Some Implications of the WP-Bailey Tree

James Mc Laughlin; Peter Zimmer

arXiv:1901.04840·math.NT·January 16, 2019·Adv. Appl. Math.

Some Implications of the WP-Bailey Tree

James Mc Laughlin, Peter Zimmer

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TL;DR

This paper explores special cases of WP-Bailey chains to derive new transformations of basic hypergeometric series, introduces two novel WP-Bailey pairs, and discusses their implications for Rogers-Ramanujan type identities.

Contribution

It introduces two new WP-Bailey pairs and derives novel transformations for basic hypergeometric series, expanding the understanding of WP-Bailey chains.

Findings

01

Derived new transformations of hypergeometric series.

02

Introduced two new WP-Bailey pairs.

03

Explored implications for Rogers-Ramanujan identities.

Abstract

We consider a special case of a WP-Bailey chain of George Andrews, and use it to derive a number of curious transformations of basic hypergeometric series. We also derive two new WP-Bailey pairs, and use them to derive some additional new transformations for basic hypergeometric series. Finally, we briefly consider the implications of WP-Bailey pairs\\ $(α_{n} (a, k)$ , $β_{n} (a, k))$ , in which $α_{n} (a, k)$ is independent of $k$ , for generalizations of identities of the Rogers-Ramanujan type.

Equations148

β_{n} (a, k)

β_{n} (a, k)

= \frac{( k / a , k ; q ) _{n}}{( a q , q ; q ) _{n}} j = 0 \sum n \frac{( q ^{- n} ) _{j} ( k q ^{n} ) _{j}}{( a q ^{1 - n} / k ) _{j} ( a q ^{n + 1} ) _{j}} (\frac{q a}{k})^{j} α_{j} (a, k) .

α_{n}^{'} (a, k)

α_{n}^{'} (a, k)

β_{n}^{'} (a, k)

a s \times j = 0 \sum n \frac{( 1 - c q ^{2 j} ) ( ρ _{1} , ρ _{2} ) _{j} ( k / c ) _{n - j} ( k ) _{n + j}}{( 1 - c ) ( k ρ _{1} / a , k ρ _{2} / a ) _{n} ( q ) _{n - j} ( q c ) _{n + j}} (\frac{k}{c})^{j} β_{j} (a, c),

\tilde{α}_{n} (a, k)

\tilde{α}_{n} (a, k)

\tilde{β}_{n} (a, k)

(a; q)_{n - r} = \frac{q ^{r (r + 1) /2} ( a ; q ) _{n}}{( q ^{1 - n} / a ; q ) _{r} ( - a q ^{n} ) ^{r}} .

(a; q)_{n - r} = \frac{q ^{r (r + 1) /2} ( a ; q ) _{n}}{( q ^{1 - n} / a ; q ) _{r} ( - a q ^{n} ) ^{r}} .

β_{n} (a, k) = j = 0 \sum n \frac{( k / a ) _{n - j} ( k ) _{n + j}}{( q ) _{n - j} ( a q ) _{n + j}} α_{j} (a, k),

β_{n} (a, k) = j = 0 \sum n \frac{( k / a ) _{n - j} ( k ) _{n + j}}{( q ) _{n - j} ( a q ) _{n + j}} α_{j} (a, k),

n = 0 \sum N \frac{( 1 - k q ^{2 n} ) ( ρ _{1} , ρ _{2} , k a q ^{N + 1} / ρ _{1} ρ _{2} , q ^{- N} ; q ) _{n}}{( 1 - k ) ( k q / ρ _{1} , k q / ρ _{2} , ρ _{1} ρ _{2} q ^{- N} / a , k q ^{1 + N} ; q ) _{n}} q^{n} β_{n} (a, k) = \frac{( k q , k q / ρ _{1} ρ _{2} , a q / ρ _{1} , a q / ρ _{2} ; q ) _{N}}{( k q / ρ _{1} , k q / ρ _{2} , a q / ρ _{1} ρ _{2} , a q ; q ) _{N}} a s d a d a s d a s d ab v v b v m v mbn v bn v d a ss d a s \times n = 0 \sum N \frac{( ρ _{1} , ρ _{2} , k a q ^{N + 1} / ρ _{1} ρ _{2} , q ^{- N} ; q ) _{n}}{( a q / ρ _{1} , a q / ρ _{2} , a q ^{1 + N} , ρ _{1} ρ _{2} q ^{- N} / k ; q ) _{n}} (\frac{a q}{k})^{n} α_{n} (a, k),

n = 0 \sum N \frac{( 1 - k q ^{2 n} ) ( ρ _{1} , ρ _{2} , k a q ^{N + 1} / ρ _{1} ρ _{2} , q ^{- N} ; q ) _{n}}{( 1 - k ) ( k q / ρ _{1} , k q / ρ _{2} , ρ _{1} ρ _{2} q ^{- N} / a , k q ^{1 + N} ; q ) _{n}} q^{n} β_{n} (a, k) = \frac{( k q , k q / ρ _{1} ρ _{2} , a q / ρ _{1} , a q / ρ _{2} ; q ) _{N}}{( k q / ρ _{1} , k q / ρ _{2} , a q / ρ _{1} ρ _{2} , a q ; q ) _{N}} a s d a d a s d a s d ab v v b v m v mbn v bn v d a ss d a s \times n = 0 \sum N \frac{( ρ _{1} , ρ _{2} , k a q ^{N + 1} / ρ _{1} ρ _{2} , q ^{- N} ; q ) _{n}}{( a q / ρ _{1} , a q / ρ _{2} , a q ^{1 + N} , ρ _{1} ρ _{2} q ^{- N} / k ; q ) _{n}} (\frac{a q}{k})^{n} α_{n} (a, k),

n = 0 \sum N \frac{( q ^{- N} ; q ) _{n}}{( q ^{- N} k ^{2} / a ^{2} ; q ) _{n}} q^{n} β_{n} (a, k) = \frac{( q a / k , q a ^{2} / k ; q ) _{N}}{( q a , q a ^{2} / k ^{2} ; q ) _{N}} \times n = 0 \sum N \frac{( q ^{N + 1} a ^{2} / k , q ^{- N} ; q ) _{n} ( k ; q ) _{2 n}}{( a q ^{1 + N} , q ^{- N} k / a ; q ) _{n} ( q a ^{2} / k ; q ) _{2 n}} (\frac{a q}{k})^{n} α_{n} (a, k) .

n = 0 \sum N \frac{( q ^{- N} ; q ) _{n}}{( q ^{- N} k ^{2} / a ^{2} ; q ) _{n}} q^{n} β_{n} (a, k) = \frac{( q a / k , q a ^{2} / k ; q ) _{N}}{( q a , q a ^{2} / k ^{2} ; q ) _{N}} \times n = 0 \sum N \frac{( q ^{N + 1} a ^{2} / k , q ^{- N} ; q ) _{n} ( k ; q ) _{2 n}}{( a q ^{1 + N} , q ^{- N} k / a ; q ) _{n} ( q a ^{2} / k ; q ) _{2 n}} (\frac{a q}{k})^{n} α_{n} (a, k) .

