LAUSR

Abstract This paper investigates an extended form of a beta function B p , q ( x , y ) . We first study the convergence problem of the function B p , 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 B p , q ( x , y ) . Subsequently, we consider the convergence problem of the extended Hurwitz–Lerch zeta function Φ λ , 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 Φ λ , μ ; ν ( z , s , a ; p , q ) . Lastly, we apply Fourier analysis to the function z a Φ μ ; ν ( z , s , a ; p , q ) and obtain a Lindelof–Wirtinger type expansion. Some interesting and promising results are also illustrated.