q\) | \(-q\) | \(\infty\) | \(1/(q^2;q^2)_{n}\) | \( (-1)^r q^{r^2} (1-q^{2r+1})/(1-q) \) | \(q^2\) |
| E(6) | \(q\) | \(-q\) | \(\infty\) | \(q^n/(q^2;q^2)_{n}\) | \( (-1)^r q^{r(r-1)} (1+q^{2r})/(1-q) \) | \(q^2\) |
| E(7) | \(q\) | \(-q\) | \(0 \) | \((-1)^n q^{-n}/(q^2;q^2)_{n}\) | \( (-1)^r q^{r^2} (q^{-r} + q^{r+1} )/(1-q) \) | \(q^2\) | F(1) | \(1\) | \( q^{1/2} \) | \( \infty \) | \( 1/[(q^{1/2};q)_{n} (q;q)_n ]\) | \( q^{r^2 - r/2} (1+q^r) \) | \(q\) | F(2) | \( q \) | \( q^{1/2} \) | \( \infty \) | \( 1/[(q^{3/2};q)_{n} (q;q)_n ]\) | \( q^{r^2 - r/2} (1+q^{r+1/2})/(1+q^{1/2}) \) | \( q^2 \) | F(3) | \(1\) | \( q^{1/2} \) | \( 0 \) | \( q^{-n/2}/[(q^{1/2};q)_{n} (q;q)_n ]\) | \( q^{-r/2} (1+q^r) \) | \(q\) | F(4) | \( q \) | \( q^{1/2} \) | \( 0 \) | \( q^{-n/2}/[(q^{3/2};q)_{n} (q;q)_n ]\) | \( q^{- r/2} (1+q^{r+1/2})/(1+q^{1/2}) \) | \( q^2 \) | H(1) | \(1\) | \( -q^{1/2} \) | \( q^{1/2} \) | \((-1;q)_{n}/[(q;q^2)_{n} (q;q)_n ]\) | \( q^{r(r-1)/2} (1+q^r) \) | \(q\) | H(2) | \(1\) | \( -q^{1/2} \) | \( \infty \) | \( (-q^{1/2};q)_{n}^{-1} (q;q)_n^{-1} \) | \( (-1)^r q^{r(r-1)/2} (1+q^r) \) | \(q\) | H(3) | \(q\) | \( 0 \) | \( 0 \) | \( (-1)^n q^{-n(n+3)/2} / (q;q)_n \) | \( (-1)^r q^{-r(r-3)/2} (1+q^{3r} ) \) | \(q\) | H(4) | \( (-1)^n q^{-n(n+1)/2} / (q;q)_n \) | \( (-1)^r q^{-r(r-1)/2} (1+q^{r} ) \) | \(q\) | H(5) | \( q \) | \( q \) | \( 0 \) | \( q^{-n} / (q;q)_n^2 \) | \( q^{-r} (1 - q^{2r} ) \) | \(q\) | H(6) | \( 1 / (q;q)_n^2 \) | \( 0 \) | \(q\) | H(7) | \( q \) | \( -q \) | \( \infty \) | \( (-1)^n / (q^2;q^2)_n \) | \( 2 (-1)^r \) | \(q\) | H(8) | \( \) | \( \) | \( \) | \( (-1)^n q^{-n} / (q^2;q^2)_n \) | \( (-1)^r q^{-r} (1+q^{2r}) \) | \(q\) | H(9) | \( q \) | \( -q \) | \( \infty \) | \( q^{n} / (q;q)_n^2 \) | \( q^{r(r+1)} (1 - q^{-2r}) \) | \(q\) | H(10) | \( q \) | \( q \) | \( -q \) | \( 2 (1+q^n) / (q;q)_n^2 \) | \( q^{r(r-1)/2} (1 - q^{r}) \) | \(q\) | H(11) | \( \) | \( \) | \( \) | \( 