scholarly journals Clausen’s Series 3F2(1) with Integral Parameter Differences

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1783
Author(s):  
Kwang-Wu Chen
Keyword(s):  
Positive Integer ◽  
Closed Form ◽  
Open Problem ◽  
Group Symmetry ◽  
Complex Numbers ◽  
Integral Parameter ◽  
Arbitrary Complex

Ebisu and Iwassaki proved that there are three-term relations for 3F2(1) with a group symmetry of order 72. In this paper, we apply some specific three-term relations for 3F2(1) to partially answer the open problem raised by Miller and Paris in 2012. Given a known value 3F2((a,b,x),(c,x+1),1), if f−x is an integer, then we construct an algorithm to obtain 3F2((a,b,f),(c,f+n),1) in an explicit closed form, where n is a positive integer and a,b,c and f are arbitrary complex numbers. We also extend our results to evaluate some specific forms of p+1Fp(1), for any positive integer p≥2.

scholarly journals Definite Integral of Algebraic, Exponential and Hyperbolic Functions Expressed in Terms of Special Functions

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1425
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Closed Form ◽  
Special Function ◽  
Special Functions ◽  
Complex Numbers ◽  
Definite Integral ◽  
Hyperbolic Functions ◽  
Closed Form Solutions ◽  
Definite Integrals ◽  
Arbitrary Complex ◽  
Famous Book

While browsing through the famous book of Bierens de Haan, we came across a table with some very interesting integrals. These integrals also appeared in the book of Gradshteyn and Ryzhik. Derivation of these integrals are not listed in the current literature to best of our knowledge. The derivation of such integrals in the book of Gradshteyn and Ryzhik in terms of closed form solutions is pertinent. We evaluate several of these definite integrals of the form ∫0∞(a+y)k−(a−y)keby−1dy, ∫0∞(a+y)k−(a−y)keby+1dy, ∫0∞(a+y)k−(a−y)ksinh(by)dy and ∫0∞(a+y)k+(a−y)kcosh(by)dy in terms of a special function where k, a and b are arbitrary complex numbers.

scholarly journals Derivation of Logarithmic and Logarithmic Hyperbolic Tangent Integrals Expressed in Terms of Special Functions

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 687 ◽  
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Closed Form ◽  
Special Function ◽  
Special Functions ◽  
Complex Numbers ◽  
General Function ◽  
Hyperbolic Tangent ◽  
Closed Form Solutions ◽  
Definite Integrals ◽  
Arbitrary Complex

The derivation of integrals in the table of Gradshteyn and Ryzhik in terms of closed form solutions is always of interest. We evaluate several of these definite integrals of the form ∫ 0 ∞ log ( 1 ± e − α y ) R ( k , a , y ) d y in terms of a special function, where R ( k , a , y ) is a general function and k, a and α are arbitrary complex numbers, where R e ( α ) > 0 .

scholarly journals NORMALITY AND SHARED SETS

2009 ◽  
Vol 86 (3) ◽  
pp. 339-354 ◽  
Author(s):  
MINGLIANG FANG ◽  
LAWRENCE ZALCMAN
Keyword(s):  
Positive Integer ◽  
Meromorphic Functions ◽  
Complex Numbers ◽  
Finite Complex ◽  
Shared Sets ◽  
Zeros Of Functions

AbstractLet ℱ be a family of meromorphic functions defined in D, all of whose zeros have multiplicity at least k+1. Let a and b be distinct finite complex numbers, and let k be a positive integer. If, for each pair of functions f and g in ℱ, f(k) and g(k) share the set S={a,b}, then ℱ is normal in D. The condition that the zeros of functions in ℱ have multiplicity at least k+1 cannot be weakened.

scholarly journals Ternary SU(3)-group symmetry and its possible applications in hadron-quark substructure. Towards a new spinor-fermion structure

