Some new Transformations for Bailey pairs and WP-Bailey Pairs

James Mc Laughlin

arXiv:1901.05889·math.NT·January 18, 2019

Some new Transformations for Bailey pairs and WP-Bailey Pairs

James Mc Laughlin

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TL;DR

This paper introduces new transformations connecting WP-Bailey pairs and standard Bailey pairs, leading to general expansions for theta function products and Lambert series.

Contribution

It presents novel transformations for WP-Bailey pairs and their relation to standard Bailey pairs, enabling broader expansions of theta functions.

Findings

01

Derived new transformations for WP-Bailey pairs

02

Established connections to standard Bailey pairs

03

Produced general expansions for theta functions and Lambert series

Abstract

We derive several new transformations relating WP-Bailey pairs. We also consider the corresponding transformations relating standard Bailey pairs, and as a consequence, derive some quite general expansions for products of theta functions which can also be expressed as certain types of Lambert series.

Equations178

β_{n} (a, k, q)

β_{n} (a, k, q)

α_{n}^{'} (a, k)

α_{n}^{'} (a, k)

β_{n}^{'} (a, k)

\times j = 0 \sum n

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

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

n = 1 \sum \infty \frac{( q k , - q k , z ; q ) _{n} ( q ; q ) _{n - 1}}{( k , - k , q k , \frac{q k}{z} ; q ) _{n}} (\frac{q a}{z})^{n} β_{n} (a, k) - n = 1 \sum \infty \frac{( q \frac{1}{k} , - q \frac{1}{k} , \frac{1}{z} ; q ) _{n} ( q ; q ) _{n - 1}}{( \frac{1}{k} , - \frac{1}{k} , \frac{q}{k} , \frac{q z}{k} ; q ) _{n}} (\frac{q z}{a})^{n} β_{n} (\frac{1}{a}, \frac{1}{k}) - n = 1 \sum \infty \frac{( z ; q ) _{n} ( q ; q ) _{n - 1}}{( q a , \frac{q a}{z} ; q ) _{n}} (\frac{q a}{z})^{n} α_{n} (a, k) + n = 1 \sum \infty \frac{( \frac{1}{z} ; q ) _{n} ( q ; q ) _{n - 1}}{( \frac{q}{a} , \frac{q z}{a} ; q ) _{n}} (\frac{q z}{a})^{n} α_{n} (\frac{1}{a}, \frac{1}{k}) = \frac{( a - k ) ( 1 - \frac{1}{z} ) ( 1 - \frac{ak}{z} )}{( 1 - a ) ( 1 - k ) ( 1 - \frac{a}{z} ) ( 1 - \frac{k}{z} )} + \frac{z}{k} \frac{( z , \frac{q}{z} , \frac{k}{a} , \frac{q a}{k} , \frac{ak}{z} , \frac{q z}{ak} , q , q ; q ) _{\infty}}{( \frac{z}{k} , \frac{q k}{z} , \frac{z}{a} , \frac{q a}{z} , a , \frac{q}{a} , k , \frac{q}{k} ; q ) _{\infty}} .

n = 1 \sum \infty \frac{( q k , - q k , z ; q ) _{n} ( q ; q ) _{n - 1}}{( k , - k , q k , \frac{q k}{z} ; q ) _{n}} (\frac{q a}{z})^{n} β_{n} (a, k) - n = 1 \sum \infty \frac{( q \frac{1}{k} , - q \frac{1}{k} , \frac{1}{z} ; q ) _{n} ( q ; q ) _{n - 1}}{( \frac{1}{k} , - \frac{1}{k} , \frac{q}{k} , \frac{q z}{k} ; q ) _{n}} (\frac{q z}{a})^{n} β_{n} (\frac{1}{a}, \frac{1}{k}) - n = 1 \sum \infty \frac{( z ; q ) _{n} ( q ; q ) _{n - 1}}{( q a , \frac{q a}{z} ; q ) _{n}} (\frac{q a}{z})^{n} α_{n} (a, k) + n = 1 \sum \infty \frac{( \frac{1}{z} ; q ) _{n} ( q ; q ) _{n - 1}}{( \frac{q}{a} , \frac{q z}{a} ; q ) _{n}} (\frac{q z}{a})^{n} α_{n} (\frac{1}{a}, \frac{1}{k}) = \frac{( a - k ) ( 1 - \frac{1}{z} ) ( 1 - \frac{ak}{z} )}{( 1 - a ) ( 1 - k ) ( 1 - \frac{a}{z} ) ( 1 - \frac{k}{z} )} + \frac{z}{k} \frac{( z , \frac{q}{z} , \frac{k}{a} , \frac{q a}{k} , \frac{ak}{z} , \frac{q z}{ak} , q , q ; q ) _{\infty}}{( \frac{z}{k} , \frac{q k}{z} , \frac{z}{a} , \frac{q a}{z} , a , \frac{q}{a} , k , \frac{q}{k} ; q ) _{\infty}} .

