Generalized Hypergeometric Functions Research Articles

General multi-sum transformations and some implications

We give two general transformations that allows certain quite general basic hypergeometric multi-sums of arbitrary depth (sums that involve an arbitrary sequence $$\{g(k)\}$$ ), to be reduced to an infinite q-product times a single basic hypergeometric sum. Various applications are given, including summation formulae for some q orthogonal polynomials and various multi-sums that are expressible as infinite products.

Improved Abramov-Petkovšek's Reduction and Creative Telescoping for Hypergeometric Terms

The Abramov-Petkovysek reduction computes an additive decomposition of a hypergeometric term, which extends the functionality of the Gosper algorithm for indefinite hypergeometric summation. We improve the Abramov- Petkovysek reduction so as to decompose a hypergeometric term as the sum of a summable term and a non-summable one. The improved reduction does not solve any auxiliary linear difference equation explicitly. It is also more efficient than the original reduction according to computational experiments. Based on this reduction, we design a new algorithm to compute minimal telescopers for bivariate hypergeometric terms. The new algorithm can avoid the costly computation of certificates.

TWO GENERAL HYPERGEOMETRIC TRANSFORMATION FORMULAS

A large number of summation and transformation formulas involving (generalized) hypergeometric functions have been developed by many authors. Here we aim at establishing two (presumably) new general hypergeometric transformations. The results are derived by manipulating the involved series in an elementary way with the aid of certain hypergeometric summation theorems obtained earlier by Rakha and Rathie. Relevant connections of certain special cases of our main results with several known identities are also pointed out.

Alternating knots, planar graphs, and $$q$$ q -series

Recent advances in Quantum Topology assign $$q$$ -series to knots in at least three different ways. The $$q$$ -series are given by generalized Nahm sums (i.e., special $$q$$ -hypergeometric sums) and have unknown modular and asymptotic properties. We give an efficient method to compute those $$q$$ -series that come from planar graphs (i.e., reduced Tait graphs of alternating links) and compute several terms of those series for all graphs with at most 8 edges drawing several conclusions. In addition, we give a graph-theory proof of a theorem of Dasbach-Lin which identifies the coefficient of $$q^k$$ in those series for $$k=0,1,2$$ in terms of polynomials on the number of vertices, edges, and triangles of the graph.

NEW SUBCLASS OF ANALYTIC FUNCTIONS ASSOCIATED WITH THE GENERALIZED HYPERGEOMETRIC FUNCTIONS Vol. 2(2) July 2014, No. 16, pp. 163-171. IBTISAM ALDAWISH AND MASLINA DARUS

NEW SUBCLASS OF ANALYTIC FUNCTIONS ASSOCIATED WITH THE GENERALIZED HYPERGEOMETRIC FUNCTIONS Vol. 2(2) July 2014, No. 16, pp. 163-171. IBTISAM ALDAWISH AND MASLINA DARUS

Applications of hypergeometric summation theorems of kummer and dixon involving double series

Using series iteration techniques, we derive a number of general double series identities and apply each of these identities in order to deduce several hypergeometric reduction formulas involving the Srivastava-Daoust double hypergeometric function. The results presented in this article are based essentially upon the hypergeometric summation theorems of Kummer and Dixon.

Some Theorems on Generalized Hypergeometric Function and Kampe De Feriet Function

Two theorems on the sums of generalized hypergeometric functions have been established. The theorems have further been employed to prove some theorems on the sums of Kampe de Feriet functions.

Application of E-Operator in Evaluating Certain Finite Integrals

In an attempt to unify some earlier results in the theory of the functions of hypergeometric type, the author establishes some new results involving the product of generalized hypergeometric function and H-function.

Some Contour Integrals Involving Generalised Hypergeometric Function

Contour integral involving Fox's H-function and modified Bessel function of the first kind has been calculated. Some important properties and particular cases of H-function, which is a generalization of G-function, have been derived and discussed.

Fourier Series for Fox's H-Function

Two integrals involving Fox's H-function have been evaluated and used to establish two Fourier series for the H-function. On specialising the parameters, the H-function can be reduced to Meijer G-function, Mac Robert's E-function, generalised hypergeometric functions and many other higher transcedental functions1. The results established are of a general character.

General Schlesinger Systems and Their Symmetry from the View Point of Twistor theory

Isomonodromic deformation of linear differential equations on ℙ1 with regular and irregular singular points is considered from the view point of twistor theory. We give explicit form of isomonodromic deformation using the maximal abelian subgroup H of G = GLN+1(ℂ) which appeared in the theory of general hypergeometric functions on a Grassmannian manifold. This formulation enables us to obtain a group of symmetry for the nonlinear system which is an Weyl group analogue NG (H)/H.

Two hypergeometric summation theorems and Ramanujan-type series

By Watson's and Whipple's theorems, we establish two general summation formulas, which are used to produce many Ramanujan-type series for π, 1/π and π2 with some free parameters.

