This paper presents, for the first time, analytical solutions for the charge and flux as a function of the state variable in the case of the HP memristor, modeled as an ideal generic memristor with Joglekar and Prodromakis window functions for a general value of the control parameter. The solutions are in the closed form of finite sums for an arbitrary positive integer values of the control parameter. Our approach is based on the decomposition of original problems using appropriate pairs of smooth hinge functions. The generalization of this approach allows us to define a novel window function based on smooth hinge functions, which brings simpler analytical expressions for charge and flux as a function of the state variable, and faster simulations when the model is used.