n = 1 \sum \infty \frac{( 1 - k q ^{2 n} ) ( z ; q ) _{n} ( q ; q ) _{n - 1}}{( 1 - k ) ( q k , q k / z ; q ) _{n}} (\frac{q a}{z})^{n} β_{n} (a, k, q) + n = 1 \sum \infty \frac{( 1 + k q ^{2 n} ) ( z ; q ) _{n} ( q ; q ) _{n - 1}}{( 1 + k ) ( - q k , - q k / z ; q ) _{n}} (\frac{- q a}{z})^{n} β_{n} (- a, - k, q) - 2 n = 1 \sum \infty \frac{( 1 - k ^{2} q ^{4 n} ) ( z ^{2} ; q ^{2} ) _{n} ( q ^{2} ; q ^{2} ) _{n - 1}}{( 1 - k ^{2} ) ( q ^{2} k ^{2} , q ^{2} k ^{2} / z ^{2} ; q ^{2} ) _{n}} (\frac{q ^{2} a ^{2}}{z ^{2}})^{n} β_{n} (a^{2}, k^{2}, q^{2}) = n = 1 \sum \infty \frac{( z ; q ) _{n} ( q ; q ) _{n - 1}}{( q a , q a / z ; q ) _{n}} (\frac{q a}{z})^{n} α_{n} (a, k, q) + n = 1 \sum \infty \frac{( z ; q ) _{n} ( q ; q ) _{n - 1}}{( - q a , - q a / z ; q ) _{n}} (\frac{- q a}{z})^{n} α_{n} (- a, - k, q) - 2 n = 1 \sum \infty \frac{( z ^{2} ; q ^{2} ) _{n} ( q ^{2} ; q ^{2} ) _{n - 1}}{( q ^{2} a ^{2} , q ^{2} a ^{2} / z ^{2} ; q ^{2} ) _{n}} (\frac{q ^{2} a ^{2}}{z ^{2}})^{n} α_{n} (a^{2}, k^{2}, q^{2}) .

n = 1 \sum \infty \frac{( 1 - k q ^{2 n} ) ( z ; q ) _{n} ( q ; q ) _{n - 1}}{( 1 - k ) ( q k , q k / z ; q ) _{n}} (\frac{q a}{z})^{n} β_{n} (a, k, q) + n = 1 \sum \infty \frac{( 1 + k q ^{2 n} ) ( z ; q ) _{n} ( q ; q ) _{n - 1}}{( 1 + k ) ( - q k , - q k / z ; q ) _{n}} (\frac{- q a}{z})^{n} β_{n} (- a, - k, q) - 2 n = 1 \sum \infty \frac{( 1 - k ^{2} q ^{4 n} ) ( z ^{2} ; q ^{2} ) _{n} ( q ^{2} ; q ^{2} ) _{n - 1}}{( 1 - k ^{2} ) ( q ^{2} k ^{2} , q ^{2} k ^{2} / z ^{2} ; q ^{2} ) _{n}} (\frac{q ^{2} a ^{2}}{z ^{2}})^{n} β_{n} (a^{2}, k^{2}, q^{2}) = n = 1 \sum \infty \frac{( z ; q ) _{n} ( q ; q ) _{n - 1}}{( q a , q a / z ; q ) _{n}} (\frac{q a}{z})^{n} α_{n} (a, k, q) + n = 1 \sum \infty \frac{( z ; q ) _{n} ( q ; q ) _{n - 1}}{( - q a , - q a / z ; q ) _{n}} (\frac{- q a}{z})^{n} α_{n} (- a, - k, q) - 2 n = 1 \sum \infty \frac{( z ^{2} ; q ^{2} ) _{n} ( q ^{2} ; q ^{2} ) _{n - 1}}{( q ^{2} a ^{2} , q ^{2} a ^{2} / z ^{2} ; q ^{2} ) _{n}} (\frac{q ^{2} a ^{2}}{z ^{2}})^{n} α_{n} (a^{2}, k^{2}, q^{2}) .

q \frac{ψ ^{3} ( q ^{3} )}{ψ ( q )} = n = 1 \sum \infty \frac{( q ^{6} ; q ^{6} ) _{n - 1} ( q ^{2} / d ; q ^{6} ) _{n} ( - q ^{2} ) ^{n}}{( q ^{2} ; q ^{6} ) _{n} ( q ^{2} / d , q ^{3} ; q ^{3} ) _{n}} - n = 1 \sum \infty \frac{( q ^{6} ; q ^{6} ) _{n - 1} ( q / d ; q ^{6} ) _{n} ( - q ) ^{n}}{( q ; q ^{6} ) _{n} ( q / d , q ^{3} ; q ^{3} ) _{n}} + n = 1 \sum \infty \frac{( 1 - q ^{12 n - 2} ) ( q ^{6} ; q ^{6} ) _{2 n - 1} ( 1/ q ^{2} , d ; q ^{6} ) _{n} q ^{6 n^{2}}}{( 1 - 1/ q ^{2} ) ( q ^{2} ; q ^{6} ) _{2 n} ( q ^{4} / d , q ^{6} ; q ^{6} ) _{n} d ^{n}} - n = 1 \sum \infty \frac{( 1 - q ^{12 n - 1} ) ( q ^{6} ; q ^{6} ) _{2 n - 1} ( 1/ q , d ; q ^{6} ) _{n} q ^{6 n^{2} + 3 n}}{( 1 - 1/ q ) ( q ^{4} ; q ^{6} ) _{2 n} ( q ^{5} / d , q ^{6} ; q ^{6} ) _{n} d ^{n}} .

