On extended Hurwitz-Lerch zeta function

This paper investigates an extended form of a beta function Bp,q(x,y). We first study the convergence problem of the function Bp,q(x,y) and consider the completely monotonic and log-convex properties of this function. As a result, we obtain a pair of Laguerre type inequalities. Next, we provide a new double integral representation for the function Bp,q(x,y). Subsequently, we consider the convergence problem of the extended Hurwitz-Lerch zeta function Φλ,μ;ν(z,s,a;p,q) defined by its series representation. Upon using the series manipulation techniques, we obtain two series identities. We also find various integral representations for the function Φλ,μ;ν(z,s,a;p,q). Lastly, we apply Fourier analysis to the function zaΦμ;ν(z,s,a;p,q) and obtain a Lindelöf-Wirtinger type expansion. Some interesting and promising results are also illustrated.