β_{n} (a, k) = j = 0 \sum n \frac{( k / a ) _{n - j} ( k ) _{n + j}}{( q ) _{n - j} ( a q ) _{n + j}} α_{j} (a, k),

β_{n} (a, k) = j = 0 \sum n \frac{( k / a ) _{n - j} ( k ) _{n + j}}{( q ) _{n - j} ( a q ) _{n + j}} α_{j} (a, k),

n = 0 \sum \infty \frac{( 1 - k q ^{2 n} ) ( ρ _{1} , ρ _{2} ; q ) _{n}}{( 1 - k ) ( k q / ρ _{1} , k q / ρ _{2} ; q ) _{n}} (\frac{a q}{ρ _{1} ρ _{2}})^{n} β_{n} (a, k) = \frac{( k q , k q / ρ _{1} ρ _{2} , a q / ρ _{1} , a q / ρ _{2} ; q ) _{\infty}}{( k q / ρ _{1} , k q / ρ _{2} , a q / ρ _{1} ρ _{2} , a q ; q ) _{\infty}} n = 0 \sum \infty \frac{( ρ _{1} , ρ _{2} ; q ) _{n}}{( a q / ρ _{1} , a q / ρ _{2} ; q ) _{n}} (\frac{a q}{ρ _{1} ρ _{2}})^{n} α_{n} (a, k),

n = 0 \sum \infty \frac{( 1 - k q ^{2 n} ) ( ρ _{1} , ρ _{2} ; q ) _{n}}{( 1 - k ) ( k q / ρ _{1} , k q / ρ _{2} ; q ) _{n}} (\frac{a q}{ρ _{1} ρ _{2}})^{n} β_{n} (a, k) = \frac{( k q , k q / ρ _{1} ρ _{2} , a q / ρ _{1} , a q / ρ _{2} ; q ) _{\infty}}{( k q / ρ _{1} , k q / ρ _{2} , a q / ρ _{1} ρ _{2} , a q ; q ) _{\infty}} n = 0 \sum \infty \frac{( ρ _{1} , ρ _{2} ; q ) _{n}}{( a q / ρ _{1} , a q / ρ _{2} ; q ) _{n}} (\frac{a q}{ρ _{1} ρ _{2}})^{n} α_{n} (a, k),

n = 0 \sum \infty (\frac{q a ^{2}}{k ^{2}})^{n} β_{n} (a, k) = \frac{( q a / k , q a ^{2} / k ; q ) _{\infty}}{( q a , q a ^{2} / k ^{2} ; q ) _{\infty}} n = 0 \sum \infty \frac{( k ; q ) _{2 n}}{( q a ^{2} / k ; q ) _{2 n}} (\frac{q a ^{2}}{k ^{2}})^{n} α_{n} (a, k) .

n = 0 \sum \infty (\frac{q a ^{2}}{k ^{2}})^{n} β_{n} (a, k) = \frac{( q a / k , q a ^{2} / k ; q ) _{\infty}}{( q a , q a ^{2} / k ^{2} ; q ) _{\infty}} n = 0 \sum \infty \frac{( k ; q ) _{2 n}}{( q a ^{2} / k ; q ) _{2 n}} (\frac{q a ^{2}}{k ^{2}})^{n} α_{n} (a, k) .

β_{n}

β_{n}

n = 0 \sum N \frac{( q k , - q k , y , z , k ^{2} q ^{N} / y z , q ^{- N} ; q ) _{n}}{( k , - k , q k / y , q k / z , y z q ^{1 - N} / k , k q ^{1 + N} ; q ) _{n}} q^{n} β_{n} = \frac{( 1 - k / y ) ( 1 - k / z ) ( 1 - k q ^{N} ) ( 1 - k q ^{N} / y z )}{( 1 - k ) ( 1 - k / y z ) ( 1 - k q ^{N} / y ) ( 1 - k q ^{N} / z )} \times n = 0 \sum N \frac{( y , z , k ^{2} q ^{N} / y z , q ^{- N} ; q ) _{n}}{( k / y , k / z , k q ^{N} , y z q ^{- N} / k ; q ) _{n}} α_{n} .

n = 0 \sum N \frac{( q k , - q k , y , z , k ^{2} q ^{N} / y z , q ^{- N} ; q ) _{n}}{( k , - k , q k / y , q k / z , y z q ^{1 - N} / k , k q ^{1 + N} ; q ) _{n}} q^{n} β_{n} = \frac{( 1 - k / y ) ( 1 - k / z ) ( 1 - k q ^{N} ) ( 1 - k q ^{N} / y z )}{( 1 - k ) ( 1 - k / y z ) ( 1 - k q ^{N} / y ) ( 1 - k q ^{N} / z )} \times n = 0 \sum N \frac{( y , z , k ^{2} q ^{N} / y z , q ^{- N} ; q ) _{n}}{( k / y , k / z , k q ^{N} , y z q ^{- N} / k ; q ) _{n}} α_{n} .

n = 0 \sum ⌊ N /2 ⌋ \frac{( q k , - q k , y , z , k ^{2} q ^{N} / y z , q ^{- N} ; q ) _{2 n} q ^{2 n}}{( k , - k , q k / y , q k / z , y z q ^{1 - N} / k , k q ^{1 + N} ; q ) _{2 n}} = \frac{( 1 - k / y ) ( 1 - k / z ) ( 1 - k q ^{N} ) ( 1 - k q ^{N} / y z )}{( 1 - k ) ( 1 - k / y z ) ( 1 - k q ^{N} / y ) ( 1 - k q ^{N} / z )} \times n = 0 \sum N \frac{( y , z , k ^{2} q ^{N} / y z , q ^{- N} ; q ) _{n}}{( k / y , k / z , k q ^{N} , y z q ^{- N} / k ; q ) _{n}} (- 1)^{n};

n = 0 \sum ⌊ N /2 ⌋ \frac{( q k , - q k , y , z , k ^{2} q ^{N} / y z , q ^{- N} ; q ) _{2 n} q ^{2 n}}{( k , - k , q k / y , q k / z , y z q ^{1 - N} / k , k q ^{1 + N} ; q ) _{2 n}} = \frac{( 1 - k / y ) ( 1 - k / z ) ( 1 - k q ^{N} ) ( 1 - k q ^{N} / y z )}{( 1 - k ) ( 1 - k / y z ) ( 1 - k q ^{N} / y ) ( 1 - k q ^{N} / z )} \times n = 0 \sum N \frac{( y , z , k ^{2} q ^{N} / y z , q ^{- N} ; q ) _{n}}{( k / y , k / z , k q ^{N} , y z q ^{- N} / k ; q ) _{n}} (- 1)^{n};

n = 0 \sum ⌊ N /2 ⌋ \frac{( 1 - y q ^{4 n + 1} ) ( y , - y q ^{N + 1} , q ^{- N} ; q ) _{2 n} q ^{2 n}}{( 1 - y q ) ( q ^{2} , - y q ^{1 - N} , y q ^{2 + N} ; q ) _{2 n}} = \frac{( 1 - q ) ( 1 + y )}{( 1 - q ^{N + 1} ) ( 1 + y q ^{N} )} \frac{( y q ^{2} , - q ; q ) _{N}}{( q y , - 1/ y ; q ) _{N}} y^{- N /2} .

n = 0 \sum ⌊ N /2 ⌋ \frac{( 1 - y q ^{4 n + 1} ) ( y , - y q ^{N + 1} , q ^{- N} ; q ) _{2 n} q ^{2 n}}{( 1 - y q ) ( q ^{2} , - y q ^{1 - N} , y q ^{2 + N} ; q ) _{2 n}} = \frac{( 1 - q ) ( 1 + y )}{( 1 - q ^{N + 1} ) ( 1 + y q ^{N} )} \frac{( y q ^{2} , - q ; q ) _{N}}{( q y , - 1/ y ; q ) _{N}} y^{- N /2} .

