Some Theorems on Generalized Basic Hypergeometric Series

In the present paper, we obtain numerical values for Gaussianhypergeometric summation theorems by giving particular values to theparameters $a,~b$ and the argument $x$; three summation theorems for${}_{2}F_{3}(\frac{1}{4},\frac{3}{4};\frac{1}{2},\frac{1}{2},1;x)$,three summation theorems for${}_{4}F_{3}(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{a+b}{b};1,1,\frac{a}{b};x)$,two summation theorems for${}_{4}F_{3}(\frac{1}{2},\frac{1}{3},\frac{2}{3},\frac{a+b}{b};1,1,\frac{a}{b};x)$,four summation theorems for${}_{4}F_{3}(\frac{1}{2},\frac{1}{6},\frac{5}{6},\frac{a+b}{b};1,1,\frac{a}{b};x)$and ten summation theorems for${}_{4}F_{3}(\frac{1}{2},\frac{1}{4},\frac{3}{4},\frac{a+b}{b};1,1,\frac{a}{b};x)$.

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.

A multivariable hypergeometric function considered recently by Niukkanen and Srivastava (1984, 1985) provides an interesting unification of the generalised hypergeometric function pFq of one variable, Appell and Kampe de Feriet functions of two variables, and Lauricella functions of n variables, and also of many other hypergeometric series which arise naturally in various physical and quantum chemical applications. As pointed out by Srivastava, the multivariable hypergeometric function is an obvious special case of the generalised Lauricella function of n variables, which was first introduced and studied by Srivastava and Daoust (1969). By employing such connections of this multivariable hypergeometric function with the more general multiple hypergeometric functions several interesting and useful properties of this function have been studied, many of which have not been given by Niukkanen. The author derives a number of new reduction formulae for the multivariable hypergeometric function from substantially more general identities involving multiple series with essentially arbitrary terms. Some interesting summation formulae for the multivariable hypergeometric function with x1= . . . =xn=1 and x1= . . . =xn=-1 are also presented.

In the present paper we investigate commutativity of rings with unity satisfying any one of the properties (Error rendering LaTeX formula) , (Error rendering LaTeX formula) , and for some in where are non-negative integers. Finally, under different appopriate constraints on commutators, commutativity of R has been established.

In this note, two interesting integrals involving the product of two generalized hypergeometric functions have been evaluated in terms of gamma function. The results are derived with the help of a known integral involving hypergeometric function recorded in the paper of Lavoie and Trottlier. We also give several interesting special cases of our main findings.

The article is devoted to the problem of the real numbers' decomposition to a linear combination of natural numbers' squares. First of all the theorem giving indirectly the conditions of existing natural numbers' two equal squares is formulated and proved. Basing on this theorem two algorithms designed for finding couples of integer numbers generating eąual sums of sąuares and determining the multiplicity of that couples are presented. Finally the calculation of the algorithms' classes and estimation of the calculation errors is made. Also an example output of presented algorithms was shown.

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

We present some findings for second order differential subordination in the open unit disk involving generalized hypergeometric function using the convolution operator.

Let X be a non-empty abstract set and S be a commutative semi-group of operators defined on X into itself. S is called a contractive semi-group on X if there exists a metric ρ on X such that for each TeS, T≠I, ρ(Tx,Ty)≤λ(T)ρ(x,y) for all x, yeX, where 0≤λ(T) 1 , T 2 ..., T n , possessing a common unique fixed point in X, these conditions are automatically satisfied. The resulting statement is the following generalization of the converse of contraction mapping principle: Theorem C . Let X be an abstract set with n mutually commuting mappings T 1 , T 2 ..., T n defined on X into itself such that each iteration T 1 k 1 , T 2 ..., T n k n ( where k 1 , k 2 , ..., k n are non-negative integers not all equal to zero) possesses a unique fixed point which is common to every choice of k 1 , k 2 , ..., k n . Then for each λe(0,1), there exists a complete metric ρ on X such that ρ(T i x,T i y)≤λρ(x,y) for 1≤i≤n, and for all x,yeX. This result reduces to that of C. Bessaga by taking n = 1. ( Rf: C. Bessaga, Colloquim Mathematicum VII (1959), 41-43.).

The object of this paper is to evaluate an infinite integral involving the product of Meijer's G-function (5) and Kampé de Fériet function (1) in terms of Kampé de Fériet function. A number of papers of Bailey (3,4), Ragab (7,8), Slater (9), and Srivastava (10) have appeared, evaluating an integral in terms of a hypergeometric function of two variables or in terms of an E-function. Their results are obviously the particular cases of my result. Since Meijer's G-function is the most general function of one variable which can be expressed in terms of special functions (5) and Kampé de Fériet's function being the most general hypergeometric function of two variables, the integral given by me is the most general integral ever obtained and generalizes most of the results obtained so far for the integral of Mellin type in terms of generalized hypergeometric series. This is because the Kampé de Fériet function reduces to the product of two generalized hypergeometric functions by choosing parameters suitably.

