Abstract. We introduce a new type of multiple zeta function, which we call a bilateral zeta function. We prove that the bilateral zeta function has a nice Fourier series expansion and the Barnes zeta function can be expressed as a finite sum of bilateral zeta functions. By these properties of the bilateral zeta functions, we obtain simple proofs of some formulas, for example, the reflection formula for the multiple gamma function, the inversion formula for the Dedekind η-function, Ramanujan's formula, Fourier expansion of the Barnes zeta function and multiple Iseki's formula.
We present new Pieri type formulas for Jack polynomials. As an application, we give a new derivation of higher order difference equations for interpolation Jack polynomials originally found by Knop and Sahi. We also propose Pieri formulas for interpolation Jack polynomials and intertwining relations for a kernel function for Jack polynomials.
We introduce a one parameter deformation of the Zwegers' µ-function as the image of q-Borel and q-Laplace transformations of a fundamental solution for the q-Hermite-Weber equation. We further give some formulas of our generalized µ-function, for example, forward and backward shift, translation, symmetry, a difference equation for the new parameter, and q-bilateral hypergeometric expressions. From one point of view, the continuous q-Hermite polynomials are some special cases of our µ-function, and the Zwegers' µ-function is regarded as a continuous q-Hermite polynomial of "−1 degree".
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.