_{12} W_{11} (k; y, y q, z, z q, \frac{k ^{2} q ^{N}}{y z}, \frac{k ^{2} q ^{1 + N}}{y z}, q^{- N}, q^{1 - N}, q^{2}; q^{2}; q^{2}),

_{12} W_{11} (k; y, y q, z, z q, \frac{k ^{2} q ^{N}}{y z}, \frac{k ^{2} q ^{1 + N}}{y z}, q^{- N}, q^{1 - N}, q^{2}; q^{2}; q^{2}),

_{r + 1} W_{r} (a_{1}; a_{4}, \dots a_{r + 1}; q, z) :=_{r + 1} ϕ_{r} [a_{1}, q a_{1}, - q a_{1}, a_{4}, \dots, a_{r + 1} a_{1}, - a_{1}, \frac{a _{1} q}{a _{4}}, \dots, \frac{a _{1} q}{a _{r + 1}}; q, z],

_{r + 1} W_{r} (a_{1}; a_{4}, \dots a_{r + 1}; q, z) :=_{r + 1} ϕ_{r} [a_{1}, q a_{1}, - q a_{1}, a_{4}, \dots, a_{r + 1} a_{1}, - a_{1}, \frac{a _{1} q}{a _{4}}, \dots, \frac{a _{1} q}{a _{r + 1}}; q, z],

_{r + 1} ϕ_{r} [a_{1}, a_{2}, \dots, a_{r + 1} b_{1}, \dots, b_{r}; q, x] = n = 0 \sum \infty \frac{( a _{1} ; q ) _{n} ( a _{2} ; q ) _{n} \dots ( a _{r + 1} ; q ) _{n}}{( q ; q ) _{n} ( b _{1} ; q ) _{n} \dots ( b _{r} ; q ) _{n}} x^{n} .

_{r + 1} ϕ_{r} [a_{1}, a_{2}, \dots, a_{r + 1} b_{1}, \dots, b_{r}; q, x] = n = 0 \sum \infty \frac{( a _{1} ; q ) _{n} ( a _{2} ; q ) _{n} \dots ( a _{r + 1} ; q ) _{n}}{( q ; q ) _{n} ( b _{1} ; q ) _{n} \dots ( b _{r} ; q ) _{n}} x^{n} .

n = 0 \sum N \frac{( q k , - q k , y , z , k ^{2} q ^{N} / y z , q ^{- N} ; q ) _{n} ( n + 1 ) q ^{n}}{( k , - k , q k / y , q k / z , y z q ^{1 - N} / k , k q ^{1 + N} ; q ) _{n}} = \frac{( 1 - k / y ) ( 1 - k / z ) ( 1 - k q ^{N} ) ( 1 - k q ^{N} / y z )}{( 1 - k ) ( 1 - k / y z ) ( 1 - k q ^{N} / y ) ( 1 - k q ^{N} / z )} \times n = 0 \sum N \frac{( y , z , k ^{2} q ^{N} / y z , q ^{- N} ; q ) _{n}}{( k / y , k / z , k q ^{N} , y z q ^{- N} / k ; q ) _{n}} .

n = 0 \sum N \frac{( q k , - q k , y , z , k ^{2} q ^{N} / y z , q ^{- N} ; q ) _{n} ( n + 1 ) q ^{n}}{( k , - k , q k / y , q k / z , y z q ^{1 - N} / k , k q ^{1 + N} ; q ) _{n}} = \frac{( 1 - k / y ) ( 1 - k / z ) ( 1 - k q ^{N} ) ( 1 - k q ^{N} / y z )}{( 1 - k ) ( 1 - k / y z ) ( 1 - k q ^{N} / y ) ( 1 - k q ^{N} / z )} \times n = 0 \sum N \frac{( y , z , k ^{2} q ^{N} / y z , q ^{- N} ; q ) _{n}}{( k / y , k / z , k q ^{N} , y z q ^{- N} / k ; q ) _{n}} .

_{8} ϕ_{7} [q k, - q k, y, z, a q, b q, k^{2} q^{N} / y z, q^{- N} k, - k, q k / y, q k / z, (a + b - 1) q, y z q^{1 - N} / k, k q^{1 + N}; q, q] = \frac{( 1 - k / y ) ( 1 - k / z ) ( 1 - k q ^{N} ) ( 1 - k q ^{N} / y z )}{( 1 - k ) ( 1 - k / y z ) ( 1 - k q ^{N} / y ) ( 1 - k q ^{N} / z )} \times_{6} ϕ_{5} [y, z, a, b, k^{2} q^{N} / y z, q^{- N} k / y, k / z, (a + b - 1) q, k q^{N}, y z q^{- N} / k; q, q^{2}] .

_{8} ϕ_{7} [q k, - q k, y, z, a q, b q, k^{2} q^{N} / y z, q^{- N} k, - k, q k / y, q k / z, (a + b - 1) q, y z q^{1 - N} / k, k q^{1 + N}; q, q] = \frac{( 1 - k / y ) ( 1 - k / z ) ( 1 - k q ^{N} ) ( 1 - k q ^{N} / y z )}{( 1 - k ) ( 1 - k / y z ) ( 1 - k q ^{N} / y ) ( 1 - k q ^{N} / z )} \times_{6} ϕ_{5} [y, z, a, b, k^{2} q^{N} / y z, q^{- N} k / y, k / z, (a + b - 1) q, k q^{N}, y z q^{- N} / k; q, q^{2}] .

α_{n} = \frac{( a q , b q ; q ) _{n}}{(( a + b - 1 ) q , q ; q ) _{n}} - \frac{( a q , b q ; q ) _{n - 1}}{(( a + b - 1 ) q , q ; q ) _{n - 1}} = \frac{( a , b ; q ) _{n} q ^{2 n}}{(( a + b - 1 ) q , q ; q ) _{n}} .

α_{n} = \frac{( a q , b q ; q ) _{n}}{(( a + b - 1 ) q , q ; q ) _{n}} - \frac{( a q , b q ; q ) _{n - 1}}{(( a + b - 1 ) q , q ; q ) _{n - 1}} = \frac{( a , b ; q ) _{n} q ^{2 n}}{(( a + b - 1 ) q , q ; q ) _{n}} .

β_{n} = \frac{( a q , b q ; q ) _{n}}{(( a + b - 1 ) q , q ; q ) _{n}} .

β_{n} = \frac{( a q , b q ; q ) _{n}}{(( a + b - 1 ) q , q ; q ) _{n}} .

_{8} ϕ_{7} [q k, - q k, y, z, a q, b q, k^{2} q^{N} / y z, q^{- N} k, - k, q k / y, q k / z, ab q, y z q^{1 - N} / k, k q^{1 + N}; q, q] = \frac{( 1 - k / y ) ( 1 - k / z ) ( 1 - k q ^{N} ) ( 1 - k q ^{N} / y z )}{( 1 - k ) ( 1 - k / y z ) ( 1 - k q ^{N} / y ) ( 1 - k q ^{N} / z )} \times_{6} ϕ_{5} [y, z, a, b, k^{2} q^{N} / y z, q^{- N} k / y, k / z, ab q, k q^{N}, y z q^{- N} / k; q, q] .