q \frac{ψ ^{3} ( q ^{3} )}{ψ ( q )} = n = 1 \sum \infty \frac{( q ^{6} ; q ^{6} ) _{n - 1} ( q ^{2} / d ; q ^{6} ) _{n} ( - q ^{2} ) ^{n}}{( q ^{2} ; q ^{6} ) _{n} ( q ^{2} / d , q ^{3} ; q ^{3} ) _{n}} - n = 1 \sum \infty \frac{( q ^{6} ; q ^{6} ) _{n - 1} ( q / d ; q ^{6} ) _{n} ( - q ) ^{n}}{( q ; q ^{6} ) _{n} ( q / d , q ^{3} ; q ^{3} ) _{n}} + n = 1 \sum \infty \frac{( 1 - q ^{12 n - 2} ) ( q ^{6} ; q ^{6} ) _{2 n - 1} ( 1/ q ^{2} , d ; q ^{6} ) _{n} q ^{6 n^{2}}}{( 1 - 1/ q ^{2} ) ( q ^{2} ; q ^{6} ) _{2 n} ( q ^{4} / d , q ^{6} ; q ^{6} ) _{n} d ^{n}} - n = 1 \sum \infty \frac{( 1 - q ^{12 n - 1} ) ( q ^{6} ; q ^{6} ) _{2 n - 1} ( 1/ q , d ; q ^{6} ) _{n} q ^{6 n^{2} + 3 n}}{( 1 - 1/ q ) ( q ^{4} ; q ^{6} ) _{2 n} ( q ^{5} / d , q ^{6} ; q ^{6} ) _{n} d ^{n}} .

(a; q)_{n}

(a; q)_{n}

(a_{1}, a_{2}, \dots, a_{j}; q)_{n}

(a; q)_{\infty}

(a_{1}, a_{2}, \dots, a_{j}; q)_{\infty}

\frac{( a q , a q / b c , a q / b d , a q / b e , a q / c d , a q / ce , a q / d e , q , q / a ; q ) _{\infty}}{( a q / b , a q / c , a q / d , a q / e , q / b , q / c , q / d , q / e , q a ^{2} / b c d e ; q ) _{\infty}} = n = - \infty \sum \infty \frac{( 1 - a q ^{2 n} ) ( b , c , d , e ; q ) _{n}}{( 1 - a ) ( a q / b , a q / c , a q / d , a q / e ; q ) _{n}} (\frac{q a ^{2}}{b c d e})^{n} = 1 + n = 1 \sum \infty \frac{( 1 - a q ^{2 n} ) ( b , c , d , e ; q ) _{n}}{( 1 - a ) ( a q / b , a q / c , a q / d , a q / e ; q ) _{n}} (\frac{q a ^{2}}{b c d e})^{n} + n = 1 \sum \infty \frac{( 1 - 1/ a q ^{2 n} ) ( b / a , c / a , d / a , e / a ; q ) _{n}}{( 1 - 1/ a ) ( q / b , q / c , q / d , q / e ; q ) _{n}} (\frac{q a ^{2}}{b c d e})^{n},

\frac{( a q , a q / b c , a q / b d , a q / b e , a q / c d , a q / ce , a q / d e , q , q / a ; q ) _{\infty}}{( a q / b , a q / c , a q / d , a q / e , q / b , q / c , q / d , q / e , q a ^{2} / b c d e ; q ) _{\infty}} = n = - \infty \sum \infty \frac{( 1 - a q ^{2 n} ) ( b , c , d , e ; q ) _{n}}{( 1 - a ) ( a q / b , a q / c , a q / d , a q / e ; q ) _{n}} (\frac{q a ^{2}}{b c d e})^{n} = 1 + n = 1 \sum \infty \frac{( 1 - a q ^{2 n} ) ( b , c , d , e ; q ) _{n}}{( 1 - a ) ( a q / b , a q / c , a q / d , a q / e ; q ) _{n}} (\frac{q a ^{2}}{b c d e})^{n} + n = 1 \sum \infty \frac{( 1 - 1/ a q ^{2 n} ) ( b / a , c / a , d / a , e / a ; q ) _{n}}{( 1 - 1/ a ) ( q / b , q / c , q / d , q / e ; q ) _{n}} (\frac{q a ^{2}}{b c d e})^{n},

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

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

n = 0 \sum \infty \frac{( a , q a , - q a , b , c , d ; q ) _{n}}{( q , a , - a , \frac{a q}{b} , \frac{a q}{c} , \frac{a q}{d} ; q ) _{n}} (\frac{a q}{b c d})^{n} = \frac{( a q , a q / b c , a q / b d , a q / c d ; q ) _{\infty}}{( a q / b , a q / c , a q / d , a q / b c d ; q ) _{\infty}} .

n = 0 \sum \infty \frac{( a , q a , - q a , b , c , d ; q ) _{n}}{( q , a , - a , \frac{a q}{b} , \frac{a q}{c} , \frac{a q}{d} ; q ) _{n}} (\frac{a q}{b c d})^{n} = \frac{( a q , a q / b c , a q / b d , a q / c d ; q ) _{\infty}}{( a q / b , a q / c , a q / d , a q / b c d ; q ) _{\infty}} .

n = 0 \sum \infty \frac{( a ; q ) _{n}}{( q ; q ) _{n}} z^{n} = \frac{( a z ; q ) _{\infty}}{( z ; q ) _{\infty}} .

