Dissection of the quintuple product, with applications

This work considers the m -dissection (for m ≢ 0 ( mod 3 ) ) of the general quintuple product Q ( z , q ) = ( z , q / z , q ; q ) ∞ ( q z 2 , q / z 2 ; q 2 ) ∞ . Multiple novel applications arise from this m -dissection. For example, we derive the general partition identity D S ( m n + ( m 2 − 1 ) / 24 ) = ( − 1 ) ( m + 1 ) / 6 b m ( n ) , for all n ≥ 0 , where m ≡ 5 ( mod 6 ) is a square-free positive integer relatively prime to 6; D S ( n ) is defined, for S the set of positive integers containing no multiples of m , to be the number of partitions of n into an even number of distinct parts from S minus the number of partitions of n into an odd number of distinct parts from S ; and b m ( n ) denotes the number of m -regular partitions of n . The dissections allow us to prove a conjecture of Hirschhorn concerning the 2 n -dissection of ( q ; q ) ∞ , as well as determine the pattern of the sign changes of the coefficients a n of the infinite product ( q 2 k − 1 ; q 2 k − 1 ) ∞ ( q p ; q p ) ∞ 2 = ∑ n = 0 ∞ a n q n , k ≥ 1 , p ≥ 5 a prime. This covers a recent result of Bringmann et al. that corresponds to the case k = 1 and p = 5 . • A formula for the m -dissection of the general quintuple product Q ( z , q ) : = ( z 2 , q / z 2 , q ; q ) ∞ ( − z , − q / z ; q ) ∞ • A proof of Hirschhorn's conjecture for the 2 n dissection of ( q ; q ) ∞ . • Extension the formula of Evans and Ramanathan for the m-dissection of ( q ; q ) ∞ , where gcd ( m , 6 ) = 1 , to also include m ≡ 2 , 4 ( mod 6 ) . • Some new kinds of partition identities. • Some new results on the periodicity of signs in the series expansion of various families of eta quotients