For a given prime p, we study the properties of the p-dissection identities of Ramanujan's theta functions ψ(q) and f (−q), respectively. Then as applications, we find many infinite family of congruences modulo 2 for some ℓ-regular partition functions, especially, for ℓ = 2, 4,5,8,13,16. Moreover, based on the classical congruences for p(n) given by Ramanujan, we obtain many more congruences for some ℓ-regular partition functions.
We introduce the Cauchy augmentation operator for basic hypergeometric series. Heine's 2 φ 1 transformation formula and Sears' 3 φ 2 transformation formula can be easily obtained by the symmetric property of some parameters in operator identities. The Cauchy operator involves two parameters, and it can be considered as a generalization of the operator T (bD q ). Using this operator, we obtain extensions of the Askey-Wilson integral, the Askey-Roy integral, Sears' two-term summation formula, as well as the qanalogs of Barnes' lemmas. Finally, we find that the Cauchy operator is also suitable for the study of the bivariate Rogers-Szegö polynomials, or the continuous big q-Hermite polynomials.
The celebrated quintuple product identity follows surprisingly from an almost-trivial algebraic identity, which is the limiting case of the terminating q-Dixon formula.
In view of the modular equation of fifth order, we give a simple proof of Keith's conjecture which is some infinite families of congruences modulo 3 for the 9-regular partition function. Meanwhile, we derive some new congruences modulo 3 for the 9-regular partition function.
An overpartition of [Formula: see text] is a partition of [Formula: see text] in which the first occurrence of a number may be overlined. Then, the rank of an overpartition is defined as its largest part minus its number of parts. Let [Formula: see text] be the number of overpartitions of [Formula: see text] with rank congruent to [Formula: see text] modulo [Formula: see text]. In this paper, we study the rank differences of overpartitions [Formula: see text] for [Formula: see text] or [Formula: see text] and [Formula: see text]. Especially, we obtain some relations between the generating functions of the rank differences modulo 4 and 8 and some mock theta functions. Furthermore, we derive some equalities and inequalities on ranks of overpartitions modulo 4 and 8.
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