_{8} ϕ_{7} [q k, - q k, y, z, a q, b q, k^{2} q^{N} / y z, q^{- N} k, - k, q k / y, q k / z, ab q, y z q^{1 - N} / k, k q^{1 + N}; q, q] = \frac{( 1 - k / y ) ( 1 - k / z ) ( 1 - k q ^{N} ) ( 1 - k q ^{N} / y z )}{( 1 - k ) ( 1 - k / y z ) ( 1 - k q ^{N} / y ) ( 1 - k q ^{N} / z )} \times_{6} ϕ_{5} [y, z, a, b, k^{2} q^{N} / y z, q^{- N} k / y, k / z, ab q, k q^{N}, y z q^{- N} / k; q, q] .

α_{n} = \frac{( a q , b q ; q ) _{n}}{( ab q , q ; q ) _{n}} - \frac{( a q , b q ; q ) _{n - 1}}{( ab q , q ; q ) _{n - 1}} = \frac{( a , b ; q ) _{n} q ^{n}}{( ab q , q ; q ) _{n}} .

α_{n} = \frac{( a q , b q ; q ) _{n}}{( ab q , q ; q ) _{n}} - \frac{( a q , b q ; q ) _{n - 1}}{( ab q , q ; q ) _{n - 1}} = \frac{( a , b ; q ) _{n} q ^{n}}{( ab q , q ; q ) _{n}} .

n = 0 \sum N \frac{( q k , - q k , y , z , k ^{2} q ^{N} / y z , q ^{- N} ; q ) _{n} q ^{(m n^{2} + (p + 2) n) /2}}{( k , - k , q k / y , q k / z , y z q ^{1 - N} / k , k q ^{1 + N} ; q ) _{n} ( - q ^{(m + p) /2} ; q ^{m} ) _{n}} = \frac{( 1 - k / y ) ( 1 - k / z ) ( 1 - k q ^{N} ) ( 1 - k q ^{N} / y z )}{( 1 - k ) ( 1 - k / y z ) ( 1 - k q ^{N} / y ) ( 1 - k q ^{N} / z )} \times (1 - q^{(m - p) /2} n = 1 \sum N \frac{( y , z , k ^{2} q ^{N} / y z , q ^{- N} ; q ) _{n} q ^{(m n^{2} + (p - 2 m) n) /2}}{( k / y , k / z , k q ^{N} , y z q ^{- N} / k ; q ) _{n} ( - q ^{(m + p) /2} ; q ^{m} ) _{n}}) .

n = 0 \sum N \frac{( q k , - q k , y , z , k ^{2} q ^{N} / y z , q ^{- N} ; q ) _{n} q ^{(m n^{2} + (p + 2) n) /2}}{( k , - k , q k / y , q k / z , y z q ^{1 - N} / k , k q ^{1 + N} ; q ) _{n} ( - q ^{(m + p) /2} ; q ^{m} ) _{n}} = \frac{( 1 - k / y ) ( 1 - k / z ) ( 1 - k q ^{N} ) ( 1 - k q ^{N} / y z )}{( 1 - k ) ( 1 - k / y z ) ( 1 - k q ^{N} / y ) ( 1 - k q ^{N} / z )} \times (1 - q^{(m - p) /2} n = 1 \sum N \frac{( y , z , k ^{2} q ^{N} / y z , q ^{- N} ; q ) _{n} q ^{(m n^{2} + (p - 2 m) n) /2}}{( k / y , k / z , k q ^{N} , y z q ^{- N} / k ; q ) _{n} ( - q ^{(m + p) /2} ; q ^{m} ) _{n}}) .

α_{n} = \frac{q ^{(m n^{2} + p n) /2}}{( - q ^{(m + p) /2} ; q ^{m} ) _{n}} - \frac{q ^{(m (n - 1)^{2} + p (n - 1)) /2}}{( - q ^{(m + p) /2} ; q ^{m} ) _{n - 1}} = - q^{(m - p) /2} \frac{q ^{(m n^{2} + (p - 2 m) n) /2}}{( - q ^{(m + p) /2} ; q ^{m} ) _{n}} .

α_{n} = \frac{q ^{(m n^{2} + p n) /2}}{( - q ^{(m + p) /2} ; q ^{m} ) _{n}} - \frac{q ^{(m (n - 1)^{2} + p (n - 1)) /2}}{( - q ^{(m + p) /2} ; q ^{m} ) _{n - 1}} = - q^{(m - p) /2} \frac{q ^{(m n^{2} + (p - 2 m) n) /2}}{( - q ^{(m + p) /2} ; q ^{m} ) _{n}} .

n = 0 \sum N \frac{( q k , - q k , y , z , \frac{k ^{2} q ^{N}}{y z} , q ^{- N} ; q ) _{n}}{( k , - k , \frac{q k}{y} , \frac{q k}{z} , \frac{y z q ^{1 - N}}{k} , k q ^{1 + N} ; q ) _{n}} \times \frac{( a p ^{2} ; p ^{2} ) _{n} ( b P ^{2} ; P ^{2} ) _{n} ( c R ^{2} ; R ^{2} ) _{n} ( \frac{a Q ^{2}}{b c} ; Q ^{2} ) _{n} q ^{n}}{( \frac{P QR}{p} ; \frac{P QR}{p} ) _{n} ( \frac{a pP Q}{c R} ; \frac{pP Q}{R} ) _{n} ( \frac{a pQR}{b P} ; \frac{pQR}{P} ) _{n} ( \frac{b c pP R}{Q} ; \frac{pP R}{Q} ) _{n}} = \frac{( 1 - \frac{k}{y} ) ( 1 - \frac{k}{z} ) ( 1 - k q ^{N} ) ( 1 - \frac{k q ^{N}}{y z} )}{( 1 - k ) ( 1 - \frac{k}{y z} ) ( 1 - \frac{k q ^{N}}{y} ) ( 1 - \frac{k q ^{N}}{z} )} n = 0 \sum N \frac{( y , z , \frac{k ^{2} q ^{N}}{y z} , q ^{- N} ; q ) _{n}}{( \frac{k}{y} , \frac{k}{z} , k q ^{N} , \frac{y z q ^{- N}}{k} ; q ) _{n}} \times \frac{( 1 - a p ^{n} P ^{n} Q ^{n} R ^{n} ) ( 1 - b \frac{p ^{n} P ^{n}}{Q ^{n} R ^{n}} ) ( 1 - \frac{P ^{n} Q ^{n}}{c p ^{n} R ^{n}} ) ( 1 - \frac{a p ^{n} Q ^{n}}{b c P ^{n} R ^{n}} )}{( 1 - a ) ( 1 - b ) ( 1 - \frac{1}{c} ) ( 1 - \frac{a}{b c} )} \times \frac{( a ; p ^{2} ) _{n} ( b ; P ^{2} ) _{n} ( c ; R ^{2} ) _{n} ( \frac{a}{b c} ; Q ^{2} ) _{n} R ^{2 n}}{( \frac{P QR}{p} ; \frac{P QR}{p} ) _{n} ( \frac{a pP Q}{c R} ; \frac{pP Q}{R} ) _{n} ( \frac{a pQR}{b P} ; \frac{pQR}{P} ) _{n} ( \frac{b c pP R}{Q} ; \frac{pP R}{Q} ) _{n}};