n = 0 \sum \infty \frac{( a ; q ) _{n}}{( q ; q ) _{n}} z^{n} = \frac{( a z ; q ) _{\infty}}{( z ; q ) _{\infty}} .

n = 0 \sum \infty \frac{z ^{n}}{( q ; q ) _{n}}

n = 0 \sum \infty \frac{z ^{n}}{( q ; q ) _{n}}

n = 0 \sum \infty \frac{q ^{n (n - 1) /2} ( - z ) ^{n}}{( q ; q ) _{n}}

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

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

n = 1 \sum \infty \frac{( q k , - q k , z ; q ) _{n} ( y q ; q ) _{n - 1}}{( k , - k , q k / y , q k / z ; q ) _{n}} (\frac{q a}{y z})^{n} β_{n} - \frac{( q k , q k / y z , q a / y , q a / z ; q ) _{\infty}}{( q k / y , q k / z , q a , q a / y z ; q ) _{\infty}} n = 1 \sum \infty \frac{( z ; q ) _{n} ( y q ; q ) _{n - 1}}{( q a / y , q a / z ; q ) _{n}} (\frac{q a}{y z})^{n} α_{n} = \frac{1}{1 - y} (\frac{( q k , q k / y z , q a / y , q a / z ; q ) _{\infty}}{( q k / y , q k / z , q a , q a / y z ; q ) _{\infty}} - 1) .

n = 1 \sum \infty \frac{( q k , - q k , z ; q ) _{n} ( y q ; q ) _{n - 1}}{( k , - k , q k / y , q k / z ; q ) _{n}} (\frac{q a}{y z})^{n} β_{n} - \frac{( q k , q k / y z , q a / y , q a / z ; q ) _{\infty}}{( q k / y , q k / z , q a , q a / y z ; q ) _{\infty}} n = 1 \sum \infty \frac{( z ; q ) _{n} ( y q ; q ) _{n - 1}}{( q a / y , q a / z ; q ) _{n}} (\frac{q a}{y z})^{n} α_{n} = \frac{1}{1 - y} (\frac{( q k , q k / y z , q a / y , q a / z ; q ) _{\infty}}{( q k / y , q k / z , q a , q a / y z ; q ) _{\infty}} - 1) .

\frac{( q k , q k / y z , q a / y , q a / z ; q ) _{\infty}}{( q k / y , q k / z , q a , q a / y z ; q ) _{\infty}} = n = 0 \sum \infty \frac{( q k , - q k , k , y , z , k / a ; q ) _{n}}{( k , - k , q k / y , q k / z , q a , q ; q ) _{n}} (\frac{q a}{y z})^{n} .

\frac{( q k , q k / y z , q a / y , q a / z ; q ) _{\infty}}{( q k / y , q k / z , q a , q a / y z ; q ) _{\infty}} = n = 0 \sum \infty \frac{( q k , - q k , k , y , z , k / a ; q ) _{n}}{( k , - k , q k / y , q k / z , q a , q ; q ) _{n}} (\frac{q a}{y z})^{n} .

f (a, k, z, q) := n = 1 \sum \infty \frac{( q k , - q k , k , z , k / a ; q ) _{n}}{( k , - k , q k , q k / z , q a ; q ) _{n} ( 1 - q ^{n} )} (\frac{q a}{z})^{n} .

f (a, k, z, q) := n = 1 \sum \infty \frac{( q k , - q k , k , z , k / a ; q ) _{n}}{( k , - k , q k , q k / z , q a ; q ) _{n} ( 1 - q ^{n} )} (\frac{q a}{z})^{n} .

f (a, k, z, q) = - f (k, a, z, q) .

f (a, k, z, q) = - f (k, a, z, q) .

\frac{1}{1 - y} (\frac{( q k , q k / y z , q a / y , q a / z ; q ) _{\infty}}{( q k / y , q k / z , q a , q a / y z ; q ) _{\infty}} - 1) = \frac{1}{1 - y} (1 - \frac{( q a , q a / y z , q k / y , q k / z ; q ) _{\infty}}{( q a / y , q a / z , q k , q k / y z ; q ) _{\infty}}) \frac{( q k , q k / y z , q a / y , q a / z ; q ) _{\infty}}{( q k / y , q k / z , q a , q a / y z ; q ) _{\infty}} .

\frac{1}{1 - y} (\frac{( q k , q k / y z , q a / y , q a / z ; q ) _{\infty}}{( q k / y , q k / z , q a , q a / y z ; q ) _{\infty}} - 1) = \frac{1}{1 - y} (1 - \frac{( q a , q a / y z , q k / y , q k / z ; q ) _{\infty}}{( q a / y , q a / z , q k , q k / y z ; q ) _{\infty}}) \frac{( q k , q k / y z , q a / y , q a / z ; q ) _{\infty}}{( q k / y , q k / z , q a , q a / y z ; q ) _{\infty}} .

g (a, k, y, z, q) := n = 0 \sum \infty \frac{( q k , - q k , k , y , z , k / a ; q ) _{n}}{( k , - k , q k / y , q k / z , q a , q ; q ) _{n}} (\frac{q a}{y z})^{n},

g (a, k, y, z, q) := n = 0 \sum \infty \frac{( q k , - q k , k , y , z , k / a ; q ) _{n}}{( k , - k , q k / y , q k / z , q a , q ; q ) _{n}} (\frac{q a}{y z})^{n},

g (a, k, y, z, q) = \frac{1}{g ( k , a , y , z , q )} .

