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 degree 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 analytically 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.

Keywords:

• Generalized hypergeometric functions
• integral representations
• Jacobi functions
• Jacobi polynomials
• multi-integral representations
• Rodrigues-type relations

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 33C45
• 44A20
• 42C05
• 33C05

Disclosure statement

No potential conflict of interest was reported by the authors.