Class Of Multivariable Polynomials Research Articles

Optimizing Hypergraph-Based Polynomials Modeling Job-Occupancy in Queuing with Redundancy Scheduling

We investigate two classes of multivariate polynomials with variables indexed by the edges of a uniform hypergraph and coefficients depending on certain patterns of unions of edges. These polynomials arise naturally to model job-occupancy in some queuing problems with redundancy scheduling policies. The question, posed by Cardinaels, Borst, and van Leeuwaarden in [Redundancy Scheduling with Locally Stable Compatibility Graphs, arXiv preprint, 2020], is to decide whether their global minimum over the standard simplex is attained at the uniform probability distribution. By exploiting symmetry properties of these polynomials we can give a positive answer for the first class and partial results for the second one, where we in fact show a stronger convexity property of these polynomials over the simplex.

Modified Saigo fractional integral operators involving multivariable H-function and general class of multivariable polynomials

In this paper, we establish modified Saigo fractional integral operators involving the product of a general class of multivariable polynomials and the multivariable H-function. The results established here are of general nature and provide extension of some results obtained recently by Saxena et al.

On a General Class of Multiple Eulerian Integrals with Multivariable Gimel-Function

Recently, Raina and Srivastava [11] and Srivastava and Hussain [16] have provided closed-form expressions for a number of a Eulerian integral about the multivariable H-functions. Motivated by these recent works, we aim at evaluating a general multiple Eulerian integrals involving the product of multivariable I-function defined by Prathima et al. [10], a class of multivariable polynomials, a generalization of the Mittag-Leffler functions and multivariable Gimel-function.

Fractional Integration of the Product of two Multivariable Gimel-Functions and a General Class of Polynomials

A significantly large number of earlier works on the subject of fractional calculus give the interesting account of the theory and applications of fractional calculus operators in many different areas of mathematical analysis (such as ordinary and partial differential equations, integral equations, special functions, the summation of series, etc.). The object of the present paper is to study and develop the Saigo-Maeda operators. First, we establish four results that give the images of the product of two multivariable Gimel-functions and a general class of multivariable polynomials in Saigo- Maeda operators. On account of the general nature of the Saigo-Maeda operators, multivariable Gimel-functions and a class multivariable polynomials a large number of new and known theorems involving Riemann-Liouville and Erdelyi- Kober fractional integral operators and several special functions.

English

English

English

English

English

English

Selberg Integral Involving the Product of Multivariable Special Functions

The Selberg integral was an integral first evaluated by Selberg in 1944. The aim of the present paper is to estimate generalized Selberg integral. It involves the product of the general class of multivariable polynomials, multivariable I-function and modified multivariable H-function. The result is believed to be new and is capable of giving a large number of integrals involving a variety of functions and polynomials as its cases. We shall see several corollaries and particular cases at the end.

An Unified Study of Some Multiple Integrals

The object of this paper is to derive three unified fractional derivatives formulae for the Saigo-Maeda operators of fractional integration. The first formula deals with the product of a general class of multivariable polynomials and the multivariable Aleph- function. The second concerns the multivariable polynomials and two multivariable Aleph-functions with the help of the Leibniz rule for fractional derivatives. The last relation also implies the product of a class of multivariable polynomials and the multivariable Aleph-function but it is obtained by the application of the first formula twice and it implicates two independents variables instead of one. The polynomials and the functions have their arguments of the type are quite general nature. These formulae, besides being on very general character have been put in a compact form avoiding the occurrence of infinite series and thus making them put in applications. Our findings provide unifications and extensions of some (known and new) results. We shall give several corollaries and particular cases.

Unified Fractional Derivative Formulae for the Multivariable Aleph-Function

In this paper, we first evaluate unified finite multiple integrals whose integrand involves the product of the generalized hypergeometric function, general class of multivariable polynomials , the series expansion of multivariable A-function, a sequence of functions and the multivariable I-function. The arguments occurring in the integrand involve the product of factors of the form while that of , occurring herein involves a finite series of such coefficients. On account of the most general nature of the functions happening in the integrand of our integral, a large number of new and known integrals can be obtained from it merely by specializing the functions and parameters involved here. At the end, we shall see two corollaries.

