A unified elementary approach to the Dyson, Morris, Aomoto, and Forrester constant term identities

We introduce an elementary method to give unified proofs of the Dyson, Morris, and Aomoto identities for constant terms of Laurent polynomials. These identities can be expressed as equalities of polynomials and thus can be proved by verifying them for sufficiently many values, usually at negative integers where they vanish. Our method also proves some special cases of the Forrester conjecture.     

A new approach to constant term identities and Selberg-type integrals

Selberg-type integrals that can be turned into constant term identities for Laurent polynomials arise naturally in conjunction with random matrix models in statistical mechanics. Built on a recent idea of Karasev and Petrov we develop a general interpolation based method that is powerful enough to establish many such identities in a simple manner. The main consequence is the proof of a conjecture of Forrester related to the Calogero–Sutherland model. In fact we prove a more general theorem, which includes Aomoto's constant term identity at the same time. We also demonstrate the relevance of the method in additive combinatorics.     

The determinant expressions of some theta constant identities

In this paper, we derive two interesting theta constant identities by considering determinant structures. One of the two identities which we obtained is given by the Pfaffian, and the other is expressed by the resultant.     

Compact gradient shrinking Ricci solitons with positive curvature operator

In this article, we first derive several identities on a compact shrinking Ricci soliton. We then show that a compact gradient shrinking soliton must be Einstein, if it admits a Riemannian metric with positive curvature operator and satisfies an integral inequality. Furthermore, such a soliton must be of constant curvature.     

Radical theory in varieties of near-rings in which the constants form

Not all the good properties of the Kurosh-Amitsur radical theory in the variety of associative rings are preserved in the bigger variety of near-rings. In the smaller and better behaved variety of O-symmetric near-rings the theory is much more satisfactory. In this note we show that many of the results of the 0-symmetric near-ring case can be extended to a much bigger variety of near-rings which, amongst others, includes all the O-symmetric as well as the constant near-rings. The varieties we shall consider are varieties of near-rings, called Fuchs varieties, in which the constants form an ideal. The good arithmetic of such varieties makes it possible to derive more explicit conditions.

(i) for the subvariety of constant near-rings to be a semisimple class (or equivalently, to have attainable identities),

(ii) for semisimple classes to be hereditary.

We shall prove that the subvariety of 0-symmetric near-rings has attainable identities in a Fuchs variety, and extend the theory of overnilpotent radicals of 0-symmetric near-rings to the largest Fuchs variety F

The near-ring construction of [7] will play a decisive role in our investigations.     

Analogues of Jacobi’s derivative formula II

In this work, we use Jacobi’s derivative formula to obtain analogues to theta constants with rational characteristics. Furthermore, we show that the analogues yield many product-series identities by only using the elementary fact that a holomorphic elliptic function is a constant.     

Proofs for certain q-trigonometric identities of Gosper

Applying an addition formula of Liu (2007), we deduce certain Jacobi theta function identities. From these results we confirm several q-trigonometric identities conjectured by Gosper (2001). Another conjectured identity on the constant Π_q is also settled.

     

On Boundary Stability of Wave Equations with Variable Coefficients

In this paper, we consider the boundary stabilization of the wave equation with variable coefficients by Riemmannian geometry method subject to a different geometric condition which is motivated by the geometric multiplier identities. Several (multiplier) identities (inequalities) which have been built for constant wave equation by Kormornik and Zuazua are generalized to the variable coefficient case by some computational techniques in Riemmannian geometry, so that the precise estimates on the exponential decay rate are derived from those inequalitities. Also, the exponential decay for the solutions of semilinear wave equation with variable coefficients is obtained under natural growth and sign assumptions on the nonlinearity. Our method is rather general and can be adapted to other evolution systems with variable coefficients (e.g. elasticity plates) as well.     

Some identities involving Inverse Gamma integrals

The Ramanujan Journal - In this paper we will give some identities related with the Fransén–Robinson constant and the Inverse Gamma function. The main result is to use Riemann...     

