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

Szmytkowski, Radosław

Integral Transforms Spec. Funct. 28, No. 9, 645-662 (2017).

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.

Reviewer: Khristo N. Boyadzhiev (Ada)

MSC:

33C05	Classical hypergeometric functions, \({}_2F_1\)
33B30	Higher logarithm functions

Keywords:

Legendre functions; Legendre polynomials; parameter derivatives; dilogarithm