On a General Class of Multiple Eulerian Integrals with Multivariable I-Functions

Recently, Raina and Srivastava and Srivastava and Hussainhave provided closed-form expressions for a number of a Eulerian integral involving multivariable H-functions. Motivated by these recent works, we aim at evaluating a general class of multiple Eulerian integrals concerning the product of two multivariable I-functions defined by Prathima et al. [6], a class of multivariable polynomials and the spheroidal function. These integrals will serve as a capital formula from which one can deduce numerous integrals.

English

English

On a General Class of Multiple Eulerian Integrals with Multivariable A-Functions

Recently, Raina and Srivastava and Srivastava and Hussain have provided closed-form expressions for a number of an Eulerian integral involving multivariable H-functions. Motivated by these recent works, we aim at evaluating a general class of multiple Eulerian integrals concerning the product of two multivariable A-functions defined by Gautam et Asgar [4], a class of multivariable polynomials and the extension of the Hurwitz-Lerch Zeta function. These integrals will serve as a fundamental formula from which one can deduce numerous useful integrals.

On a General Class of Multiple Eulerian Integrals with Multivariable Aleph-Functions

Recently, Raina and Srivastava [5] and Srivastava and Hussain [12] have provided closed-form expressions for a number of a Eulerian integral involving multivariable H-functions. Motivated by these recent works, we aim at evaluating a general class of multiple Eulerians integral involving the product of two multivariable Aleph-functions, a class of multivariable polynomials and the general sequence of functions. These integrals will serve as a capital formula from which one can deduce numerous integrals.

English

English

English

English

Marichev-Saigo Integral Operators Involving the Product of K-Function and Multivariable Polynomials

The aim of this paper to establish the Marichev-Saigo-Maeda fractional integration formula to the product of the K-function with the general class of multivariable polynomials. The results are presented in terms of the Wright generalized hypergeometric function. Corresponding assertions in terms of Saigo, Erdelyi-Kober, Riemann-Liouville, and Weyl type of fractional integrals are also presented. Further, we point out also their relevance.

An algebraic geometry version of the Kakeya problem

We propose an algebraic geometry framework for the Kakeya problem. We conjecture that for any polynomials f,g∈Fq0[x,y] and any Fq/Fq0, the image of the map Fq3→Fq3 given by (s,x,y)↦(s,sx+f(x,y),sy+g(x,y)) has size at least q34−O(q5/2) and prove the special case when f=f(x),g=g(y). We also prove it in the case f=f(y),g=g(x) under the additional assumption f′(0)g′(0)≠0 when f,g are both affine polynomials. Our approach is based on a combination of Cauchy–Schwarz and Lang–Weil. The algebraic geometry inputs in the proof are various results concerning irreducibility of certain classes of multivariate polynomials.

Generalized fractional integral operators and the multivariable H-function

The main object of the present paper is to establish new fractional integral formulas (of Marichev-Saigo-Maeda type) involving the products of the multivariable H-functions and the first class of multivariable polynomials due to Srivastava and Garg. All the results derived here are of general character and can yield a number of (new and known) results in the theory of fractional calculus.

Certain unified fractional integrals and derivatives for a product of Aleph function and a general class of multivariable polynomials

Saigo and Maeda (Transform Methods and Special Functions, Varna, Bulgaria, pp. 386-400, 1996) introduced and investigated certain generalized fractional integral and derivative operators involving the Appell function . Here we aim at presenting four unified fractional integral and derivative formulas of Saigo and Maeda type, which are involved in a product of ℵ-function and a general class of multivariable polynomials. The main results, being of general nature, are shown to be some unification and extension of many known formulas given, for example, by Saigo and Maeda (Transform Methods and Special Functions, Varna, Bulgaria, pp. 386-400, 1996), Saxena et al. (Kuwait J. Sci. Eng. 35(1A):1-20, 2008), Srivastava and Garg (Rev. Roum. Phys. 32:685-692, 1987), Srivastava et al. (J. Math. Anal. Appl.193:373-389, 1995) and so on. Our main results are also shown to be further specialized to yield a large number of known and (presumably) new formulas involving, for instance, Saigo fractional calculus operators, several special functions such as H-function, I-function, Mittag-Leffler function, generalized Wright hypergeometric function, generalized Bessel-Maitland function. MSC:26A33, 33E20, 33C45, 33C60, 33C70.