2 q^n (1+q^n) / (q;q)_n^2 \) | \( q^{r(r+1)/2} (1 - q^{-r}) \) | \(q\) | H(12) | \( q^2 \) | \( q \) | \( 0 \) | \( (q^2;q)_n^{-1} (q;q)_n^{-1} \) | \( q^{-r} (1 - q^{2r+1}) \) | \(q^2 \) | H(13) | \( \) | \( \) | \( \) | \( (q^2;q)_n^{-1} (q;q)_n^{-1} \) | \( 0 \) | \(q^2 \) | H(14) | \( q^2 \) | \( q \) | \( \infty \) | \( q^n (q^2;q)_n^{-1} (q;q)_n^{-1} \) | \( -q^{r^2-1} (1 - q^{2r+1}) \) | \(q^2 \) | H(15) | \( q^2 \) | \( q \) | \( q^{3/2} \) | \( (q^{1/2};q)_n (q^2;q)_n^{-1} (q;q)_n^{-1} (q^{3/2};q)_n^{-1} \) | \( (-1)^r (q^{r^2/2} + q^{(r+1)^2/2} ) \) | \(q^2 \) | H(16) | \( \) | \( \) | \( \) | \( q^n (q^{1/2};q)_n (q^2;q)_n^{-1} (q;q)_n^{-1} (q^{3/2};q)_n^{-1} \) | \( (-1)^r (q^{r^2/2+r} + q^{(r^2-1)/2} ) \) | \(q^2 \) | H(17) | \( 1 \) | \( 0 \) | \( \infty \) | \( 0 \) | \( (-1)^r q^{r(r-1)/2} (1 + q^r ) \) | \( q \) | H(18) | \( q \) | \( q^{1/2} \) | \( q \) | \( (1 - q^{1/2}) (1 - q^{n+1/2})^{-1} (q;q)_n^{-2} \) | \( (-1)^r q^{r^2/2} (1 + q^{r+1/2}) / (1 + q^{1/2}) \) | \( q^2 \) | H(19) | \( q \) | \( q^{1/2} \) | \( -q \) | \( (-q^{1/2};q)_n (q^{3/2};q)_n^{-1} (q^2;q^2)_n^{-1} \) | \( q^{r^2/2} (1 + q^{r+1/2}) / (1 + q^{1/2}) \) | \( q^2 \) |
Groups C, E (partial), G, and I
In (BBS), set \( b = q^{-n/2} \), \(c= q^{(1-n)/2} \) and replace \( q \) by \(q^2\) throughout. This yields \[ \sum_{r = -\lfloor n/2 \rfloor}^{\lfloor n/2 \rfloor} \frac{(1-aq^{4r}) (q^{-n};q)_{2r} a^{2r} q^{2nr+r} (d,e;q^2)_r} {(1-a) (aq^{n+1};q)_{2r} (aq^2/d, aq^2/e;q^2)_r (de)^r} = \frac{(q^2/q, aq/d, aq/e, aq^2/[de]; q^2)_\infty } { (q, q^2/d, q^2/e, a^2q/[de];q^2)_\infty } \frac{(q, aq, a^2q/[de];q^2)_n}{(aq;q^2)_n (aq/d,aq/d;q)_n}, \] where \(a\) has a a value such that the series terminates below at \( r = -\lfloor n/2 \rfloor \).
Historical note: Group C, E, and G Bailey pairs are transformations due to Rogers (1917), translated into Bailey pair language by Slater. Group I pairs are due to Slater (1951, 1952).
