On the derivatives ${\mathrm{\partial }}^2{P}_{\nu }(z)\mathrm{/}\mathrm{\partial }{\nu }^2$ and $\mathrm{\partial }{Q}_{\nu }(z)\mathrm{/}\mathrm{\partial }\nu$ of the Legendre functions with respect to their degrees

We provide closed-form expressions for the degree-derivatives $[{\mathrm{\partial }}^2{P}_{\nu }(z)\mathrm{/}\mathrm{\partial }{\nu }^2{]}_{\nu =n}$ and $[\partial Q_{\nu}(z)/\partial\nu]{\nu=n}$,with$z\in$\mathbb{C}$$and$n\in$\mathbb{N}${0}$,where$P_{\nu}(z)$and$Q_{\nu}(z)$ are the Legendre functions of the first and the second kind, respectively. For $[{\mathrm{\partial }}^2{P}_{\nu }(z)\mathrm{/}\mathrm{\partial }{\nu }^2{]}_{\nu =n}$, we find that $${\displaystyle \frac{{\mathrm{\partial }}^2{P}_{\nu }(z)}{\mathrm{\partial }{\nu }^2}{\mid }_{\nu =n}=-2P_n(z){\text{Li}}_2\frac{1-z}{2}+B_n(z)\ln \frac{z+1}{2}+C_n(z),}$$ where ${\text{Li}}_2[(1-z)\mathrm{/}2]$ is the dilogarithm function, $P_n(z)$ is the Legendre polynomial, while $B_n(z)$ and $C_n(z)$ are certain polynomials in $z$ of degree $n$. For $[\mathrm{\partial }{Q}_{\nu }(z)\mathrm{/}\mathrm{\partial }\nu {]}_{\nu =n}$ and $z\in \mathbb{C}\setminus [-1,1]$, we derive $${\displaystyle \frac{\mathrm{\partial }{Q}_{\nu }(z)}{\mathrm{\partial }\nu }{\mid }_{\nu =n}=-P_n(z){\text{Li}}_2\frac{1-z}{2}-\frac{1}{2}{P}_n(z)\ln \frac{z+1}{2}\ln \frac{z-1}{2}+\frac{1}{4}{B}_n(z)\ln \frac{z+1}{2}-\frac{(-1{)}^n}{4}{B}_n(-z)\ln \frac{z-1}{2}-\frac{{\pi }^2}{6}{P}_n(z)+\frac{1}{4}{C}_n(z)-\frac{(-1{)}^n}{4}{C}_n(-z)\mathrm{.}}$$ A counterpart expression for $[\mathrm{\partial }{Q}_{\nu }(x)\mathrm{/}\mathrm{\partial }\nu {]}_{\nu =n}$, applicable when $x\in (-1,1)$, is also presented. Explicit representations of the polynomials $B_n(z)$ and $C_n(z)$ as linear combinations of the Legendre polynomials are given.