We present a mathematical approach to the worm-like chain model of semiflexible polymers. Our method is built on a novel generating function from which all the properties of the model can be derived. Moreover, this approach satisfies the local inextensibility constraint exactly. In this paper, we focus on the lowest order contribution to the generating function and derive explicit analytical expressions for the characteristic function, polymer propagator, single chain structure factor, and mean square end-to-end distance. These analytical expressions are valid for polymers with any degree of stiffness and contour length. We find that our calculations are able to capture the fully flexible and infinitely stiff limits of the aforementioned quantities exactly while providing a smooth and approximate crossover behavior for intermediate values of the stiffness of the polymer backbone. In addition, our results are in very good quantitative agreement with the exact and approximate results of five other treatments of semiflexible polymers.