g (a, k, y, z, q) = \frac{1}{g ( k , a , y , z , q )} .

f (a, k, z, q) = n = 1 \sum \infty \frac{k q ^{n}}{1 - k q ^{n}} + n = 1 \sum \infty \frac{q ^{n} a / z}{1 - q ^{n} a / z} - n = 1 \sum \infty \frac{a q ^{n}}{1 - a q ^{n}} - n = 1 \sum \infty \frac{q ^{n} k / z}{1 - q ^{n} k / z} .

f (a, k, z, q) = n = 1 \sum \infty \frac{k q ^{n}}{1 - k q ^{n}} + n = 1 \sum \infty \frac{q ^{n} a / z}{1 - q ^{n} a / z} - n = 1 \sum \infty \frac{a q ^{n}}{1 - a q ^{n}} - n = 1 \sum \infty \frac{q ^{n} k / z}{1 - q ^{n} k / z} .

G (y) := \frac{( q k , q k / y z , q a / y , q a / z ; q ) _{\infty}}{( q k / y , q k / z , q a , q a / y z ; q ) _{\infty}}

G (y) := \frac{( q k , q k / y z , q a / y , q a / z ; q ) _{\infty}}{( q k / y , q k / z , q a , q a / y z ; q ) _{\infty}}

f (a, k, z, q)

f (a, k, z, q)

= y \to 1 lim \frac{G ( y ) - G ( 1 )}{1 - y} = - G^{'} (1) .

G^{\prime}(1)=G(1)\frac{d\log G(y)}{dy}\bigg{|}_{y=1}=\frac{d\log G(y)}{dy}\bigg{|}_{y=1}.

G^{\prime}(1)=G(1)\frac{d\log G(y)}{dy}\bigg{|}_{y=1}=\frac{d\log G(y)}{dy}\bigg{|}_{y=1}.

f (a, k, z, q) + f (- a, - k, z, q) = 2 f (a^{2}, k^{2}, z^{2}, q^{2}) .

f (a, k, z, q) + f (- a, - k, z, q) = 2 f (a^{2}, k^{2}, z^{2}, q^{2}) .

n = 1 \sum \infty \frac{x q ^{n}}{1 - x q ^{n}} + n = 1 \sum \infty \frac{( - x ) q ^{n}}{1 - ( - x ) q ^{n}} = n = 1 \sum \infty (\frac{x q ^{n}}{1 - x q ^{n}} + \frac{( - x ) q ^{n}}{1 - ( - x ) q ^{n}}) = n = 1 \sum \infty \frac{2 x ^{2} q ^{2 n}}{1 - x ^{2} q ^{2 n}} = 2 n = 1 \sum \infty \frac{x ^{2} q ^{2 n}}{1 - x ^{2} q ^{2 n}}

n = 1 \sum \infty \frac{x q ^{n}}{1 - x q ^{n}} + n = 1 \sum \infty \frac{( - x ) q ^{n}}{1 - ( - x ) q ^{n}} = n = 1 \sum \infty (\frac{x q ^{n}}{1 - x q ^{n}} + \frac{( - x ) q ^{n}}{1 - ( - x ) q ^{n}}) = n = 1 \sum \infty \frac{2 x ^{2} q ^{2 n}}{1 - x ^{2} q ^{2 n}} = 2 n = 1 \sum \infty \frac{x ^{2} q ^{2 n}}{1 - x ^{2} q ^{2 n}}

j = 0 \sum m - 1 f (a ω^{j}, k ω^{j}, z, q) = m f (a^{m}, k^{m}, z^{m}, q^{m}) .

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Full text

Some new Transformations for Bailey pairs and WP-Bailey Pairs

James Mc Laughlin

Mathematics Department

25 University Avenue

West Chester University, West Chester, PA 19383

[email protected]

Abstract.

We derive several new transformations relating WP-Bailey pairs. We also consider the corresponding transformations relating standard Bailey pairs, and as a consequence, derive some quite general expansions for products of theta functions which can also be expressed as certain types of Lambert series.

Key words and phrases:

Bailey pairs, WP-Bailey Chains, WP-Bailey pairs, Lambert Series, Basic Hypergeometric Series, q-series, theta series

2000 Mathematics Subject Classification:

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

1. Introduction

Andrews [1], building on previous work of Bressoud [8] and Singh [14], defined a WP-Bailey pair to be a pair of sequences $(\alpha_{n}(a,k,q)$ , $\beta_{n}(a,k,q))$ (if the context is clear, we occasionally suppress the dependence on some or all of $a$ , $k$ and $q$ ) satisfying $\alpha_{0}(a,k,q)$ $=\beta_{0}(a,k,q)=1$ , and for $n>0$ ,

[TABLE]

If $k=0$ , then the pair of sequences $(\alpha_{n}(a,q),\beta_{n}(a,q))$ is called a Bailey pair with respect to $a$ .

In the same paper Andrews showed that if the pair $(\alpha_{n}(a,k),\,\beta_{n}(a,k))$ satisfies (1.1), then so does $(\alpha_{n}^{\prime}(a,k),\,\beta_{n}^{\prime}(a,k))$ where

[TABLE]

with $c=kyz/aq$ . Andrews [1] also described a second method for deriving new WP-Bailey pairs from existing pairs, but this second method will not concern us in the present paper.