This paper continues the study of recursion formulas of multivariable hypergeometric functions. In [Recursion formulas for multivariable hypergeometric functions, Asian-Eur. J. Math. 8(4) (2015) Article ID: 1550082, 50 pp.], the authors have given the recursion formulas for three-variable Lauricella functions, three Srivastava’s triple hypergeometric functions and four [Formula: see text]-variable Lauricella functions. We present here the recursion formulas for the general triple hypergeometric function.

The multivariable hypergeometric function $$F_{q_0 :q_1 ;...;q_n }^{P_0 :P_1 ;...;P_n } \left( {\begin{array}{*{20}c} {x_1 } \\ \vdots \\ {x_n } \\ \end{array} } \right),$$ considered recently by A. W. Niukkanen and H.M. Srivastava, is known to provide an interesting unification of the generalized hypergeometric functionp F q of one variable, Appell and Kampe de Feriet functions of two variables, and Lauricella functions ofn variables, as also of many other hypergeometric series which arise naturally in various physical, astrophysical, and quantum chemical applications. Indeed, as already pointed out by Srivastava, this multivariable hypergeometric function is an obvious special case of the generalized Lauricella function ofn variables, which was first introduced and studied by Srivastava and M. C. Daoust. By employing such fruitful connections of this multivariable hypergeometric function with much more general multiple hypergeometric functions studied in the literature rather systematically and widely, Srivastava presented several interesting and useful properties of this function, most of which did not appear in the work of Niukkanen. The object of this sequel to Srivastava's work is to derive a further reduction formula for the multivariable hypergeometric function from substantially more general identities involving multiple series with essentially arbitrary terms. Some interesting connections of the results considered here with those given in the literature, and some indication of their applicability, are also provided.

Gathering a group of mobile agents is a fundamental task in the field of distributed and mobile systems. This can be made drastically more difficult to achieve when some agents are subject to faults, especially the Byzantine ones that are known as being the worst faults to handle. In this paper we study, from a deterministic point of view, the task of Byzantine gathering in a network modeled as a graph. In other words, despite the presence of Byzantine agents, all the other (good) agents, starting from {possibly} different nodes and applying the same deterministic algorithm, have to meet at the same node in finite time and stop moving. An adversary chooses the initial nodes of the agents (the number of agents may be larger than the number of nodes) and assigns a different positive integer (called label) to each of them. Initially, each agent knows its label. The agents move in synchronous rounds and can communicate with each other only when located at the same node. Within the team, f of the agents are Byzantine. A Byzantine agent acts in an unpredictable and arbitrary way. For example, it can choose an arbitrary port when it moves, can convey arbitrary information to other agents and can change its label in every round, in particular by forging the label of another agent or by creating a completely new one. Besides its label, which corresponds to a local knowledge, an agent is assigned some global knowledge denoted by GK that is common to all agents. In literature, the Byzantine gathering problem has been analyzed in arbitrary n-node graphs by considering the scenario when GK=(n,f) and the scenario when GK=f. In the first (resp. second) scenario, it has been shown that the minimum number of good agents guaranteeing deterministic gathering of all of them is f+1 (resp. f+2). However, for both these scenarios, all the existing deterministic algorithms, whether or not they are optimal in terms of required number of good agents, have the major disadvantage of having a time complexity that is exponential in n and L, where L is the value of the largest label belonging to a good agent. In this paper, we seek to design a deterministic solution for Byzantine gathering that makes a concession on the proportion of Byzantine agents within the team, but that offers a significantly lower complexity. We also seek to use a global knowledge whose the length of the binary representation (that we also call size) is small. In this respect, assuming that the agents are in a strong team i.e., a team in which the number of good agents is at least some prescribed value that is quadratic in f, we give positive and negative results. On the positive side, we show an algorithm that solves Byzantine gathering with all strong teams in all graphs of size at most n, for any integers n and f, in a time polynomial in n and the length |l_{min}| of the binary representation of the smallest label of a good agent. The algorithm works using a global knowledge of size O(log log log n), which is of optimal order of magnitude in our context to reach a time complexity that is polynomial in n and |l_{min}|. Indeed, on the negative side, we show that there is no deterministic algorithm solving Byzantine gathering with all strong teams, in all graphs of size at most n, in a time polynomial in n and |l_{min}| and using a global knowledge of size o(log log log n).

In this study, we analytically investigate hypergeometric functions having some properties such as convexity and starlike. We fundamentally focus on obtaining desired conditions on the parameters \(a,b\), and $c$ in order that a hypergeometric function to be in various subclasses of starlike and convex functions of order \(\alpha=2^{-r}\) and order \(\alpha=2^{-r}\) type $\beta=2^{-1}$, with $r$ is a positive integer.

The aim of this paper is to establish certain integrals involving product of the Aleph function with Srivastava’s polynomials and Fox-Wright’s Generalized Hypergeometric function. Being unified and general in nature, these integrals yield a number of known and new results as special cases. For the sake of illustration, four corollaries are also recorded here as special case of our main results.