Properties of the Lebesgue function for Lagrange interpolation on equidistant nodes are investigated. It is proved that the Lebesgue function can be formulated both in terms of a hypergeometric function 2F1 and Jacobi polynomials. Moreover an integral expression of the Lebesgue function is also obtained. Finally, the asymptotic behavior of the Lebesgue constant is studied.

We prove the following theorems: 1) The Laurent expansions in ε of the Gauss hypergeometric functions 2F1(I1 + aε, I2 + bε; I3 + p q + cε; z), 2F1(I1 + p q + aε, I2 + p q + bε; I3+ p q + cε; z) and 2F1(I1+ p q +aε, I2+ bε; I3 + p q + cε; z), where I1, I2, I3, p, q are arbitrary integers, a, b, c are arbitrary numbers and ε is an infinitesimal parameter, are expressible in terms of multiple poly...

The fast computation of the Gauss hypergeometric function 2F1 with all its parameters complex is a difficult task. Although the 2F1 function verifies numerous analytical properties involving power series expansions whose implementation is apparently immediate, their use is thwarted by instabilities induced by cancellations between very large terms. Furthermore, small areas of the complex plane,...

The hypergeometric functions are one of the most important and special in mathematics. They generalization exponential functions. Particularly ordinary function 2F1(a, b; c; z) is represented by series a solution to second order differential equation. Similarly, Laplace transform form integral that converts linear equations algebraic equations. This paper aims study convergence some Moreover, r...

We present two algorithms for computing hypergeometric solutions of second order linear differential operators with rational function coefficients. Our first algorithm searches for solutions of the form exp( ∫ r dx) · 2F1(a1, a2; b1; f) (1) where r, f ∈ Q(x), and a1, a2, b1 ∈ Q. It uses modular reduction and Hensel lifting. Our second algorithm tries to find solutions in the form

To solve globally bounded order 3 linear differential equations with rational function coefficients, this thesis introduces a partial 3F2-solver (Section 3.2) and F1-solver (Chapter 4) where 3F2 is the hypergeometric function 3F2(a1, a2, a3; b1, b2 |x) and F1 is the Appell’s F1(a, b1, b2, c |x, y). To investigate the relations among order 3 multivariate hypergeometric functions, this thesis pre...

We calculate the precession of Keplerian orbits under the influence of arbitrary central-force perturbations. Our result is in the form of a one-dimensional integral that is straightforward to evaluate numerically. We demonstrate the effectiveness of our formula for the case of the Yukawa potential. We obtain analytic results for potentials of the form V (r) = αr and V (r) = α ln(r/λ) in terms ...

Gauss’s hypergeometric function gives a modular parameterization of period integrals of elliptic curves in Legendre normal form E(λ) : y = x(x− 1)(x− λ). We study a modular function which “measures” the variation of periods for the isomorphic curves E(λ) and E ( λ λ−1 ) , and we show that it padically “interpolates” the cusp form for the “congruent number” curve E(2), the case where these pairs...

The main object of this paper is to consider the asymptotic distribution of the zeros of certain classes of the Gauss hypergeometric polynomials. Some classical analytic methods and techniques are used here to analyze the behavior of the zeros of the Gauss hypergeometric polynomials, 2F1(−n, a;−n+ b; z), where n is a nonnegative integer. Owing to the connection between the classical Jacobi poly...