| Label | \( a \) | \( d \) | \( e \) | \(\beta_n\) | \(\alpha_{2r}\) | \(\alpha_{2r+1}\) | \(x\) |
|---|---|---|---|---|---|---|---|
| C(1) | \( 1 \) | \( 1 \) | \( \infty \) | \( (q;q^2)_n^{-1} (q;q)_n^{-1} \) | \( (-1)^r q^{r(3r-1)/2} (1+q^{r}) \) | \( 0 \) | \( q \) |
| C(2) | \( q^2 \) | \( q^2 \) | \( \infty \) | \( q^n (q;q^2)_n^{-1} (q;q)_n^{-1} \) | \( (-1)^r q^{3r^2-r} (1+q^{2r}) \) | \( (-1)^{r+1} q^{3r^2+r} (1-q^{4r+2}) \) | \( q \) |
| C(3) | \( q^2 \) | \( q^2 \) | \( \infty \) | \( (q^3;q^2)_n^{-1} (q;q)_n^{-1} \) | \( (-1)^r q^{3r^2+r} \) | \( (-1)^{r+1} q^{3r^2+5r+2} \) | \( q^2 \) |
| C(4) | \( q^2 \) | \( q^2 \) | \( \infty \) | \( q^n (q^3;q^2)_n^{-1} (q;q)_n^{-1} \) | \( (-1)^r q^{3r^2+3r} \) | \( (-1)^{r+1} q^{3r^2+3r} \) | \( q^2 \) |
| C(5) | \( 1 \) | \( 1 \) | \( 0 \) | \( q^{n(n-1)/2} (q;q^2)_n^{-1} (q;q)_n^{-1} \) | \( (-1)^r q^{r^2-r} (1+q^{2r}) \) | \( 0 \) | \( q \) |
| C(6) | \( q^2 \) | \( q^2 \) | \( 0 \) | \( q^{n(n-1)/2} (q^3;q^2)_n^{-1} (q;q)_n^{-1} \) | \( (-1)^r q^{r^2-r} \) | \( (-1)^{r+1} q^{r^2+3r+2} \) | \( q^2 \) |
| C(7) | \( q^2 \) | \( q^2 \) | \( 0 \) | \( q^{n(n+1)/2} (q^3;q^2)_n^{-1} (q;q)_n^{-1} \) | \( (-1)^r q^{r^2+r} \) | \( (-1)^{r+1} q^{r^2+5r+2} \) | \( q^2 \) |
| E(1) | \( q \) | \( \infty \) | \( \infty \) | \( 1/(q^2;q^2)_n \) | \( 2q^{4r^2} \) | \( -2q^{4r^2+4r+1} \) | \( q \) |
| E(2) | \( q \) | \( 0 \) | \( 0 \) | \( (-1)^n/(q^2;q^2)_n \) | \( 2 \) | \( -2 \) | \( q \) |
| E(4) | \( q \) | \( \infty \) | \( \infty \) | \( q^n/(q^2;q^2)_n \) | \( q^{4r^2-2r} (1+q^{4r}) \) | \( -q^{4r^2+2r}(1+q^{4r+2}) \) | \( q \) |
| E(5) | \( q \) | \( 0 \) | \( 0 \) | \( (-1)^n q^{-n} /(q^2;q^2)_n \) | \( q^{-2r} + q^{2r} \) | \( -q^{2r+1} - q^{-2r-1} \) | \( q \) |
| G(1) | \( q \) | \( -q^{3/2} \) | \( \infty \) | \( (q^2;q^2)_n^{-1} (-q^{1/2};q)_n^{-1} \) | \( q^{3r^2}(q^{r/2} + q^{-r/2}) \) | \( -q^{3r^2 + 7r/2 + 1} - q^{3r^2 + 5r/2 + 1/2} \) | \( q \) |
| G(2) | \( q \) | \( -q^{1/2} \) | \( \infty \) | \( (q^2;q^2)_n^{-1} (-q^{3/2};q)_n^{-1} \) | \( q^{3r^2+r/2}(1-q^{2r+1/2})(1-q^{1/2})^{-1} \) | \( q^{3r^2-r/2}(1-q^{-2r+1/2})(1-q^{1/2})^{-1} \) | \( q^2 \) |