These two constructions allow a “tree” of WP-Bailey pairs to be generated from a single WP-Bailey pair. The implications of these two branches were further investigated by Andrews and Berkovich in [2]. Spiridonov [16] derived an elliptic generalization of Andrews first WP-Bailey chain. Four additional branches were added to the WP-Bailey tree by Warnaar [18], two of which had generalizations to the elliptic level. More recently, Liu and Ma [10] introduced the idea of a general WP-Bailey chain, and added one new branch to the WP-Bailey tree. In [12], the authors added three new WP-Bailey chains.

It is not difficult to show (see Corollary 1 in [13], for example) that the WP-Bailey chain at (1.2) implies that if $(\alpha_{n},\beta_{n})$ satisfy (1.1), then subject to suitable convergence conditions,

[TABLE]

In the present paper we prove some new relations for WP-Bailey pairs. These include the following.

Theorem 1.

If $(\alpha_{n}(a,k),\beta_{n}(a,k))$ is a WP-Bailey pair, then subject to suitable convergence conditions,

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Theorem 2.

If $(\alpha_{n}(a,k,q),\beta_{n}(a,k,q))$ is a WP-Bailey pair, then subject to suitable convergence conditions,

[TABLE]

We find some similar relations for standard Bailey pairs and derive some interesting consequences. For example, recall that $\psi(q)=\sum_{n=0}^{\infty}q^{n(n+1)/2}$

$=(q^{2};q^{2})_{\infty}/(q;q^{2})_{\infty}$ is Ramanujan’s theta function (see [5, page 36], for example). If $d\not=q^{3n\pm 1}$ , then

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We show that similar results hold for many other theta products.

We use the standard notations:

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We will make use of Bailey’s $\,{}_{6}\psi_{6}$ summation formula [17].

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where the second equality follows from the definition

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We also recall Jackson’s summation formula for a very-well-poised ${}_{6}\phi_{5}$ series [9, p. 356, Eq. (II. 20)] (which follows upon setting $e=a$ in (1.6)):

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Finally, we make use of the $q$ -Binomial Theorem [9, page 8],

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For later use, we note the special cases

[TABLE]

which following respectively from setting $a=0$ , and replacing $z$ with $z/a$ and then letting $a\to\infty$ . Unless stated otherwise, we assume $|q|<1$ .

2. Proofs of the Main Identities

The next transformation follows easily from the identity at (1.3).

Lemma 1.

If $(\alpha_{n}(a,k,q),\beta_{n}(a,k,q))$ is a WP-Bailey pair, then subject to suitable convergence conditions,

[TABLE]

Proof.

Rewrite (1.3) as

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The left side of (2.1) follows upon letting $y\to 1$ on the left side of (2.2). From (1.8) it can be seen that

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Upon making this substitution in the right side of (2.2), we get the right side of (2.1), after setting $(y;q)_{n}/(1-y)=(yq;q)_{n-1}$ , then letting $y\to 1$ and finally setting $(q;q)_{n-1}/(q;q)_{n}=1/(1-q^{n})$ . ∎

For later use we note that the series on the right side of (2.1) has the following properties. We define

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Lemma 2.

If $f(a,k,z,q)$ is as defined at (2.4), $|qa|,|qk|<|z|$ and none of the denominators below vanish, then

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Proof.

This follows easily upon writing

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From the proof of Lemma 1, it can be see that the result of letting $y\to 1$ on the left side above is $f(a,k,z,q)$ . On the other hand, the infinite product following the “ $-$ ” sign on the right side above is the product on the left side above with $a$ and $k$ interchanged, so that the result of letting $y\to 1$ on the right side is $-f(k,a,z,q)$ . ∎

We remark in passing that the expansion at (2.3) and the similar expansion of the reciprocal of this product imply that if

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then

[TABLE]

We next express $f(a,k,z,q)$ as a sum of Lambert series.

Lemma 3.

If $f(a,k,z,q)$ is as defined at (2.4), $|qa|<|z|$ and none of the denominators below vanish, then

[TABLE]

Proof.

If we define

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we see that $G(1)=1$ and

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That $-G^{\prime}(1)$ equals the right side of (2.7) follows by logarithmically differentiating the infinite products in $G(y)$ , noting that

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∎

Lemma 4.

If $f(a,k,z,q)$ is as defined at (2.4), $|qa|<|z|$ and none of the denominators below vanish, then

[TABLE]

Proof.

Use (2.7) to write $f(a,k,z,q)+f(-a,-k,z,q)$ in terms of Lambert series. Then use the elementary identity

[TABLE]

to combine pairs of Lambert series into single Lambert series, thus deriving the right side of (2.8) ∎

Remark: By somewhat similar reasoning, one can show that if $m\geq 2$ is a positive integer and $\omega$ is a primitive $m$ -root of unity, then

[TABLE]

Proof of Theorem 2.

Use (2.1) (noting that the series on the right side is $f(a,k,z,q)$ ) to substitute for $f(a,k,z,q)$ , $f(-a,-k,z,q)$ and $f(a^{2},k^{2},z^{2},q^{2})$ in (2.8), and the results follows after a little rearrangement of terms. ∎

One could easily insert specific WP-Bailey pairs in (1.5) to provide explicit identities, but we leave that to the reader. We also note that letting $k\to 0$ in Theorem 2 gives a result for standard Bailey pairs.