n = 0 \sum N \frac{( q k , - q k , y , z , \frac{k ^{2} q ^{N}}{y z} , q ^{- N} ; q ) _{n}}{( k , - k , \frac{q k}{y} , \frac{q k}{z} , \frac{y z q ^{1 - N}}{k} , k q ^{1 + N} ; q ) _{n}} \times \frac{( a p ^{2} ; p ^{2} ) _{n} ( b P ^{2} ; P ^{2} ) _{n} ( c R ^{2} ; R ^{2} ) _{n} ( \frac{a Q ^{2}}{b c} ; Q ^{2} ) _{n} q ^{n}}{( \frac{P QR}{p} ; \frac{P QR}{p} ) _{n} ( \frac{a pP Q}{c R} ; \frac{pP Q}{R} ) _{n} ( \frac{a pQR}{b P} ; \frac{pQR}{P} ) _{n} ( \frac{b c pP R}{Q} ; \frac{pP R}{Q} ) _{n}} = \frac{( 1 - \frac{k}{y} ) ( 1 - \frac{k}{z} ) ( 1 - k q ^{N} ) ( 1 - \frac{k q ^{N}}{y z} )}{( 1 - k ) ( 1 - \frac{k}{y z} ) ( 1 - \frac{k q ^{N}}{y} ) ( 1 - \frac{k q ^{N}}{z} )} n = 0 \sum N \frac{( y , z , \frac{k ^{2} q ^{N}}{y z} , q ^{- N} ; q ) _{n}}{( \frac{k}{y} , \frac{k}{z} , k q ^{N} , \frac{y z q ^{- N}}{k} ; q ) _{n}} \times \frac{( 1 - a p ^{n} P ^{n} Q ^{n} R ^{n} ) ( 1 - b \frac{p ^{n} P ^{n}}{Q ^{n} R ^{n}} ) ( 1 - \frac{P ^{n} Q ^{n}}{c p ^{n} R ^{n}} ) ( 1 - \frac{a p ^{n} Q ^{n}}{b c P ^{n} R ^{n}} )}{( 1 - a ) ( 1 - b ) ( 1 - \frac{1}{c} ) ( 1 - \frac{a}{b c} )} \times \frac{( a ; p ^{2} ) _{n} ( b ; P ^{2} ) _{n} ( c ; R ^{2} ) _{n} ( \frac{a}{b c} ; Q ^{2} ) _{n} R ^{2 n}}{( \frac{P QR}{p} ; \frac{P QR}{p} ) _{n} ( \frac{a pP Q}{c R} ; \frac{pP Q}{R} ) _{n} ( \frac{a pQR}{b P} ; \frac{pQR}{P} ) _{n} ( \frac{b c pP R}{Q} ; \frac{pP R}{Q} ) _{n}};

n = 0 \sum N \frac{( q k , - q k , y , z , \frac{k ^{2} q ^{N}}{y z} , q ^{- N} ; q ) _{n}}{( k , - k , \frac{q k}{y} , \frac{q k}{z} , \frac{y z q ^{1 - N}}{k} , k q ^{1 + N} ; q ) _{n}} \frac{( a q ^{m} , b q ^{m} , c q ^{m} , \frac{a q ^{m}}{b c} ; q ^{m} ) _{n}}{( \frac{a}{c} q ^{m} , \frac{a}{b} q ^{m} , b c q ^{m} , q ^{m} ; q ^{m} ) _{n}} q^{n} = \frac{( 1 - \frac{k}{y} ) ( 1 - \frac{k}{z} ) ( 1 - k q ^{N} ) ( 1 - \frac{k q ^{N}}{y z} )}{( 1 - k ) ( 1 - \frac{k}{y z} ) ( 1 - \frac{k q ^{N}}{y} ) ( 1 - \frac{k q ^{N}}{z} )} \times n = 0 \sum N \frac{( y , z , \frac{k ^{2} q ^{N}}{y z} , q ^{- N} ; q ) _{n}}{( \frac{k}{y} , \frac{k}{z} , k q ^{N} , \frac{y z q ^{- N}}{k} ; q ) _{n}} \frac{( q ^{m} a , - q ^{m} a , a , b , c , \frac{a}{b c} ; q ^{m} ) _{n} q ^{mn}}{( a , - a , \frac{a}{c} q ^{m} , \frac{a}{b} q ^{m} , b c q ^{m} , q ^{m} ; q ^{m} ) _{n}} .

n = 0 \sum N \frac{( q k , - q k , y , z , \frac{k ^{2} q ^{N}}{y z} , q ^{- N} ; q ) _{n}}{( k , - k , \frac{q k}{y} , \frac{q k}{z} , \frac{y z q ^{1 - N}}{k} , k q ^{1 + N} ; q ) _{n}} \frac{( a q ^{m} , b q ^{m} , c q ^{m} , \frac{a q ^{m}}{b c} ; q ^{m} ) _{n}}{( \frac{a}{c} q ^{m} , \frac{a}{b} q ^{m} , b c q ^{m} , q ^{m} ; q ^{m} ) _{n}} q^{n} = \frac{( 1 - \frac{k}{y} ) ( 1 - \frac{k}{z} ) ( 1 - k q ^{N} ) ( 1 - \frac{k q ^{N}}{y z} )}{( 1 - k ) ( 1 - \frac{k}{y z} ) ( 1 - \frac{k q ^{N}}{y} ) ( 1 - \frac{k q ^{N}}{z} )} \times n = 0 \sum N \frac{( y , z , \frac{k ^{2} q ^{N}}{y z} , q ^{- N} ; q ) _{n}}{( \frac{k}{y} , \frac{k}{z} , k q ^{N} , \frac{y z q ^{- N}}{k} ; q ) _{n}} \frac{( q ^{m} a , - q ^{m} a , a , b , c , \frac{a}{b c} ; q ^{m} ) _{n} q ^{mn}}{( a , - a , \frac{a}{c} q ^{m} , \frac{a}{b} q ^{m} , b c q ^{m} , q ^{m} ; q ^{m} ) _{n}} .