| G(3) | \( q \) | \( -q^{3/2} \) | \( \infty \) | \( q^n (q^2;q^2)_n^{-1} (-q^{1/2};q)_n^{-1} \) | \( q^{3r^2}(q^{r/2} + q^{-r/2}) \) | \( -q^{3r^2 + 3r/2} - q^{3r^2 + 9r/2 + 3/2} \) | \( q \) |
| G(4) | \( q \) | \( -q^{3/2} \) | \(0 \) | \( (-1)^n q^{n^2/2} (q^2;q^2)_n^{-1} (-q^{1/2};q)_n^{-1} \) | \( q^{r^2}(q^{r/2} + q^{-r/2}) \) | \( -q^{r^2 + r/2} - q^{r^2 + 3r/2 + 1/2} \) | \( q \) |
| G(5) | \( q \) | \( -q^{1/2} \) | \(0 \) | \( (-1)^n q^{n^2/2} (q^2;q^2)_n^{-1} (-q^{3/2};q)_n^{-1} \) | \( (-1)^r q^{r^2 - r/2} (1-q^{2r+1/2}) / (1-q^{1/2}) \) | \( (-1)^r q^{r^2 + r/2} (1-q^{-2r+1/2}) / (1-q^{1/2}) \) | \( q^2 \) |
| G(6) | \( q \) | \( -q^{1/2} \) | \(0 \) | \( (-1)^n q^{n^2/2-n} (q^2;q^2)_n^{-1} (-q^{1/2};q)_n^{-1} \) | \( q^{r^2}(q^{3r/2} + q^{-3r/2}) \) | \( -q^{r^2+5r/2+1} - q^{r^2-r/2 -1/2} \) | \( q \) |
| I(1) | \( q \) | \( q^{3/2} \) | \(0 \) | \( q^{n^2/2-n} (q^2;q^2)_n^{-1} (q^{1/2};q)_n^{-1} \) | \( (-1)^r q^{r^2}(q^{3r/2} + q^{-3r/2}) \) | \( (-1)^r q^{r^2}(q^{(r+1)/2} - q^{5r/2+1} \) | \( q \) |
| I(2) | \( q \) | \( q^{3/2} \) | \(0 \) | \( q^{n^2/2} (q^2;q^2)_n^{-1} (q^{1/2};q)_n^{-1} \) | \( (-1)^r q^{r^2}(q^{r/2} + q^{-r/2}) \) | \( (-1)^{r+1} q^{r^2}(q^{(r/2} + q^{3r/2+1/2}) \) | \( q \) |
| I(3) | \( q \) | \( q^{3/2} \) | \( \infty \) | \( (q^2;q^2)_n^{-1} (q^{1/2};q)_n^{-1} \) | \( (-1)^r q^{3r^2}(q^{r/2} + q^{-r/2}) \) | \( (-1)^{r+1} q^{3r^2}(q^{7r/2+1} - q^{5r/2+1/2}) \) | \( q \) |
| I(4) | \( q \) | \( q^{3/2} \) | \( \infty \) | \( q^n (q^2;q^2)_n^{-1} (q^{1/2};q)_n^{-1} \) | \( (-1)^r q^{3r^2}(q^{3r/2} + q^{-3r/2}) \) | \( (-1)^{r+1} q^{3r^2}(q^{3r/2} - q^{9r/2+3/2}) \) | \( q \) |
| I(5) | \( q \) | \( q \) | \( q^2 \) | \( (1+q^n)^{-1} (q;q)_n^{-2} \) | \( q^{2r^2 - r} \) | \( -q^{2r^2+3r+1} \) | \( q \) |
| I(6) | \( q \) | \( q \) | \( q^2 \) | \( q^n (1+q^n)^{-1} (q;q)_n^{-2} \) | \( q^{2r^2 + r} \) | \( -q^{2r^2+r} \) | \( q \) |
| I(7) | \( q \) | \( q^{3/2} \) | \( -q^{3/2} \) | \( (-1;q^2)_n / (q;q)_{2n} \) | \( (-1)^r q^{2r^2} (q^r + q^{-r} ) \) | \( (-1)^r q^{2r^2} (q^r - q^{3r+1} ) \) | \( q \) |
| I(8) | \( q \) | \( q^{3/2} \) | \( -q^{3/2} \) | \( q^n (-1;q^2)_n / (q;q)_{2n} \) | \( (-1)^r q^{2r^2} (q^r + q^{-r} ) \) | \( (-1)^r q^{2r^2} (q^{r+1} - q^{r} ) \) | \( q \) |
| I(9) | \( 1 \) | \( 1 \) | \( q\) | \( (-q;q)_{n-1} (q;q^2)_n^{-1} (q;q)_n^{-1} \) | \( q^{2r^2} (q^r + q^{-r} ) \) | \( 0 \) | \( q \) |
