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On the derivatives $$\partial^2P_\nu(z)/\partial\nu^2$$ and $$\partial Q_\nu(z)/\partial\nu$$ of the Legendre functions with respect to their degrees. (English) Zbl 1376.33006
The Legendre function $$P_{\lambda } (x)$$ is defined through the differential equation
$$(1-x^{2} )\frac{d^{2} }{dx^{2} } P_{\lambda } (x)-2x\frac{d}{dx} P_{\lambda } (x)+\lambda (\lambda +1)P_{\lambda } (x)=0$$, where $$\lambda$$ is a parameter. When $$\lambda =n$$, a nonnegative integer, $$P_{n} (x)$$ is the classical Legendre polynomial.
In two previous papers [J. Phys. A, Math. Gen. 39, No. 49, 15147–15172 (2006; Zbl 1112.33009)] and also [J. Phys. A, Math. Theor. 40, No. 49, 14887–14891 (2007; Zbl 1125.33303)] the author computed the first derivative $$\left. \frac{d}{d\lambda } P_{\lambda } (x)\right|_{\lambda =n}$$. In the present paper it is shown that $$\left. \frac{d^{2} }{d\lambda ^{2} } P_{\lambda } (x)\right|_{\lambda =n} =-2P_{n} (x)\operatorname{Li}_{2} \frac{1-x}{2} +B_{n} (x)\ln \frac{1+x}{2} +C_{n} (x)$$, where $$\operatorname{Li}_{2}$$ is the dilogarithm and $$B_{n} (x)$$, $$C_{n} (x)$$ are certain polynomials defined as linear combinations of the Legendre polynomials. A similar result is obtained for the Legendre functions of the second kind $$Q_{\lambda } (x)$$. Namely, $$\left. \frac{d}{d\lambda } Q_{\lambda } (x)\right|_{\lambda =n}$$ is computed in explicit form.

##### MSC:
 33C05 Classical hypergeometric functions, $${}_2F_1$$ 33B30 Higher logarithm functions
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