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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, ${P}_{\nu }\left(z\right)$, with respect to its degree $\nu$. At first, we provide two contour integral representations for $\partial {P}_{\nu }\left(z\right)/\partial \nu$. Then, we proceed to investigate the case of ${\left[\partial {P}_{\nu }\left(z\right)/\partial \nu \right]}_{\nu =n}$, with $n$ being an integer; this case is met in some physical and engineering problems. Since it holds that ${\left[\partial {P}_{{\nu }^{\text{'}}}\left(z\right)/\partial {\nu }^{\text{'}}\right]}_{{\nu }^{\text{'}}=-\nu -1}=-{\left[\partial {P}_{{\nu }^{\text{'}}}\left(z\right)/\partial {\nu }^{\text{'}}\right]}_{{\nu }^{\text{'}}=\nu }$, we focus on the sub-case of $n$ being a non-negative integer. We show that

${\frac{\partial {P}_{\nu }\left(z\right)}{\partial \nu }|}_{\nu =n}={P}_{n}\left(z\right)ln\frac{z+1}{2}+{R}_{n}\left(z\right)\phantom{\rule{1.em}{0ex}}\left(n\in ℕ\right)$

where ${R}_{n}\left(z\right)$ is a polynomial in $z$ of degree $n$. We present alternative derivations of several known explicit expressions for ${R}_{n}\left(z\right)$ and also add some new. A generating function for ${R}_{n}\left(z\right)$ is also constructed. Properties of the polynomials ${V}_{n}\left(z\right)=\left[{R}_{n}\left(z\right)+{\left(-1\right)}^{n}{R}_{n}\left(-z\right)\right]/2$ and ${W}_{n-1}\left(z\right)=-\left[{R}_{n}\left(z\right)-{\left(-1\right)}^{n}{R}_{n}\left(-z\right)\right]/2$ are also investigated. It is found that ${W}_{n-1}\left(z\right)$ is the Christoffel polynomial, well known from the theory of the Legendre function of the second kind, ${Q}_{n}\left(z\right)$. As examples of applications of the results obtained, we present non-standard derivations of some representations of ${Q}_{n}\left(z\right)$, 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