Multi-integral representations for Jacobi functions of the first and second kind

Abstract

One may consider the generalization of Jacobi polynomials and the Jacobi function of the second kind to a general function where the index is allowed to be a complex number instead of a non-negative integer. These functions are referred to as Jacobi functions. In a similar fashion as associated Legendre functions, these break into two categories, functions which are analytically continued from the real line segment \((-1,1)\) and those continued from the real ray \((1,+\infty)\).

Using properties of Gauss hypergeometric functions, we derive multi-derivative and multi-integral representations for the Jacobi functions of the first and second kind.

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@article{arXiv:2308.13652,
author={Cohl, H. S. and Roberto S. Costas-Santos},
title={Multi-integral representations for Jacobi functions of the first and second kind},
journal={Arab Journal of Basic and Applied Sciences},
volume={30},
number={1},
pages={583--592},
year={2023},
publisher={Taylor & Francis},
DOI={10.1080/25765299.2023.2268911},
URL={https://doi.org/10.1080/25765299.2023.2268911},
ZBL={arXiv:2308.13652},
}