| I(10) | \( q^2 \) | \( q^2 \) | \( q^2 \) | \( (1-q) (1-q^{2n+1})^{-1} (q;q)_n^{-2} \) | \( q^{2r^2} \) | \( -q^{2r^2+4r+2} \) | \( q^2 \) |
| I(11) | \( q^2 \) | \( q^2 \) | \( q^2 \) | \( (1-q) (1-q^{2n+1})^{-1} (q;q)_n^{-2} \) | \( q^{2r^2} \) | \( -q^{2r^2+2r} \) | \( q^2 \) |
| I(12) | \( q^2 \) | \( q^2 \) | \( -q^2 \) | \( (-q;q^2)_n (q^3;q^2)_n^{-1} (q^2;q^2)_n^{-1} \) | \( (-1)^r q^{2r^2} \) | \( (-1)^{r+1} q^{2(r+1)^2} \) | \( q^2 \) |
| I(13) | \( q^2 \) | \( q^2 \) | \( -q^2 \) | \( q^n (-q;q^2)_n (q^3;q^2)_n^{-1} (q^2;q^2)_n^{-1} \) | \( (-1)^r q^{2r^2+2r} \) | \( (-1)^{r+1} q^{2r^2+2r} \) | \( q^2 \) |
| I(14) | \( 1 \) | \( 1 \) | \( q \) | \( (-q^2;q^2)_{n-1} (q;q^2)_n^{-1} (q;q)_n^{-1} (-q;q)_{n-1}^{-1} \) | \( (-1)^r (1+q^{2r}) q^{2r^2 - r} \) | \( 0 \) | \( q \) |
| I(15) | \( q \) | \( q^{1/2} \) | \( q^{3/2} \) | \( (q;q^2)_{n} (q^2;q^2)_n^{-1} (q^{1/2};q)_n^{-1} (q^{3/2};q)_{n}^{-1} \) | \( (1+q^{2r+1/2}) q^{2r^2} / (1+q^{1/2}) \) | \( (1+q^{2r+3/2}) q^{2r^2} / (1+q^{1/2}) \) | \( q^2 \) |
| I(16) | \( q \) | \( q^{1/2} \) | \( -q^{3/2} \) | \( (q;q^2)_{n} (q^2;q^2)_n^{-1} (-q^{1/2};q)_n^{-1} (q^{3/2};q)_{n}^{-1} \) | \( (1+q^{2r+1/2})(-1)^r q^{2r^2} / (1+q^{1/2}) \) | \( (1+q^{2r+3/2}) (-1)^r q^{2r^2} / (1+q^{1/2}) \) | \( q^2 \) |
| I(17) | \( q^2 \) | \( q^{2} \) | \( q^3 \) | \( (q^2;q^2)_{n} (q^3;q^2)_n^{-1} (q;q)_n^{-1} (q^{2};q)_{n}^{-1} \) | \( (1+q^{2r+1}) q^{2r^2+r} / (1+q) \) | \( 0 \) | \( q^3 \) |
| I(18) | \( q^2 \) | \( q^{2} \) | \( -q \) | \( (-q^2;q^2)_{n} (q^3;q^2)_n^{-1} (q;q)_n^{-1} (-q^{2};q)_{n}^{-1} \) | \( (1-q^{2r+1})(-1)^r q^{2r^2+r} / (1-q) \) | \( 0 \) | \( q^3 \) |
Group K
In (BBS), set \( b = q^{-n/4} \), \( c = q^{(1-n)/4} \), \( d = q^{(2-n)/4} \), \( e = q^{(3-n)/4} \), and replace \( q \) by \(q^4\) throughout. This yields \[ \sum_{r = -\lfloor n/4 \rfloor}^{\lfloor n/4 \rfloor} \frac{(1-aq^{8r}) (q^{-n};q)_{4r} a^{2r} q^{4nr-2r} a^{2r}} {(1-a) (aq^{n+1};q)_{4r} } = \frac{(q^4/a, a/q, a, aq; q^4)_\infty } { (q, q^2, q^3, a^2/q^2;q^4)_\infty } \frac{(aq, q;q)_n (-a/q;q^2)_n}{(a;q)_{2n}} \] where \(a\) has a value such that the series terminates below at \( r = -\lfloor n/4 \rfloor \).
