On asymptotic properties of the Legendre functions (Q2880021)

In the present paper, the so-called asymptotic representation of the special functions are studied. The goal of the work is to investigate the behavior of the Legendre functions \(P_{\mu}^{\nu}(t)\) on the set \(t\in (1, +\infty).\) The main result here is the following. Suppose that \(\varepsilon\in (0, \pi),\) then we have some expression for \(P_{i\lambda}-\frac{1}{2}^{\mu}\) at \(\lambda\rightarrow\infty,\) \(| \arg\,\lambda| \leq \pi-\varepsilon.\) To note that some particular cases of the representation mentioned above were known earlier. The results of the paper can be applied to the different problems of mathematical analysis and function theory.