Corollary 1.

If $(\alpha_{n}(a,q),\beta_{n}(a,q))$ is a Bailey pair with respect to $a$ , then subject to suitable convergence conditions,

[TABLE]

Once again we leave it to the reader to produce particular identities, by inserting specific Bailey pairs.

Lemma 5.

If $f(a,k,z,q)$ is as defined at (2.4), $|qa|<|z|<|a/q|$ and none of the denominators vanish, then

[TABLE]

Proof.

One can check (preferably with a computer algebra system) that

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so that

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Similarly, it can be shown that

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From the remarks above and (1.7),

[TABLE]

where the last equality follows from (1.6), with $b=a$ , $c=k$ , $d=a/z$ , $e=k/z$ and $ak/z$ instead of $a$ . Some further easy manipulations gives the final result. ∎

Remark: The proof that the sum of Lambert series above combine to give the stated infinite product was first given by Andrews, Lewis and Liu in [4] (using a different labeling for the parameters) in a different context, so they did not have our reciprocity result for the basic hypergeometric series $f(a,k,z,q)$ .

Note that substituting the expression for $f(a,k,z,q)$ from (2.4) into (2.10) leads to the identity

[TABLE]

an identity which does not appear to follow directly from Bailey’s formula at (1.6). We are now ready to prove Theorem 1.

Proof of Theorem 1.

In the identity at (2.1), note that the right side equals $f(a,k,z,q)$ . Now replace $a$ with $1/a$ , $k$ with $1/k$ , $z$ with $1/z$ , and subtract the resulting identity from the original identity. The left side of the resulting identity is the left side of (1.4), while the right side is $f(a,k,z,q)-f(1/a,1/k,1/z,q)$ , which by (2.10) is the right side of (1.4). ∎

Any WP-Bailey that is inserted into (1.4) will lead to a summation formula for basic hypergeometric series. We give two example as illustrations.

Corollary 2.

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Proof.

Insert Singh’s WP-Bailey pair [14],

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into (1.4). ∎

Corollary 3.

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Proof.

Insert the WP-Bailey pair (see [2, (3.3) - (3.4)]

[TABLE]

into (1.4). ∎

3. Applications to Bailey Pairs

If we let $k\to 0$ in Lemmas 1, 2 and 3, we get the following result.

Theorem 3.

If $(\alpha_{n},\beta_{n})$ is a Bailey pair with respect to $a$ , then subject to suitable convergence conditions,

[TABLE]

where

[TABLE]

Note that the first two representations for $f_{1}(a,z,q)$ follow from (3.1), upon inserting, respectively, the “unit” Bailey pair

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and the “trivial” Bailey pair

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However, here and subsequently, we prefer to write these representations explicitly. Upon letting $z\to\infty$ the following identity results.

Corollary 4.

If $(\alpha_{n},\beta_{n})$ is a Bailey pair with respect to $a$ , then subject to suitable convergence conditions,

[TABLE]

where

[TABLE]

As is well known, many theta products/series can be represented as sums of Lambert series of the type immediately above. The other representations of $f_{2}(a,q)$ now let these theta functions be represented in two different ways as basic hypergeometric series.

Let

[TABLE]

Here we are using the notation for this series employed in [7].

Corollary 5.

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Proof.

The following result is Entry 18.2.8 of Ramanujan’s Lost Notebook (see [3, page 402]):

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Use (3.3) (with $q$ replaced with $q^{3}$ and $a=q^{-1}$ and $a=q^{-2}$ , respectively) to replace each of the Lambert series with, in turn, each of the other two representations of $f_{2}(a,q^{3})$ , and the result follows. ∎

Remark: It is clear that a quite general statement concerning $a(q)$ may be deduced from (3.4) by a similar argument. Indeed, if $(\alpha_{n}(a,q)$ , $\beta_{n}(a,q))$ is any Bailey pair in which $a$ is a free parameter, then

[TABLE]

As an example, if we insert the Bailey pair of Slater [15, Equation (4.1), page 469],

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in (3.6), we get, for any values for $c$ and $d$ that do not make any denominator vanish, that

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If we let $c,d\to\infty$ in this identity we get that

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A similar situation will hold for some of the other identities given below. Recall (see [5, page 36])

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Corollary 6.

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For any values for $c$ and $d$ that do not make any denominator vanish,

[TABLE]

Proof.

By Entry 8 (i) in chapter 17 of [5],

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We omit the remainder of the arguments, since they parallel those for the identities involving $a(q)$ above. ∎

If we let $c,d\to\infty$ in (3.11), we get the identity

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Corollary 7.

If $(\alpha_{n},\beta_{n})$ is a Bailey pair with respect to $a$ , then subject to suitable convergence conditions,

[TABLE]

where

[TABLE]

Proof.

Let $z\to-1$ in (3.1) and simplify. ∎

One reason we single out this special case is that many theta products/series can also be expressed in terms of Lambert series of the type just above. We consider one example. Recall (see [5, page 36]) that

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By Entry 34 (p.284) in chapter 36 of Ramanujan’s notebooks (see [6, page 374]),

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Upon replacing $q$ with $q^{3}$ and $a$ with $q^{-2}$ and then $a$ with $q^{-1}$ in Corollary 7 and combining the various serious appropriately, we get the following identities.

