Summary: We study the derivative of the Legendre function of the first kind, , with respect to its degree . At first, we provide two contour integral representations for . Then, we proceed to investigate the case of , with being an integer; this case is met in some physical and engineering problems. Since it holds that , we focus on the sub-case of being a non-negative integer. We show that
where is a polynomial in of degree . We present alternative derivations of several known explicit expressions for and also add some new. A generating function for is also constructed. Properties of the polynomials and are also investigated. It is found that is the Christoffel polynomial, well known from the theory of the Legendre function of the second kind, . As examples of applications of the results obtained, we present non-standard derivations of some representations of , sum to closed forms some Legendre series, evaluate some definite integrals involving Legendre polynomials and also derive an explicit representation of the indefinite integral of the Legendre polynomial squared.