We use a method based on the division algorithm to determine all the values of the real parameters b and c for which the hypergeometric polynomials 2F1(−n, b; c; z) have n real, simple zeros. Furthermore, we use the quasi-orthogonality of Jacobi polynomials to determine the intervals on the real line where the zeros are located.

and Dn denotes the derivative operator ∂/∂x1, . . . ,∂xn. The operators in (1.1) provide multidimensional generalizations to the well-known one-dimensional Riemann-Liouville andWeyl fractional integral operators defined in [5] (see also [1]). The paper [7] considers several formulas and interesting properties of (1.1). By invoking the Gauss hypergeometric function 2F1(α,β;γ;x), the following ge...

We obtain inverse factorial-series solutions of second-order linear difference equations with a singularity of rank one at infinity. It is shown that the Borel plane of these series is relatively simple, and that in certain cases the asymptotic expansions incorporate simple resurgence properties. Two examples are included. The second example is the large a asymptotics of the hypergeometric func...

The Gauss hypergeometric function 2F1(a, b, c; z) can be computed by using the power series in powers of z, z/(z − 1), 1 − z, 1/z, 1/(1 − z), (z − 1)/z. With these expansions 2F1(a, b, c; z) is not completely computable for all complex values of z. As pointed out in Gil, et al. [2007, §2.3], the points z = e±iπ/3 are always excluded from the domains of convergence of these expansions. Bühring [...

It is shown that Ramanujan’s cubic transformation of the Gauss hypergeometric function 2F1 arises from a relation between modular curves, namely the covering of X0(3) by X0(9). In general, when 2 N 7, the N-fold cover of X0(N) by X0(N) gives rise to an algebraic hypergeometric transformation. The N = 2, 3, 4 transformations are arithmetic–geometric mean iterations, but the N = 5, 6, 7 transform...

We present an efficient implementation of hypergeometric functions in arbitraryprecision interval arithmetic. The functions 0F1, 1F1, 2F1 and 2F0 (or the Kummer U -function) are supported for unrestricted complex parameters and argument, and by extension, we cover exponential and trigonometric integrals, error functions, Fresnel integrals, incomplete gamma and beta functions, Bessel functions, ...

Let A be the class of analytic functions in the open unit disk U . For complex numbers a, b and c (c 6= 0,−1,−2, ....), let I c be the operator defined on A by (I c (f))(z) = z2F1(a, b; c; z) ∗ f(z) where 2F1(a, b; c; z) is the Gaussian hypergeometric function. The function f in A is said to be in the class k − SP c if I a,b c (f) is a k-parabolic starlike function. For this class the Fekete-Sz...

in terms of power means and other related means have precipitated the search for similar bounds for the more general 2F1(α, β; γ; r). In an early paper, B. C. Carlson considered the approximation of the hypergeometric mean values ( 2F1(−a, b; b + c; r)) in terms of means of order t, given by Mt(s, r) := {(1 − s) + s(1 − r)t}1/t. In this note, a refinement of one such approximation is establishe...

ess as: 2), http: athemat s.2012.0 Abstract Two Gauss functions are said to be contiguous if they are alike except for one pair of parameters, and these differ by unity. Contiguous relations are of great use in extending numerical tables of the function. In this paper we will introduce a new method for computing such types of relations. a 2012 Egyptian Mathematical Society. Production and hosti...

In investigating the properties of a certain class of homogeneous polynomials, we discovered an identity satisfied by their coefficients which involves simple 2F1 Gauss hypergeometric functions. This result appears to be new and we supply a direct proof. The simplicity of the identity is suggestive of a deeper result.