We evaluate sums of certain classes of new series involving the Riemann zeta function by using the theory of the double gamma function and a property of the gamma function. Relevant connections with various known results are also pointed out.

Asymptotic expansions of the Barnes double zeta-functionformula hereand the double gamma-function Γ2(α, (1, w)), with respect to the parameter w, are proved. An application to Hecke L-functions of real quadratic fields is also discussed.

Lots of formulas for series of zeta function have been developed in many ways. We show how we can apply the theory of the double gamma function, which has recently been revived according to the study of determinants of Laplacians, to evaluate some series involving the Riemann zeta function.

A new integral representation for the Barnes double gamma function is derived. This is canonical in the sense that solutions to a class of functional difference equations of first order with trigonometrical coefficients can be expressed in terms of the Barnes function. The integral representation given here makes these solutions very simple to compute. Several well-known difference equations are solved by this method, and a treatment of the linear theory for moving contact line flow in an inviscid fluid wedge is given.

A new and exact solution is obtained for the diffraction of an E -polarized electromagnetic plane wave by an imperfectly conducting wedge of arbitrary angle. The original boundaryvalue problem is reduced to the solution of an ordinary difference equation. This equation is solved in terms of the double gamma function defined by Barnes (1899). If the wedge angle is equal to pπ/2q where p and q are relatively prime integers, with p odd, the difference equation is soluble in a simple closed form. The resulting solution for the field components is then comparatively simple. The present theoretical results show very good agreement with experimental results in the shadow region for normal incidence on a 16° wedge.