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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.

33C45Orthogonal polynomials and functions of hypergeometric type
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