k = 0 \sum n \frac{( 1 - a p ^{k} P ^{k} Q ^{k} R ^{k} ) ( 1 - b \frac{p ^{k} P ^{k}}{Q ^{k} R ^{k}} ) ( 1 - \frac{P ^{k} Q ^{k}}{c p ^{k} R ^{k}} ) ( 1 - \frac{a p ^{k} Q ^{k}}{b c P ^{k} R ^{k}} )}{( 1 - a ) ( 1 - b ) ( 1 - \frac{1}{c} ) ( 1 - \frac{a}{b c} )} \times \frac{( a ; p ^{2} ) _{k} ( b ; P ^{2} ) _{k}}{( \frac{P QR}{p} ; \frac{P QR}{p} ) _{k} ( \frac{a pP Q}{c R} ; \frac{pP Q}{R} ) _{k}} \frac{( c ; R ^{2} ) _{k} ( \frac{a}{b c} ; Q ^{2} ) _{k}}{( \frac{a pQR}{b P} ; \frac{pQR}{P} ) _{k} ( \frac{b c pP R}{Q} ; \frac{pP R}{Q} ) _{k}} R^{2 k} = \frac{( a p ^{2} ; p ^{2} ) _{n} ( b P ^{2} ; P ^{2} ) _{n} ( c R ^{2} ; R ^{2} ) _{n} ( \frac{a Q ^{2}}{b c} ; Q ^{2} ) _{n}}{( \frac{P QR}{p} ; \frac{P QR}{p} ) _{n} ( \frac{a pP Q}{c R} ; \frac{pP Q}{R} ) _{n} ( \frac{a pQR}{b P} ; \frac{pQR}{P} ) _{n} ( \frac{b c pP R}{Q} ; \frac{pP R}{Q} ) _{n}},

k = 0 \sum n \frac{( 1 - a p ^{k} P ^{k} Q ^{k} R ^{k} ) ( 1 - b \frac{p ^{k} P ^{k}}{Q ^{k} R ^{k}} ) ( 1 - \frac{P ^{k} Q ^{k}}{c p ^{k} R ^{k}} ) ( 1 - \frac{a p ^{k} Q ^{k}}{b c P ^{k} R ^{k}} )}{( 1 - a ) ( 1 - b ) ( 1 - \frac{1}{c} ) ( 1 - \frac{a}{b c} )} \times \frac{( a ; p ^{2} ) _{k} ( b ; P ^{2} ) _{k}}{( \frac{P QR}{p} ; \frac{P QR}{p} ) _{k} ( \frac{a pP Q}{c R} ; \frac{pP Q}{R} ) _{k}} \frac{( c ; R ^{2} ) _{k} ( \frac{a}{b c} ; Q ^{2} ) _{k}}{( \frac{a pQR}{b P} ; \frac{pQR}{P} ) _{k} ( \frac{b c pP R}{Q} ; \frac{pP R}{Q} ) _{k}} R^{2 k} = \frac{( a p ^{2} ; p ^{2} ) _{n} ( b P ^{2} ; P ^{2} ) _{n} ( c R ^{2} ; R ^{2} ) _{n} ( \frac{a Q ^{2}}{b c} ; Q ^{2} ) _{n}}{( \frac{P QR}{p} ; \frac{P QR}{p} ) _{n} ( \frac{a pP Q}{c R} ; \frac{pP Q}{R} ) _{n} ( \frac{a pQR}{b P} ; \frac{pQR}{P} ) _{n} ( \frac{b c pP R}{Q} ; \frac{pP R}{Q} ) _{n}},

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Full text

Some Implications of the WP-Bailey Tree

James Mc Laughlin

Mathematics Department

Anderson Hall

West Chester University, West Chester, PA 19383

[email protected]

and

Peter Zimmer

Mathematics Department

Anderson Hall

West Chester University, West Chester, PA 19383

[email protected]

Abstract.

We consider a special case of a WP-Bailey chain of George Andrews, and use it to derive a number of curious transformations of basic hypergeometric series.

We also derive two new WP-Bailey pairs, and use them to derive some additional new transformations for basic hypergeometric series.

Finally, we briefly consider the implications of WP-Bailey pairs

$(\alpha_{n}(a,k)$ , $\beta_{n}(a,k))$ , in which $\alpha_{n}(a,k)$ is independent of $k$ , for generalizations of identities of the Rogers-Ramanujan type.

Key words and phrases:

Q-Series, Rogers-Ramanujan Type Identities, Bailey chains, WP-Bailey pairs

2000 Mathematics Subject Classification:

Primary: 33D15. Secondary:11B65, 05A19.

1. Introduction

Andrews, building on prior work of Bressoud [4] and Singh [9], in [1] defined a WP-Bailey pair to be a pair of sequences $(\alpha_{n}(a,k),\,\beta_{n}(a,k))$ satisfying

[TABLE]

Andrews also showed in [1] that there were two distinct ways to construct new WP-Bailey pairs from a given pair. If $(\alpha_{n}(a,k),\,\beta_{n}(a,k))$ satisfy (1.1), then so do $(\alpha_{n}^{\prime}(a,k),\,\beta_{n}^{\prime}(a,k))$ and $(\tilde{\alpha}_{n}(a,k),\,\tilde{\beta}_{n}(a,k))$ , where

[TABLE]

with $c=k\rho_{1}\rho_{2}/aq$ for the pair above, and

[TABLE]

These two constructions allow a “tree” of WP-Bailey pairs to be generated from a single WP-Bailey pair. Andrews and Berkovich [2] further investigated these two branches of the WP-Bailey tree, in the process deriving many new transformations for basic hypergeometric series. Spiridonov [11] derived an elliptic generalization of Andrews first WP-Bailey chain, and Warnaar [14] 111In a note added after submitting the paper [14], Warnaar remarks that he had discovered many more transformations for basic and elliptic WP-Bailey pairs, and gives two further examples of chains that hold at the elliptic level. added four new branches to the WP-Bailey tree, two of which had generalizations to the elliptic level. More recently, Liu and Ma [7] introduced the idea of a general WP-Bailey chain (as a solution to a system of linear equations), and added one new branch to the WP-Bailey tree.

In the present paper, we derive two new WP-Bailey pairs, one of them restricted in the sense that it is necessary to set $k=q$ . We then insert these in some of the WP-Bailey chains listed above to derive new transformations of basic hypergeometric series.

We also consider a special case ( $k=aq$ ) of the first WP-Bailey chain of Andrews, and show how it leads some unusual transformations of series.

We also briefly consider the special case of a WP-Bailey pair $(\alpha_{n}(a),$ $\beta_{n}(a,k))$ , where the $\alpha_{n}$ are independent of $k$ . We show how a such pair may give rise to a generalization of a Slater-type identity deriving from the standard Bailey pair $(\alpha_{n}(a),\beta_{n}(a,0))$ .

2. Finite Basic Hypergeometric Identities deriving from Simple WP-Bailey pairs

In this section we derive some unusual transformations from (1.2) by inserting “simple” WP-Bailey pairs (see below for the definition).

We reformulate the constructions at (1.2) and (for later use) (1.3) as transformations relating the original WP-Bailey pair $(\alpha_{n}(a,k),\,\beta_{n}(a,k))$ . We first recall the following elementary transformation:

[TABLE]

Theorem 1.

If $(\alpha_{n}(a,k),\beta_{n}(a,k))$ satisfy

[TABLE]

then

[TABLE]

and

[TABLE]

Proof.

The identity at (2.3) follows from (1.2), after substituting for $\alpha^{\prime}_{n}(a,k)$ in (1.1), employing (2.1), then setting the two expressions for $\beta^{\prime}_{n}(a,k)$ equal, and finally replacing $c$ with $k$ . The identity at (2.4) follows from (1.3), after setting $k=qa^{2}/c$ , using (2.1), and finally replacing $c$ with $k$ . ∎

For later use we also note the following corollary, which is immediate upon letting $N\to\infty$ .

Corollary 1.

If $(\alpha_{n}(a,k),\beta_{n}(a,k))$ satisfy

[TABLE]

then

[TABLE]

and

[TABLE]

Remark: At several places throughout the paper we replace $a$ with $x/q$ , $\rho_{1}$ with $y$ and $\rho_{2}$ with $z$ , in order to more easily make comparisons with a transformation due to Bailey (4.4) later.

We now consider some applications of the following corollary.

Corollary 2.