Corollary 8.

[TABLE]

If $(\alpha_{n}(a,q),\beta_{n}(a,q))$ is a Bailey pair in which $a$ is a free parameter, then

[TABLE]

If $d\not=q^{3n\pm 1}$ , then

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Proof.

Identities (3.15) - (3.16) follow directly from Corollary 8. The identity at (3.17) follows upon inserting the Bailey pair (see Corollary 2.13 in [11])

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into (3.16), and (3.18) is a consequence of letting $d\to\infty$ in (3.17). ∎

4. The Lambert series $\sum_{n=1}^{\infty}\frac{aq^{n}}{1-aq^{n}}$ again

Define

[TABLE]

From Corollary 4 it can be seen that $L_{a}(q)$ can be variously represented as

[TABLE]

where $(\alpha_{n},\beta_{n})$ is a Bailey pair with respect to $a$ . We subsequently noticed that it was possible to give two additional representations of $L_{a}(q)$ .

Corollary 9.

[TABLE]

Proof.

Let $k\to 0$ and $z\to\infty$ in (1.3), and rearrange to get

[TABLE]

where $(\alpha_{n},\beta_{n})$ is a Bailey pair with respect to $a$ . The result of letting $y\to 1$ on the left side of (4.3) is the left side of (3.2), and hence equals $-L_{a}(q)$ , from the final representation of $f_{2}(a,q)$ in Corollary 4.

Thus (4.3) now gives that

[TABLE]

In the first case we use the second identity at (1.10) to get that

[TABLE]

by L’Hospital’s rule. The proof for the other representation follows similarly, using the first identity at (1.10). ∎

These expressions for $L_{a}(q)$ may also be used to write any theta product that is expressible in terms of such Lambert series in terms of $q$ -series similar to those in Corollary 9. Recall that $\phi(q)$ is defined at (3.9).

Corollary 10.

[TABLE]

Proof.

By **Entry 18.2.16 (formula (1.21), p.353; formula (3.51),

p.357)** in Ramanujan’s Lost Notebook (see [3, page 405]),

[TABLE]

The proofs now follow as a consequence of Corollary 9, with $q$ replaced with $q^{3}$ , and $a$ taking, in turn, the values $-1/q^{2}$ and $-1/q$ . ∎

Remark: Ramanujan gives a number of other examples of theta products expressible as sums of Lambert series of the types considered in the present paper. The methods of the present paper could also be applied to those theta products, but we refrain from further examples, leaving these for the reader’s own entertainment.

Bibliography18

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

1[1] Andrews G. E., 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., 2001, 30, Kluwer Acad. Publ., Dordrecht
2[2] Andrews G.E., Berkovich A., The WP-Bailey tree and its implications, J. London Math. Soc.(2), 2002, 66 , no. 3, 529–549.
3[3] Andrews G.E., Berndt B.C., Ramanujan’s Lost Notebook, Part I, Springer, 2005.
4[4] Andrews G.E., Lewis R., Liu Z.G., An identity relating a theta function to a sum of Lambert series, Bull. London Math. Soc., 2001, 33, no. 1, 25–31.
5[5] Berndt B. C., Ramanujan’s Notebooks, Part III, Springer-Verlag, New York, 1991
6[6] Berndt B. C., Ramanujan’s Notebooks, Part V, Springer-Verlag, New York, 1998
7[7] Borwein J. M., Borwein P. B., A cubic counterpart of Jacobi’s identity and the AGM. Trans. Amer. Math. Soc., 1991, 323 no. 2, 691–701
8[8] Bressoud D., Some identities for terminating q 𝑞 q -series, Math. Proc. Cambridge Philos. Soc., 1981, 89, no. 2, 211–223

TL;DR

Contribution

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Abstract

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Some new Transformations for Bailey pairs and WP-Bailey Pairs

Abstract.

Key words and phrases:

2000 Mathematics Subject Classification:

1. Introduction

Theorem 1**.**

Theorem 2**.**

2. Proofs of the Main Identities

Lemma 1**.**

Proof.

Lemma 2**.**

Proof.

Lemma 3**.**

Proof.

Lemma 4**.**

Proof.

Proof of Theorem 2.

Corollary 1**.**

Lemma 5**.**

Proof.

Proof of Theorem 1.

Corollary 2**.**

Proof.

Corollary 3**.**

Proof.

3. Applications to Bailey Pairs

Theorem 3**.**

Corollary 4**.**

Corollary 5**.**

Proof.

Corollary 6**.**

Proof.

Corollary 7**.**

Proof.

Corollary 8**.**

Proof.

4. The Lambert series ∑n=1∞aqn1−aqn\sum_{n=1}^{\infty}\frac{aq^{n}}{1-aq^{n}}∑n=1∞​1−aqnaqn​ again

Corollary 9**.**

Proof.

Corollary 10**.**

Proof.

Theorem 1.

Theorem 2.

Lemma 1.

Lemma 2.

Lemma 3.

Lemma 4.

Corollary 1.

Lemma 5.

Corollary 2.

Corollary 3.

Theorem 3.

Corollary 4.

Corollary 5.

Corollary 6.

Corollary 7.

Corollary 8.

4. The Lambert series $\sum_{n=1}^{\infty}\frac{aq^{n}}{1-aq^{n}}$ again

Corollary 9.

Corollary 10.