New infinite q-product expansions with vanishing coefficients

Motivated by results of Hirschhorn, Tang, and Baruah and Kaur on vanishing coefficients (in arithmetic progressions) in a new class of infinite product which have appeared recently, we further examine such infinite products, and find that many such results on vanishing coefficients may be grouped into families. For example, one result proved in the present paper is that if b ∈ { 1 , 2 , ⋯ , 9 , 10 } and the sequence { r n } is defined by ( q 8 b , q 11 - 8 b ; q 11 ) ∞ 3 ( q 11 - b , q 11 + b ; q 22 ) ∞ = : ∑ n = - 756 ∞ r n q n , then r 11 n + 6 b 2 + b = 0 for all n . Further, if b ∈ { 1 , 3 , 5 , 7 , 9 } , then r 11 n + 4 b 2 + b = 0 for all n also. Each particular value of b gives a specific result, such as the following (for b = 1 ): if the sequences { a n } is defined by ∑ n = 0 ∞ a n q n : = ( q 3 , q 8 ; q 11 ) ∞ 3 ( q 10 , q 12 ; q 22 ) ∞ , then a 11 n + 5 = a 11 n + 7 = 0 .