Let $N$ be a positive integer. Suppose the sequences $\{\alpha_{n}\}$ and $\{\beta_{n}\}$ are related by

[TABLE]

Then

[TABLE]

Proof.

Let $\rho_{1}=y$ , $\rho_{2}=z$ and $a=k/q$ in (2.2) and (2.3) in Theorem 1. ∎

It is interesting that the simple condition relating $\alpha_{n}$ and $\beta_{n}$ at (2.7) (we might call such pairs “simple” Bailey pairs) should have several non-trivial consequences. We had previously discovered (in [8]) the result that follows from Corollary 2 upon letting $N\to\infty$ (found by another method), so in fact many of the results in [8] has a finite counterpart. We give some examples.

Corollary 3.

Let $N$ be a positive integer. Then

[TABLE]

Proof.

Let $\alpha_{n}=(-1)^{n}$ in Corollary 2 to get (2.9). The identity at (2.10) follows from (2.9), after setting $k=qy$ and $z=-q\sqrt{y}$ , using the finite form of the $q$ -Dixon sum ((II.14) on page 355 of [5]) to sum the series on the right side, and finally using the identity at (I.9) on page 351 of [5] to rearrange two of the $q$ -products in this sum. ∎

Remark: The left side in (2.9) above could be expressed as

[TABLE]

where the additional $q$ -products are inserted here to give the series the form of a ${}_{r+1}\,W_{r}$ series. The identity at (2.9), which can thus be regarded as a transformation between a ${}_{12}\,\phi_{11}$ series with base $q^{2}$ and a ${}_{5}\phi_{4}$ series with base $q$ , does not appear to be a special case of other similar transformations due to Bailey (which involve ${}_{12}\,\phi_{11}$ series with base $q$ - see [5], pages 46–47) or Jain and Verma [6] (which involve ${}_{12}\,\phi_{11}$ series with base $q$ or $q^{3}$ ).

Recall that a $\,{}_{r+1}W_{r}$ series is defined by

[TABLE]

where, as usual, an ${}_{r+1}\phi_{r}$ basic hypergeometric series is defined by

[TABLE]

Corollary 4.

Let $N$ be a positive integer. Then

[TABLE]

Proof.

Let $\alpha_{n}=1$ in Corollary 2. ∎

Corollary 5.

[TABLE]

Proof.

In Corollary 2, define $\alpha_{0}=1$ , and for $n>0$ ,

[TABLE]

The sum for $\beta_{n}$ is easily seen to telescope to give

[TABLE]

and the result follows. ∎

Corollary 6.

[TABLE]

Proof.

This time, in Corollary 2, define $\alpha_{0}=1$ , and for $n>0$ ,

[TABLE]

The result follows as above. ∎

Corollary 7.

Let $N$ and $m$ be positive integers and let $p$ be an integer. Then

[TABLE]

Proof.

In Corollary 2 set $\alpha_{0}=1$ and, for $n>0$ ,

[TABLE]

∎

Corollary 8.

Let $P$ , $p$ , $Q$ , $q$ , $R$ , $a$ , $b$ , $c$ , $k$ , $y$ and $z$ be complex numbers such that none of the denominators below vanish. Then

[TABLE]

Proof.

We use the special case $m=0$ , $d=1$ of the identity of Subbarao and Verma labeled (2.2) in [12], namely,

[TABLE]

and then, in (2.8) above, let $\alpha_{i}$ be the $i$ -th term in the sum above, and let $\beta_{n}$ be the quantity on the right side above.

The identity at (2.16) follows upon setting $P=Q=p=R=q^{m/2}$ and simplifying. ∎

Apart from the first chain of Andrews, there is only one other WP-Bailey chain, of those seven alluded to in the introduction, that leads to a non-trivial transformation similar to that in Corollary 2 (the consequences of letting $k=aq$ in the other chains being entirely trivial). This is the third chain of Warnaar (Theorem 2.5 in [14]). This chain implies that if $(\alpha_{n}(a,k),\,\beta_{n}(a,k))$ satisfy (1.1), then

[TABLE]

Upon setting $m=aq$ , we get that if $\beta_{n}=\sum_{j=0}^{n}\alpha_{j}$ , then

[TABLE]

However, we do not pursue the consequences of this transformation further here.

3. New WP-Bailey Pairs

We next exhibit two new WP-Bailey pairs.

Lemma 1.

The pair $(\alpha_{n}^{(1)}(a,k),\beta_{n}^{(1)}(a,k))$ is a WP-Bailey pair, where

[TABLE]

Proof.

[TABLE]

The third equality follows from the $q$ -Pfaff-Saalschütz sum,

[TABLE]

∎

Lemma 2.

The pair $(\alpha_{n}^{(2)}(a,q),\beta_{n}^{(2)}(a,q))$ is a WP-Bailey pair, where

[TABLE]

Remark: Note that this pair is restricted in that it is necessary to set $k=q$ for (1.1) to hold.

Proof.

[TABLE]

The third equality follows from a $q$ -analogue of Whipple’s ${}_{3}F_{2}$ sum,

[TABLE]

and upon setting $a=q^{-n}$ ,

[TABLE]

∎

We had initially thought that the following WP-Bailey pair was totally new also.

[TABLE]

However, this pair can be derived from an existing WP-Bailey pair using a result of Warnaar [14].

Lemma 3 (Warnaar, [14]).

For $a$ and $k$ indeterminates the following equations are equivalent:

[TABLE]

If $a$ and $k$ are interchanged in the second equation, we see that Lemma 3 implies the following.

Corollary 9.

*If $(\alpha_{n}(a,k),\beta_{n}(a,k))$ are a WP-Bailey pair, then so are

$(\alpha_{n}^{\prime}(a,k),\beta_{n}^{\prime}(a,k))$ , where*

[TABLE]

For the present purposes, we may call the pair $(\alpha_{n}^{\prime}(a,k),\beta_{n}^{\prime}(a,k))$ the dual of the pair $(\alpha_{n}(a,k),\beta_{n}(a,k))$ . Thus, for example, our pair at (3.5) is the dual of the WP-Bailey pair of Andrews and Berkovich in [2]:

[TABLE]

We had initially thought that the WP-Bailey pair in Lemma 1 was the dual of the following pair of Andrews and Berkovich:

[TABLE]

However, the dual of the pair in Lemma 1 is the pair

[TABLE]

while the dual of the pair at (3.7) is the pair

[TABLE]

Inserting the new WP-Bailey pairs in any of the existing WP-Bailey chains will lead to transformations relating basic hypergeometric series. We believe the following transformations of basic hypergeometric series to be new.

Corollary 10.

[TABLE]

Proof.

Insert the WP-Bailey pair at (3.1) into (2.3) and set $\rho_{1}=y$ and $\rho_{2}=z$ . ∎

Substituting the pair at (3.1) into (2.4) leads to the following result.

Corollary 11.

[TABLE]

Corollary 12.

[TABLE]

Proof.

Insert the WP-Bailey pair at (3.3) into (2.3), set $\rho_{1}=y$ and $\rho_{2}=z$ and replace $k$ with $q$ . For (3.13), set $z=q/y$ , let $N\to\infty$ and use (3.4) to sum the resulting right side. ∎

Corollary 13.

[TABLE]

Proof.

