We describe a strategy for rigorous arbitrary-precision evaluation of Legendre polynomials on the unit interval and its application in the generation of Gauss-Legendre quadrature rules. Our focus is on making the evaluation practical for a wide range of realistic parameters, corresponding to the requirements of numerical integration to an accuracy of about 100 to 100 000 bits. Our algorithm com...

The roots of the general equation of degree n satisfy an A-hypergeometric system of diierential equations in the sense of Gel'fand, Kapranov and Zelevinsky. We construct the n distinct A-hypergeometric series solutions for each of the 2 n?1 triangulations of the Newton segment. This works over any eld whose characteristic is relatively prime to the lengths of the segments in the triangulation. ...

In this paper, we study an exact solution of the asymmetric simple exclusion process on a periodic lattice of finite sites with two typical updates, i.e., random and parallel. Then, we find that the explicit formulas for the partition function and the average velocity are expressed by the Gauss hypergeometric function. In order to obtain these results, we effectively exploit the recursion formu...

Epstein-Hubbell 1963 elliptic-type integrals occur in radiation field problems. In this paper, we consider a generalization of the elliptic-type integrals introduced by Kalla and Tuan 1996 and developed by Salman 2006. We give some recurrence relations for this generalized special function. Some of these relations can be used for analytic continuation to different domains. Asymptotic relations ...

The main purpose of this paper is to compute all irreducible spherical functions on G = SU(3) of arbitrary type δ ∈ K̂, where K = S(U(2) × U(1)) ≃ U(2). This is accomplished by associating to a spherical function Φ on G a matrix valued function H on the complex projective plane P2(C) = G/K. It is well known that there is a fruitful connection between the hypergeometric function of Euler and Gaus...

Using the C?dariu–Radu method derived from Diaz–Margolis theorem, we study existence, uniqueness and Gauss hypergeometric stability of ?-Hilfer fractional differential equations defined on compact domains. Next, show main results for unbounded To illustrate result a system, present an example.

We investigate so-called “higher” Siegel theta lifts on Lorentzian lattices in the spirit of Bruinier–Ehlen–Yang and Bruinier–Schwagenscheidt. give a series representation lift terms Gauss hypergeometric functions, evaluate as constant term Fourier involving Rankin–Cohen bracket harmonic Maass forms functions. Using higher lifts, we obtain vector-valued analogue Mertens’ result stating that hol...