Subclass of Multivalent Harmonic Functions Defi ned by Wright Generalized Hypergeometric Functions

We introduce a new class of multivalent harmonic functions defi ned by Wright generalized hypergeometric function. Coefficient estimates, extreme points, distortion bounds, and convex combination for functions belonging to this class are obtained.

HYPERDIRE, HYPERgeometric functions DIfferential REduction: MATHEMATICA-based packages for differential reduction of generalized hypergeometric functions p F p − 1 , F 1 , F 2 , F 3 , F 4

HYPERDIRE, HYPERgeometric functions DIfferential REduction: MATHEMATICA-based packages for differential reduction of generalized hypergeometric functions p F p − 1 , F 1 , F 2 , F 3 , F 4

ON A HYPERGEOMETRIC SUMMATION THEOREM DUE TO QURESHI ET AL.

We first aim at proving an interesting easily derivable summation formula. Then it is easily seen that this formula immediately yields a hypergeometric summation theorem recently added to the literature by Qureshi et al. Moreover we apply the main formulas to present some interesting summation formulas, whose special cases are also seen to yield the earlier known results.

Q-hypergeometric double sums as mock theta functions

Recently, Bringmann and Kane established two new Bailey pairs and used them to relate certain q-hypergeometric series to real quadratic fields. We show how these pairs give rise to new mock theta functions in the form of q-hypergeometric double sums. Additionally, we prove an identity between one of these sums and two classical mock theta functions introduced by Gordon and McIntosh.

Hypergeometric summation representations of the Stieltjes constants

The Stieltjes constants $\gamma_k$ appear in the regular part of the Laurent expansion of the Riemman and Hurwitz zeta functions. We demonstrate that these coefficients may be written as certain summations over mathematical constants and specialized hypergeometric functions $_pF_{p+1}$. This family of results generalizes a representation of the Euler constant in terms of a summation over values of the trigonometric integrals Si or Ci. The series representations are suitable for acceleration. As byproducts, we evaluate certain sine-logarithm integrals and present the leading asymptotic form of the particular $_pF_{p+1}$ functions.

A method for deriving hypergeometric and related identities from the H 2 Hardy norm of conformal maps

We explore a method which is implicit in a paper of Burkholder of identifying the H 2 Hardy norm of a conformal map with the explicit solution of Dirichlet's problem in the complex plane. Using the series form of the Hardy norm, we obtain an identity for the sum of a series obtained from the conformal map. We use this technique to evaluate several hypergeometric sums, as well as several sums that can be expressed as convolutions of the terms in a hypergeometric series. The most easily stated of the identities we obtain are Euler's famous Basel sum, as well as the sum We will be able to obtain the following hypergeometric reduction: A related identity is We will obtain two families of identities depending on a parameter, representative examples of which are and where C(k) is the kth Catalan number. We will also sum two series whose terms are defined by certain recurrence relations, and discuss an extension of the method to maps which are not conformal.

MORE SPECIAL FUNCTIONS TRAPPED

We extend the technique of using the trapezoidal rule for efficient evaluation of the special functions of mathematical physics given by integral representations. This technique was recently used for Bessel functions, and here we treat incomplete gamma functions and the general confluent hypergeometric function.

Diophantine approximation with partial sums of power series

For each nonzero rational number r, in [1], we considered the problem of approximating G(r) with partial sums of the series (1.1). In the case that an ≡ 1 and s = 1, we asked how well one can approximate e by the partial sums ∑n `=0 1 `! . J. Sondow [6] conjectured that exactly two of these partial sums are also convergents to the continued fraction of e. Among several results, Sondow and K. Schalm [7], proved that for almost all positive integers n, the partial sum ∑n `=0 1 `! is not a convergent to the continued fraction of e. Thus, the probability of obtaining a convergent to the continued fraction of e upon randomly choosing one of the first n partial sums of the power series of e tends to zero as n → ∞. Knowledge of the continued fraction of e, where a is a nonzero integer, and the best possible diophantine approximation of e, discovered by S. Ramanujan [5] and rediscovered by C. S. Davis [4], enabled the authors to prove in [2] that at most Oa(logM) of the first M convergents to the continued fraction of e are also partial sums of the corresponding power series. In [1], we considered general hypergeometric functions pFp(a1, . . . , ap; b1, . . . , bp; r) and showed that among their first N partial sums, no more than O(logN) are convergents to the continued fraction of pFp(a1, . . . , ap; b1, . . . , bp; r). Observe that this result includes e as a special case and that this particular corollary is a dual of the aforementioned result established in [2]. Moreover, when an is a real Dirichlet character or when {an}, n > 0, are the coefficients of an L-series attached to an elliptic curve without complex multiplication, we proved similar theorems in [1]. Lastly, we remark that in [2], we, in fact, proved Sondow’s conjecture. At the focal point of our study in [1] and in this paper are the following two definitions. For any rational number μ = a/b, with (a, b) = 1, consider the height H(μ) of μ, given by H(μ) = max{|a|, |b|}. For any real number α, and any positive real number δ, denote