Generalizations of the Jacobi identity to the case of the Lauricella function FD(N)

ABSTRACT

The paper considers the issue of constructing differential relations for the Lauricella hypergeometric function $F_D^{\left(N\right)}$. The formulas found give explicit expressions for the derivative of the product of the function $F_D^{\left(N\right)}$ and some binomials through the product of other binomials and a combination of adjacent Lauricella functions whose parameters differ by 1 or $-1$. The derived differential relations generalize the Jacobi identity and other differential formulas known in the theory of the Gauss hypergeometric function on the multidimensional case. With special parameter values $F_D^{\left(N\right)}$, the right-hand side of such Jacobi-type formulas are simplified and have the form of a product of binomials and a polynomial. Such formulas can be used to transform a Cauchy-type integral to the form of the Schwartz–Christoffel integral and have an application to the Riemann–Hilbert problem with piecewise constant coefficients.

KEYWORDS:

• Multiple hypergeometric functions
• Lauricella function $F_D^{\left(N\right)}$
• Jacobi-type differential identities

AMS CLASSIFICATIONS:

• 33C65
• 32D15
• 33C90

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the Russian Science Foundation under grant No. 22-71-10094, https://rscf.ru/project/22-71-10094/.