Historical note: Group K Bailey pairs are due to Slater (1951).
| Label | \( a \) | \( \beta_n \) | \( \alpha_{4r-2} \) | \(\alpha_{4r-1} \) | \(\alpha_{4r}\) | \(\alpha_{4r+1}\) | \(x\) |
|---|---|---|---|---|---|---|---|
| K(1) | \( q \) | \( (-q^2;q^2)_{n-1}/(q;q)_{2n} \) | \( 0 \) | \( -q^{8r^2-6r+1} \) | \(q^{8r^2-2r}(1+q^{4r})\) | \( -q^{8r^2+6r+1} \) | \(q \) |
| K(2) | \( q \) | \( q^n (-q^2;q^2)_{n-1}/(q;q)_{2n} \) | \( 0 \) | \( -q^{8r^2-2r} \) | \(q^{8r^2-2r}(1+q^{4r})\) | \( -q^{8r^2+2r} \) | \(q \) |
| K(3) | \( q^2 \) | \( (-q;q^2)_{n}/(q^2;q)_{2n} \) | \( -q^{8r^2-8r+2} \) | \( q^{8r^2} \) | \(q^{8r^2}\) | \( -q^{8r^2+8r+2} \) | \(q^2 \) |
| K(4) | \( q^2 \) | \( q^n (-q;q^2)_{n}/(q^2;q)_{2n} \) | \( -q^{8r^2-4r} \) | \( q^{8r^2-4r} \) | \( q^{8r^2+4r}\) | \( -q^{8r^2+4r} \) | \(q^2 \) |
| K(5) | \( q^3 \) | \((-q^2;q^2)_{n}/(q^3;q)_{2n} \) | \( q^{8r^2-2r} \) | \( 0 \) | \( q^{8r^2+2r}\) | \( -q^{8r^2+5r+1}(1+q^{5r+2}) \) | \(q^3 \) |
| K(6) | \( q^3 \) | \( q^n (-q^2;q^2)_{n}/(q^3;q)_{2n} \) | \( q^{8r^2-6r} \) | \( 0 \) | \( q^{8r^2+6r}\) | \( -q^{8r^2+6r}(1+q^{4r+2}) \) | \(q^3 \) |
Group L
In (BBS), set \( d = q^{-n/2} \) and \( e = -q^{n/2} \); replace \( a \) by \( a^{1/2} \) and \( q \) by \(q^{1/2}\) throughout.