Insert the WP-Bailey pair at (3.3) into (2.4), and replace $k$ with $q$ . ∎

Remark: The extra $q$ -products inserted in each of the series on the right side of (3.12) and in (3.14) are there to allow these series to be represented as ${}_{r+1}\phi_{r}$ or ${}_{r+1}W_{r}$ series.

Corollary 14.

[TABLE]

Proof.

After replacing $m$ with $k$ and employing some simple transformations for $q$ -products, (2.18) can be rewritten as

[TABLE]

The result follows, upon inserting the pair from Lemma 1, and rearranging. ∎

Inserting the new pairs in other WP-Bailey chains will lead to other, possibly new, transformations of basic hypergeometric series, but we refrain from further examples here.

4. WP-Burge Pairs

In [2], the authors termed a WP-Bailey pair $(\alpha_{n}(1,k),\beta_{n}(1,k))$ in which $\alpha_{n}(1,k)$ does not depend on $k$ a WP-Burge pair.

One such pair that they derive in [2, (7.16)] is the pair in the following theorem. For completeness, we give an alternative proof of this result.

Theorem 2.

Define

[TABLE]

Then $(\alpha_{n}(1,k),\beta_{n}(1,k))$ satisfy (1.1) (with $a=1$ ).

Proof.

We begin by recalling Bailey’s $\,{}_{6}\psi_{6}$ summation formula [13].

[TABLE]

where the second equality follows from the definition

[TABLE]

Next, set $a=-1$ , $b=-c$ and $d=-e$ , so that both sums in the final expression above become equal, to get

[TABLE]

or, upon replacing $c^{2}$ with $c$ and $e^{2}$ with $e$ ,

[TABLE]

Now replace $q$ with $\sqrt{q}$ , set $e=q^{-N}$ , $c=kq^{N}$ to get, after some minor rearrangements, that

[TABLE]

The result now follows from (1.1), after some simple manipulations. ∎

One reason this WP-Bailey pair is somewhat interesting is that the special case $k=0$ of (4.1) gives Slater’s standard Bailey pair F3 from [10]. Recall that a pair of sequences $(\alpha_{n},\beta_{n})$ is termed a Bailey pair relative to $x/q$ , if they satisfy

[TABLE]

We also recall the following result of Bailey, a particular case of the “Bailey Transform”, from his 1949 paper [3].

Theorem 3.

Subject to suitable convergence conditions, if the sequences $\{\alpha_{n}\}$ and $\{\beta_{n}\}$ satisfy (4.3), then

[TABLE]

Thus the pair at (4.1) “lifts” this standard Bailey pair to a WP-Bailey pair and lifts all the series–product identities following from F3 to more general series–product identities containing the free parameter $k$ .

More generally, let $(\alpha_{n}(a,k),\beta_{n}(a,k))$ be a WP-Bailey pair in which $\alpha_{n}(a,k)$ is independent of $k$ . Suppose further that this pair is a “lift” of a standard Bailey pair, in the sense that setting $k=0$ in the WP-Bailey pair recovers the Bailey pair. By comparing (4.4) and (2.5) (first setting $y=\rho_{1}$ , $z=\rho_{2}$ and $x=aq$ in (4.4)) we see that the two infinite series containing $\alpha_{n}=\alpha_{n}(a,k)$ are identical. This means that if particular choices of $y$ and $z$ in (4.4) lead to an identity of the Rogers-Ramanujan type, then the same choices for $\rho_{1}$ and $\rho_{2}$ in (2.5) will lead to a more general series–product identity containing an extra free parameter, namely $k$ , and this more general identity will revert back to the original identity of Rogers-Ramanujan type, upon setting $k=0$ .

The substitution of this pair at (4.1) into (2.3) (first setting $\rho_{1}=y$ , $\rho_{2}=z$ in (2.3) as above, and then setting $a=1$ ) and (2.5) (in addition, replacing $q$ with $q^{2}$ and specializing $y$ and $z$ ) leads to the following corollary.

Corollary 15.

[TABLE]

The last three identities also follow as special cases of Jackson’s ${}_{8}\phi_{7}$ summation formula. However, they do illustrate how a lift of a standard Bailey pair leads to generalizations of identities arising from this standard pair (the identities given by setting $k=0$ in the corollary above).

We will investigate this phenomenon of WP-Bailey pairs that are lifts of standard Bailey pairs further in a subsequent paper.

Bibliography14

The reference list from the paper itself. Each links out to its DOI / PubMed record.

1[1] G. E. Andrews, Bailey’s transform, lemma, chains and tree. Special functions 2000: current perspective and future directions (Tempe, AZ), 1–22, NATO Sci. Ser. II Math. Phys. Chem., 30, Kluwer Acad. Publ., Dordrecht, 2001.
2[2] G. E. Andrews and A. Berkovich, The WP-Bailey tree and its implications. J. London Math. Soc. (2) 66 (2002), no. 3, 529–549.
3[3] W. N. Bailey, Identities of the Rogers-Ramanujan type. Proc. London Math. Soc., 50 (1949) 1–10.
4[4] D. Bressoud, Some identities for terminating q 𝑞 q -series. Math. Proc. Cambridge Philos. Soc. 89 (1981), no. 2, 211–223.
5[5] G, Gasper and M. Rahman, Basic hypergeometric series. With a foreword by Richard Askey. Second edition. Encyclopedia of Mathematics and its Applications, 96. Cambridge University Press, Cambridge, 2004. xxvi+428 pp.
6[6] V. K. Jain and A. Verma, Transformations between basic hypergeometric series on different bases and identities of Rogers-Ramanujan type. J. Math. Anal. Appl. 76 (1980), no. 1, 230–269.
7[7] Q. Liu and X. Ma, On the Characteristic Equation of Well-Poised Baily Chains - To appear.
8[8] J. Mc Laughlin and P. Zimmer, Some Applications of a Bailey-type Transformation - submitted

TL;DR

Contribution

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Abstract

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Some Implications of the WP-Bailey Tree

Abstract.

Key words and phrases:

2000 Mathematics Subject Classification:

1. Introduction

2. Finite Basic Hypergeometric Identities deriving from Simple WP-Bailey pairs

Theorem 1**.**

Proof.

Corollary 1**.**

Corollary 2**.**

Proof.

Corollary 3**.**

Proof.

Corollary 4**.**

Proof.

Corollary 5**.**

Proof.

Corollary 6**.**

Proof.

Corollary 7**.**

Proof.

Corollary 8**.**

Proof.

3. New WP-Bailey Pairs

Lemma 1**.**

Proof.

Lemma 2**.**

Proof.

Lemma 3** (Warnaar, [14]).**

Corollary 9**.**

Corollary 10**.**

Proof.

Corollary 11**.**

Corollary 12**.**

Proof.

Corollary 13**.**

Proof.

Corollary 14**.**

Proof.

4. WP-Burge Pairs

Theorem 2**.**

Proof.

Theorem 3**.**

Corollary 15**.**

Theorem 1.

Corollary 1.

Corollary 2.

Corollary 3.

Corollary 4.

Corollary 5.

Corollary 6.

Corollary 7.

Corollary 8.

Lemma 1.

Lemma 2.

Lemma 3 (Warnaar, [14]).

Corollary 9.

Corollary 10.

Corollary 11.

Corollary 12.

Corollary 13.

Corollary 14.

Theorem 2.

Theorem 3.

Corollary 15.