m-Dissections of some infinite products and related identities

For integers t and m with m ≥ 5 relatively prime to 6 such that 1 ≤ t < m / 2 and gcd ( t , m ) = 1 , define Q ( t , m ) : = ( q 2 t , q m - 2 t , q m ; q m ) ∞ ( q t , q m - t ; q m ) ∞ . The m -dissection of this quintuple product is given in terms of other quintuple products. Hirschhorn’s 5-dissections of Ramanujan’s function R = R ( q ) and its reciprocal follow as special cases of this general result. Define the sequence { c n } n ≥ 0 by Q ( t , m ) ( q m ; q m ) ∞ = : ∑ n = 0 ∞ c n q n . It is proven that n = n 0 : = 1 6 ( m - 4 ) m ( m + 1 ) + ( m - 1 ) t is the largest value of n for which c n = 0 (a slight modification to the value of n 0 is needed if t = ( m - 1 ) / 2 ). Further, it is shown that for n > n 0 , the signs of the c n are periodic with period m . In addition, a formula is given for a polynomial f t , m ( q ) (deriving from the m -dissection of Q ( t , m )) from which a second polynomial g t , m ( q ) of degree m - 1 is derived, such that the coefficient of q d (either 1 or -1) in g t , m ( q ) indicates the sign of all the coefficients in the arithmetic progression c m k + d , for 0 ≤ d ≤ m - 1 and m k + d > n 0 . By using methods like those used to prove the results above, other similar results are proved. We re-derive the result of Evans and Ramanathan giving the m -dissection of ( q ; q ) ∞ in terms of quintuple products for any integer m ≡ ± 1 ( mod 6 ) . Other results include various Lambert series identities, such as the following. Define A = A ( q ) : = q 2 , q 5 ; q 7 ∞ q , q 6 ; q 7 ∞ , B = B ( q ) : = q 3 , q 4 ; q 7 ∞ q 2 , q 5 ; q 7 ∞ . Then 1 + 14 ∑ n = 1 ∞ q 3 n 1 - q 7 n - 14 ∑ n = 1 ∞ q 4 n 1 - q 7 n = A 3 + 3 B q A - 6 q B f 7 2 q 3 , q 4 ; q 7 ∞ , where f i : = ( q i ; q i ) ∞ .