Historical note: Group L Bailey pairs are due to Slater (1952).
| Label | \( a^{1/2} \) | \( b\) | \( c \) | \(\beta_n \) | \(\alpha_{r}\) | \(x\) |
|---|---|---|---|---|---|---|
| L(1) | \( 1 \) | \( 0 \) | \( 1 \) | \( (-1)^n q^{n^2/2} (q;q)_n^{-1} (-q^{1/2};q^{1/2})_{2n}^{-1} \) | \( (1+q^{r/2})(-1)^r q^{r(r-1)/4} \) | \( q \) |
| L(2) | \( 1 \) | \( -q^{1/4} \) | \( 1 \) | \( (q^{1/4};q^{1/2})_{2n} / (q;q)_{2n} \) | \( (1+q^{r/2})(-1)^r q^{r(2r-1)/4} \) | \( q \) |
| L(3) | \( 1 \) | \( \infty \) | \( 1 \) | \( (q;q)_{n}^{-1} (-q^{1/2};q^{1/2})_{2n}^{-1} \) | \( (1+q^{r/2})(-1)^r q^{r(3r-1)/4} \) | \( q \) |
| L(4) | \( -1 \) | \( 0 \) | \( iq^{1/4} \) | \( q^{n(n-1)/2}(-q^{1/2};q)_{n}^{-1} (q^{1/2};q^{1/2})_{2n}^{-1} \) | \( (1+q^{r})(-1)^r q^{r(r-2)/2} \) | \( q \) |
| L(5) | \( -1 \) | \( iq^{1/4} \) | \( -iq^{1/4} \) | \( (-1;q^{1/2} )_{2n} (-q^{1/2};q)_{n}^{-1} (q^{1/2};q^{1/2})_{2n}^{-1} \) | \( (1+q^{r}) q^{r(r-1)/2} \) | \( q \) |
| L(6) | \( -1 \) | \( iq^{1/4} \) | \( \infty \) | \( (-q^{1/2};q)_{n}^{-1} (q^{1/2};q^{1/2})_{2n}^{-1} \) | \( (1+q^{r}) (-1)^{r/2} q^{r(3r-2)/4} \frac{ 1+(-1)^r }{2} \) | \( q \) |
Group M
In (BBS), set \( b = q^{-n/4} \) , \( c = -q^{n/4} \), \( d = iq^{n/4} \), and \( e = -iq^{n/4} \); replace \( a \) by \( a^{1/4} \) and \( q \) by \(q^{1/4}\) throughout.
Historical note: Group M Bailey pairs are due to Slater (1952).
| Label | \( a^{1/4} \) | \(\beta_n \) | \(\alpha_{r}\) | \(x\) |
|---|---|---|---|---|
| M(1) | \(-1 \) | \( (q^{1/4};q^{1/2})_n / (q;q)_{2n} \) | \( (-1)^r q^{r^2/2 - r/4} (1+q^{r/2}) \) | \(q \) |
| M(2) | \(-q^{1/4} \) | \( (-q^{1/4};q^{1/2})_{2n+1} / (q;q)_{2n+1} \) | \( q^{r(2r+1)/4} (1+q^{(2r+1)/4}) /(1-q) \) | \(q^2 \) |
| M(3) | \(i q^{1/4} \) | \( (q^{1/4};q^{1/2})_{2n+1} / (q;q)_{2n+1} \) | \( (-1)^r q^{r^2/2} (q^{r/2} - q^{(3r+1)/4}) /(1-q) \) | \(q^2 \) |
References
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W. N. Bailey (1936) Series of hypergeometric type which are infinite in both directions, Quarterly J. Math. (Oxford) 7, 105–115. DOI
G. Gasper and M. Rahman (2004) Basic Hypergeometric Series, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 96, Cambridge University Press. DOI
J. McLaughlin and A. V. Sills (2008) Ramanujan–Slater type identities related to the moduli 18 and 24, J. Math. Anal Appl. 344, 765–777. DOI
L. J. Rogers (1917) On two theorems of combinatory analysis and allied identities, Proc. London Math. Soc. s2-16, 315–336. DOI
L. J. Slater (1951) A new proof of Rogers's transformations of infinite series, Proc. London Math. Soc. s2-53, 460–475. DOI
L. J. Slater (1952) Further identities of the Rogers–Ramanujan type, Proc. London Math. Soc. s2-54, 147–167. DOI