2019 ◽  
Vol 204 ◽  
pp. 02007
Author(s):  
Alexander Maslikov ◽  
Guennady Volkov
Keyword(s):  
Harmonic Functions ◽  
Complex Space ◽  
Group Symmetry ◽  
Complex Numbers ◽  
Spatial Properties ◽  
Laplace Equations ◽  
Quark Charge ◽  
Spinor Matter ◽  
Quark Substructure

The questions on the existence of the three color quark symmetry and three quark-lepton generations could have the origin associated with the new exotic symmetries outside the Cartan-Killing-Lie algebras/groups. Our long-term search for these symmetries has been began with our Calabi-Yau space classification on the basis of the n-ary algebra for the reflexive projective numbers and led us to the expansion of the binary n = 2 complex and hyper complex numbers in the framework of the n-ary complex and hyper-complex numbers with n = 3, 4, … where we constructed new Abelian and non-Abelian symmetries. We have studied then norm-division properties of the Abelian nary complex numbers and have built the infinite chain of the Abelian groups U(n–1) = [U(1) × … × U(1)](n–1). We have developed the n-ary holomorphic (polymorphic) analysis on the n-ary complex space NC{n}, which led us to the generalization of the quadratic Laplace equations for the harmonic functions. The generalized Laplace equations for the n-ary harmonic functions give us the n-th order homogeneous differential equations which are invariant with respect to the Abelian n-ary groups U(n–1) and with some new spatial properties. Further consideration of the non-Abelian n-ary hyper-complex numbers opens the infinite series of the non-Abelian TnSU(n)-Lie groups(n=3,4,…) and its corresponding tnsu(n) algebras. One of the exceptional features of these symmetry groups is the appearance of some new n-dimensional spinors that could lead to an extension of the concept of the SU(2)-spin, to the appearance of n-dimensional quantum structures -exotic “n-spinor” matter(n = 3, 4, … - maarcrions). It is natural to assume that these new exotic “quantum spinor states” could be candidates for the pra-matter of the quark-charge leptons or/and for the dark matter. We will be also interested in the detection of the exotic quantum ’n-spinor” matter in the neutrino and hadron experiments.

scholarly journals Rational Divide-and-Conquer Relations

2011 ◽  
Vol 2011 ◽  
pp. 1-14
Author(s):  
Charinthip Hengkrawit ◽  
Vichian Laohakosol ◽  
Watcharapon Pimsert
Keyword(s):  
Positive Integer ◽  
Closed Form ◽  
Rational Function ◽  
Natural Generalization ◽  
Divide And Conquer ◽  
Recursive Equation ◽  
Closed Form Solutions ◽  
Tangent Function

A rational divide-and-conquer relation, which is a natural generalization of the classical divide-and-conquer relation, is a recursive equation of the form f(bn)=R(f(n),f(n),…,f(b−1)n)+g(n), where b is a positive integer ≥2; R a rational function in b−1 variables and g a given function. Closed-form solutions of certain rational divide-and-conquer relations which can be used to characterize the trigonometric cotangent-tangent and the hyperbolic cotangent-tangent function solutions are derived and their global behaviors are investigated.

On methods of summability based on integral functions

1959 ◽  
Vol 55 (1) ◽  
pp. 23-30 ◽  
Author(s):  
D. Borwein
Keyword(s):  
Integral Function ◽  
Radius Of Convergence ◽  
Complex Numbers ◽  
Arbitrary Complex ◽  
Image Position

Suppose throughout thatand thatis an integral function. Suppose also that l, sn(n = 0,1,…) are arbitrary complex numbers and denote by ρ(ps) the radius of convergence of the series

scholarly journals Note on a Stieltjes Transform in terms of the Lerch Function

2021 ◽  
Vol 14 (3) ◽  
pp. 723-736
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Closed Form ◽  
Special Functions ◽  
Complex Numbers ◽  
Definite Integral ◽  
Fundamental Constants ◽  
Logarithmic Function ◽  
Stieltjes Transform ◽  
Closed Form Solutions ◽  
General Complex

