*(English)*Zbl 0974.33001

The Legendre functions ${P}_{n}^{m}\left(z\right)$ and ${Q}_{n}^{m}\left(z\right)$ (of the first and second kind, respectively) are solutions of the Legendre equation, which is an explicit linear second order differential equation. If $k$ and $m$ are positive integers, then these functions are called (solid) spherical functions. They can be written as Gauss hypergeometric functions and appear in solving the wave equation in spherical coordinates. The generalized associated Legendre functions, which are studied in this book, were introduced by Kuipers and Meulenbeld in 1957 and have an extra parameter. These functions ${P}_{k}^{m,n}\left(z\right)$ and ${Q}_{k}^{m,n}\left(z\right)$ satisfy the differential equation

which for $n=m$ reduces to the Legendre equation. In 25 chapters, the authors give an extensive theory of these functions, covering the classical topics which are of interest for most special functions, such as series representation (chapter 3), relations between different solutions (chapter 4), contiguous relations (chapter 5), and differential operators generated by this differential equation (chapter 6). Asymptotic formulas are worked out in the neighborhood of the singular points and as function of the parameters (chapters 7-8). Integral representations are given in chapters 10-11 and 13, and connections with other special functions (Jacobi functions, Bessel functions) are found in chapters 12 and 14. The generalized associated Legendre functions can also be used to generate integral transforms, as is shown in chapters 15-24. This particular ‘application’ covers approximately half of the book. Unfortunately, contrary to what the title of the book promises, no real applications are worked out. The preface gives some references indicating applications for some partial differential equations for the prolate ellipsoid of rotation, but this is not covered in the book. The book is mostly a collection (in a unified way) of information about these generalized associated Legendre functions, including some new results, and as such it will be useful to researchers interested in special functions and integral transforms.