A family of q-Dyson style constant term identities

By generalizing Gessel-Xin's Laurent series method for proving the Zeilberger-Bressoud q-Dyson Theorem, we establish a family of q-Dyson style constant term identities. These identities give explicit formulas for certain coefficients of the q-Dyson product, including three conjectures of Sills' as special cases and generalizing Stembridge's first layer formulas for characters of SL(n,C).     

Identities for trigonometric B-splines with an application to curve design

In this paper we investigate some properties of trigonometric B-splines. We establish a complex integral representation for these functions, which is in certain analogy to the polynomial case, but the proof of which has to be done in a different and more complicated way. Using this integral representation, we can prove some identities concerning the evaluation of a trigonometric B-spline, its derivative and its partial derivative w.r.t. the knots. Finally we show that—in the case of equidistant knots—the trigonometric B-splines of odd order form a partition of a constant, and therefore the corresponding B-spline curve possesses the convex-hull property. This is illustrated by a numerical example.     

q-Pell sequences and two identities of V.A. Lebesgue

We examine a pair of Rogers-Ramanujan type identities of Lebesgue, and give polynomial identities for which the original identities are limiting cases. The polynomial identities turn out to be q-analogs of the Pell sequence. Finally, we provide combinatorial interpretations for the identities.     

Character sum analogues of constant term identities for root systems

Certain identities connected with root systems provide explicit constant terms in Laurent series expansions of multivariable functions. Character sum analogues of these identities are given.     

On the refracted–reflected spectrally negative Lévy processes

We study a combination of the refracted and reflected Lévy processes. Given a spectrally negative Lévy process and two boundaries, it is reflected at the lower boundary while, whenever it is above the upper boundary, a linear drift at a constant rate is subtracted from the increments of the process. Using the scale functions, we compute the resolvent measure, the Laplace transform of the occupation times as well as other fluctuation identities that will be useful in applied probability including insurance, queues, and inventory management.     

ODE Methods in Non-Local Equations

Non-local equations cannot be treated using classical ODE theorems. Nevertheless, several new methods have been introduced in the non-local gluing scheme of our previous article; we survey and improve those, and present new applications as well. First, from the explicit symbol of the conformal fractional Laplacian, a variation of constants formula is obtained for fractional Hardy operators. We thus develop, in addition to a suitable extension in the spirit of Caffarelli–Silvestre, an equivalent formulation as an infinite system of second order constant coefficient ODEs. Classical ODE quantities like the Hamiltonian and Wrońskian may then be utilized. As applications, we obtain a Frobenius theorem and establish new Pohožaev identities. We also give a detailed proof for the non-degeneracy of the fast-decay singular solution of the fractional Lane–Emden equation.     

Minkowski identities for hypersurfaces in constant sectional curvature manifolds

We give a new proof of the generalized Minkowski identities relating the higher degree mean curvatures of orientable closed hypersurfaces immersed in a given constant sectional curvature manifold. Our methods rely on a fundamental differential system of Riemannian geometry introduced by the author. We develop the notion of position vector field, which lies at the core of the Minkowski identities.     

Identities for the Hankel transform and their applications

In the present paper the authors show that iterations of the Hankel transform with Kν-transform is a constant multiple of the Widder transform. Using these iteration identities, several Parseval-Goldstein type theorems for these transforms are given. By making use of these results a number of new Goldstein type exchange identities are obtained for these and the Laplace transform. The identities proven in this paper are shown to give rise to useful corollaries for evaluating infinite integrals of special functions. Some examples are also given as illustration of the results presented here.     

On a Pair of Identities from Ramanujan's Lost Notebook

Using a pair of two variable series-product identities recorded by Ramanujan in the lost notebook as inspiration, we find some new identities of similar type. Each identity immediately implies an infinite family of Rogers-Ramanujan type identities, some of which are well-known identities from the literature. We also use these identities to derive some general identities for integer partitions.     

On Half-Way AZ-Style Identities

The Ahlswede–Zhang identity is an elegant sharpening of the famous LYM-inequality. Recently, we have found a parametrised identity which implies the AZ identity and characterizes deficiencies of other inequalities in combinatorics. In this paper, we show identities of half-way extraction from AZ-style identities. These identities aim to characterize more clearly terms participating in AZ identities or LYM-style inequalities.