On the derivative of the Legendre function of the first kind with respect to its degree.

*(English)* Zbl 1112.33009
Summary: We study the derivative of the Legendre function of the first kind, $P_\nu(z)$, with respect to its degree $\nu$. At first, we provide two contour integral representations for $\partial P_\nu(z)/\partial\nu$. Then, we proceed to investigate the case of $[\partial P_\nu(z)/\partial \nu]_{\nu=n}$, with $n$ being an integer; this case is met in some physical and engineering problems. Since it holds that $[\partial P_{\nu'} (z)/\partial\nu']_{\nu'=-\nu-1}=-[\partial P_{\nu'}(z)/\partial\nu']_{\nu' =\nu}$, we focus on the sub-case of $n$ being a non-negative integer. We show that $$\left.\frac{\partial P_\nu(z)} {\partial\nu}\right \vert_{\nu=n}=P_n(z)\ln\frac{z+1}{2}+R_n(z)\quad (n\in \bbfN)$$ where $R_n(z)$ is a polynomial in $z$ of degree $n$. We present alternative derivations of several known explicit expressions for $R_n(z)$ and also add some new. A generating function for $R_n(z)$ is also constructed. Properties of the polynomials $V_n(z)=[R_n(z)+(-1)^nR_n(-z)]/2$ and $W_{n-1}(z)=-[R_n(z)-(-1)^nR_n(-z)]/2$ are also investigated. It is found that $W_{n-1}(z)$ is the Christoffel polynomial, well known from the theory of the Legendre function of the second kind, $Q_n(z)$. As examples of applications of the results obtained, we present non-standard derivations of some representations of $Q_n(z)$, 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.

##### MSC:

33C45 | Orthogonal polynomials and functions of hypergeometric type |