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

TL;DR

This paper develops multi-integral and multi-derivative representations for Jacobi functions of the first and second kind, extending their classical definitions to complex indices and analyzing their properties via hypergeometric functions.

Contribution

It introduces new multi-integral and multi-derivative formulas for Jacobi functions with complex indices, generalizing classical polynomial cases.

Findings

Derived multi-integral representations for Jacobi functions.

Established multi-derivative formulas using hypergeometric functions.

Extended Jacobi functions to complex indices with analytical continuations.

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.