Basic and bibasic identities related to divisor functions

q-Identities Related to Overpartitions and Divisor Functions

We generalize and prove conjectures of Corteel and Lovejoy, related to overpartitions and divisor functions.

n-COLOR OVERPARTITIONS, TWISTED DIVISOR FUNCTIONS, AND ROGERS-RAMANUJAN IDENTITIES

In the early 90’s Andrews discussed a certain q-series whose coefficients are determined by a twisted divisor function. We provide several other examples of this nature. All of these q-series can be interpreted combinatorially in terms of n-color overpartitions, as can some closely related series occurring in identities of the Rogers-Ramanujan type.

Fermionic representation for basic hypergeometric functions related to Schur polynomials

We present the fermionic representation for the q-deformed hypergeometric functions related to Schur polynomials. For q = 1 it is known that these hypergeometric functions are related to zonal spherical polynomials for GL(N,C)/U(N) symmetric space. We show that multivariate hypergeometric functions are tau-functions of the KP and of the two-dimensional Toda lattice hierarchies. The variables of...

Modular Divisor Functions and Quadratic Reciprocity

DIVISOR FUNCTIONS AND WEIERSTRASS FUNCTIONS ARISING FROM q-SERIES

We consider Weierstrass functions and divisor functions arising from q-series. Using these we can obtain new identities for divisor functions. Farkas [3] provided a relation between the sums of divisors satisfying congruence conditions and the sums of numbers of divisors satisfying congruence conditions. In the proof he took logarithmic derivative to theta functions and used the heat equation. ...