Theory Of Hypergeometric Series Research Articles

Four Families of Summation Formulas for 4F3(1) with Application

A collection of functions organized according to their indexing based on non-negative integers is grouped by the common factor of fixed integer N. This grouping results in a summation of N series, each consisting of functions partitioned according to this modulo N rule. Notably, when N is equal to two, the functions in the series are divided into two subseries: one containing even-indexed functions and the other containing odd-indexed functions. This partitioning technique is widely utilized in the mathematical literature and finds applications in various contexts, such as in the theory of hypergeometric series. In this paper, we employ this partitioning technique to establish four distinct families of summation formulas for F34(1) hypergeometric series. Subsequently, we leverage these summation formulas to introduce eight categories of integral formulas. These integrals feature compositions of Beta function-type integrands and F23(x) hypergeometric functions. Additionally, we highlight that our primary summation formulas can be used to derive some well-known summation results.

On Energy Laws and Stability of Runge--Kutta Methods for Linear Seminegative Problems

This paper presents a systematic theoretical framework to derive the energy identities of general implicit and explicit Runge--Kutta (RK) methods for linear seminegative systems. It generalizes the stability analysis of explicit RK methods in [Z. Sun and C.-W. Shu, SIAM J. Numer. Anal., 57 (2019), pp. 1158-1182]. The established energy identities provide a precise characterization on whether and how the energy dissipates in the RK discretization, thereby leading to weak and strong stability criteria of RK methods. Furthermore, we discover a unified energy identity for all the diagonal Pade approximations, based on an analytical Cholesky type decomposition of a class of symmetric matrices. The structure of the matrices is very complicated, rendering the discovery of the unified energy identity and the proof of the decomposition highly challenging. Our proofs involve the construction of technical combinatorial identities and novel techniques from the theory of hypergeometric series. Our framework is motivated by a discrete analogue of integration by parts technique and a series expansion of the continuous energy law. In some special cases, our analyses establish a close connection between the continuous and discrete energy laws, enhancing our understanding of their intrinsic mechanisms. Several specific examples of implicit methods are given to illustrate the discrete energy laws. A few numerical examples further confirm the theoretical properties.

New Subclass of Analytic Function Related with Generalized Conic Domain Associated with q − Differential Operator

The quantum (or q -) calculus is widely applied in various operators which include the q -difference ( q -derivative) operator, and this operator plays an important role in geometric function theory (GFT) as well as in the theory of hypergeometric series. In our present investigation, we introduce and study q -differential operator associated with q -Mittag–Leffler function which is an extension of the Salagean q -differential operator. By using this newly defined operator, we define a new subclass of analytic function and studied certain subclass of analytic function in generalized conic domain Ω k , q , γ . For this class, we investigate structural formula, coefficient estimates, sufficient condition, Fekete–Szegö problem, and also some subordination results.

On a class of analytic functions related to conic domains involving q-calculus

The theory of q-analysis has many applications in various subfields of mathematics and quantum physics. Research work in connection with function theory and q-theory together was first introduced by Ismail et al. [6]. Till now only non-significant interest in this area was shown although it deserves more attention. Exploiting this, we aim to introduce a new class of analytic functions that are closely related to the domains bounded by conic sections. The authors hope this article will motivate future researchers to work in the area of q-calculus which can find many applications in the theory of hypergeometric series and quantum theory.

Q-Wiener index of some compound trees

The Wiener index W is the sum of distances between all pairs of vertices of a connected graph. Recently, q-analogs of W were conceived, motivated by the theory of hypergeometric series. In this article formulas are obtained for computing the q-Wiener indices of some compound trees. These generalize expressions, earlier known to hold for W.

On the contiguous relations of hypergeometric series

Contiguous relations are a fundamental concept within the theory of hypergeometric series and orthogonal polynomials. Their study goes back to Gauss who gave a list of 15 “fundamental” relations for the F 1 2 hypergeometric series. In this paper we will prove some consequences of contiguous relations of F 1 2 .

New algorithms in the theory of hypergeometric series and Burchnall-Chaundy expansions

It is shown that the Burchnall-Chaundy expansions, which are of fundamental importance in the theory of Appell's functions, can easily be implemented and generalized by means of the operator factorization method, which provides a simple and universal base, both for a new theory of hypergeometric series and for the development of effective new algorithms for computer-aided symbolic transformations of these series. Five new generalized expansions are derived, including 44 Burchnall-Chaundy expansions, as well as many new expansions, some of which are related to the Horn series.

A new approach to the theory of hypergeometric series and special functions of mathematical physics

A new approach to the theory of hypergeometric series and special functions of mathematical physics

On the Evaluation of Certain Determinants

Using the properties of the generalized Hilbert matrix and familiar results from the theory of hypergeometric series, we evaluate the determinants of certain matrices whose general terms are known explicitly. In certain cases it is even possible to find the analytic expression for the general terms of the inverses.

On the evaluation of certain determinants

Using the properties of the generalized Hilbert matrix and familiar results from the theory of hypergeometric series, we evaluate the determinants of certain matrices whose general terms are known explicitly. In certain cases it is even possible to find the analytic expression for the general terms of the inverses.