Related papers: New Series Expansions of the Gauss Hypergeometric …

Each family of Gauss hypergeometric functions $$ f_n={}_2F_1(a+\epsilon_1n, b+\epsilon_2n ;c+\epsilon_3n; z), $$ for fixed $\epsilon_j=0,\pm1$ (not all $\epsilon_j$ equal to zero) satisfies a second order linear difference equation of the…

Functions that are smooth but non-periodic on a certain interval possess Fourier series that lack uniform convergence and suffer from the Gibbs phenomenon. However, they can be represented accurately by a Fourier series that is periodic on…

In the 1980's, Greene defined {\it hypergeometric functions over finite fields} using Jacobi sums. The framework of his theory establishes that these functions possess many properties that are analogous to those of the classical…

In this paper, we present and prove that the coefficients $u_n$ and $v_n$ in the series expansions $F^2(a,b;c;z) = \sum_{n=0}^\infty u_n z^n$ and $F^3(a,b;c;z) = \sum_{n=0}^\infty v_n z^n$ ($a,b,c,z \in \mathbb{C}$ and $-c \notin \mathbb{N}…

Series involving hypergeometric functions are used to derive, extend and evaluate integrals involving the product of two Bessel functions of the first kind $J_{u}(a z)$ $J_{v}(b z)$ with order $u,v$, studied by Landau et al. The method used…

Surprisingly, apart from some special cases, simple asymptotic expansions for the associated Legendre functions $P_\nu ^\mu (z)$ and $Q_\nu ^\mu (z)$ for large degree $\nu$ or large order $\mu$ are not available in the literature. The main…

Let $q$ be a prime power and $\mathbb{F}_q$ be a finite field with $q$ elements. Let $e$ and $d$ be positive integers. In this paper, for $d\geq2$ and $q\equiv1(\mathrm{mod}~ed(d-1))$, we calculate the number of points on an algebraic curve…

The generalized hypergeometric function $_qF_p$ is a power series in which the ratio of successive terms is a rational function of the summation index. The Gaussian hypergeometric functions $_2F_1$ and $_3F_2$ are most common special cases…

Sequence transformations accomplish an acceleration of convergence or a summation in the case of divergence by detecting and utilizing regularities of the elements of the sequence to be transformed. For sufficiently large indices, certain…

In this paper, we obtain various conditions on the parameters $a,\, b,\, c\,, d$ and $e$ for which the hypergeometric functions $z\,_3F_2(a,b,c;d,e;z)$ to be in the class of all close-to-convex function with respect to some well known…

The two most commonly used hypergeometric functions are the confluent hypergeometric function and the Gauss hypergeometric function. We review the available techniques for accurate, fast, and reliable computation of these two hypergeometric…

In the present paper, we study spirallikenss (including starlikeness) of the shifted hypergeometric function $f(z)=z_2F_1(a,b;c;z)$ with complex parameters $a,b,c,$ where $_2F_1(a,b;c;z)$ stands for the Gaussian hypergeometric function.…

In the past two decades, many researchers have studied {\it index $2$} Gauss sums, where the group generated by the characteristic $p$ of the underling finite field is of index $2$ in the unit group of ${\mathbb Z}/m{\mathbb Z}$ for the…

Lattice Green functions appear in lattice gauge theories, in lattice models of statistical physics and in random walks. Here, space coordinates are treated as parameters and series expansions in the mass are obtained. The singular points in…

Let p be an odd prime. In 1984, Greene introduced the notion of hypergeometric functions over finite fields. Special values of these functions have been of interest as they are related to the number of F_p points on algebraic varieties and…

The asymptotic expansion of the Kummer function ${}_1F_1(a; b; z)$ is examined as $z\to+\infty$ on the Stokes line $\arg\,z=0$. The correct form of the subdominant algebraic contribution is obtained for non-integer $a$. Numerical results…

We consider the ratio of two Gauss hypergeometric functions, in which the parameters of the numerator function differ from the respective parameters of the denominator function by integers. We derive explicit integral representations for…

We obtained the region of convergence and the summation formula for some modified generalized hypergeometric series (1.2). We also investigated rationality of the sums of the power series (1.3). As a result the series (1.4) cannot be the…

It is well-known that differentiation of hypergeometric function multiplied by a certain power function yields another hypergeometric function with a different set of parameters. Such differentiation identities for hypergeometric functions…

We present the Mathematica package HypExp which allows to expand hypergeometric functions $_JF_{J-1}$ around integer parameters to arbitrary order. At this, we apply two methods, the first one being based on an integral representation, the…