In this work the authors derive the Stieltjes transform of the logarithmic function in terms of the Lerch function. This transform is used to derive closed form solutions involving fundamental constants and special functions. Specifically we derive the definite integral given by\[\int_{0}^{\infty} \frac{(1-b x)^m \log ^k(c (1-b x))+(b x+1)^m \log ^k(c (b x+1))}{a+x^2}dx\]where $a,b,c,m$ and $k$ are general complex numbers subject to the restrictions given in connection with the formulas.

scholarly journals Separating the k-party communication complexity hierarchy: an application of the Zarankiewicz problem

2011 ◽  
Vol Vol. 13 no. 4 ◽  
Author(s):  
Thomas P. Hayes
Keyword(s):  
Positive Integer ◽  
Open Problem ◽  
Communication Complexity ◽  
60Th Birthday ◽  
Equal Size ◽  
Special Issue ◽  
Algorithms And Complexity ◽  
International Audience ◽  
Zarankiewicz Problem ◽  
Complexity Hierarchy

special issue in honor of Laci Babai's 60th birthday: Combinatorics, Groups, Algorithms, and Complexity International audience For every positive integer k, we construct an explicit family of functions f : \0, 1\(n) -\textgreater \0, 1\ which has (k + 1) - party communication complexity O(k) under every partition of the input bits into k + 1 parts of equal size, and k-party communication complexity Omega (n/k(4)2(k)) under every partition of the input bits into k parts. This improves an earlier hierarchy theorem due to V. Grolmusz. Our construction relies on known explicit constructions for a famous open problem of K. Zarankiewicz, namely, to find the maximum number of edges in a graph on n vertices that does not contain K-s,K-t as a subgraph.

scholarly journals On some properties of determinants with complex elements and their practical application

2021 ◽  
pp. 67-77
Author(s):  
Mykhailo Fys ◽  
Roman Kvit ◽  
Tetyana Salo
Keyword(s):  
Closed Form ◽  
Imaginary Part ◽  
Complex Numbers ◽  
System Of Equations ◽  
Order Complex ◽  
Symmetric Polynomials ◽  
Practical Application ◽  
The Real ◽  
The Given ◽  
Selection Of

The formulas presented in this paper make it possible to select the real and imaginary part of the determinant value of the n -th order complex quantity, greatly simplifying the process of its deployment. Moreover, its module is given by the determinant of the 2n -th order, the elements of which are the real and imaginary parts of complex numbers. This makes it possible to analyze analytically the process described using determinants with complex numbers. The real and imaginary parts are also determined by the sum of determinants already with n rows and columns, the elements of which make up complex elements. The terms of this sum are solutions of a system of equations represented in closed form using symmetric polynomials, the arguments of which are its coefficients. Part of this combination is expressed by two determinants of the n -th order, the elements of which are the sum and difference of the real and imaginary parts of the elements. This significantly reduces the number of arithmetic operations during the deployment of a complex determinant and the selection of its real and imaginary parts. The given numerical example confirms the feasibility of this approach.

scholarly journals Periods of Ehrhart Coefficients of Rational Polytopes

10.37236/6059 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Tyrrell B. McAllister ◽  
Hélène O. Rochais
Keyword(s):  
Positive Integer ◽  
Open Problem ◽  
Polynomial Function ◽  
Integer Lattice ◽  
Lattice Points ◽  
Periodic Functions

Let $\mathcal{P} \subset \mathbb{R}^{n}$ be a polytope whose vertices have rational coordinates. By a seminal result of E. Ehrhart, the number of integer lattice points in the $k$th dilate of $\mathcal{P}$ ($k$ a positive integer) is a quasi-polynomial function of $k$ — that is, a "polynomial" in which the coefficients are themselves periodic functions of $k$. It is an open problem to determine which quasi-polynomials are the Ehrhart quasi-polynomials of rational polytopes. As partial progress on this problem, we construct families of polytopes in which the periods of the coefficient functions take on various prescribed values.

Sign in / Sign up

Export Citation Format

Share Document