2023/08/25 by Cohl, Howard S., Costas-Santos, Roberto S.
#33C05 #33C45 #42C05 #44A20 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · doi:10.48550/arxiv.2308.13652
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, ∞). Using properties of Gauss hypergeometric functions, we derive multi-derivative and multi-integral representations for the Jacobi functions of the first and second kind.