A Hybrid Mean Value Involving Dedekind Sums and the Generalized Kloosterman Sums

In this paper, we use the mean value theorem of Dirichlet L -functions and the properties of Gauss sums and Dedekind sums to study the hybrid mean value problem involving Dedekind sums and the general Kloosterman sums and give an interesting identity for it.

Trapezium-Type Inequalities for an Extension of Riemann–Liouville Fractional Integrals Using Raina’s Special Function and Generalized Coordinate Convex Functions

In this paper, the authors analyse and study some recent publications about integral inequalities related to generalized convex functions of several variables and the use of extended fractional integrals. In particular, they establish a new Hermite–Hadamard inequality for generalized coordinate ϕ-convex functions via an extension of the Riemann–Liouville fractional integral. Furthermore, an interesting identity for functions with two variables is obtained, and with the use of it, some new extensions of trapezium-type inequalities using Raina’s special function via generalized coordinate ϕ-convex functions are developed. Various special cases have been studied. At the end, a brief conclusion is given as well.

Some New Fractional Trapezium-Type Inequalities for Preinvex Functions

In this paper, authors the present the discovery of an interesting identity regarding trapezium-type integral inequalities. By using the lemma as an auxiliary result, some new estimates with respect to trapezium-type integral inequalities via general fractional integrals are obtained. It is pointed out that some new special cases can be deduced from the main results. Some applications regarding special means for different real numbers are provided as well. The ideas and techniques described in this paper may stimulate further research.

Some Types of Identities Involving the Legendre Polynomials

In this paper, a new non-linear recursive sequence is firstly introduced. Then, using this sequence, a computational problem involving the convolution of the Legendre polynomial is studied using the basic and combinatorial methods. Finally, we give an interesting identity.

A New Identity Involving the Chebyshev Polynomials

In this paper, firstly, we introduced a second order non-linear recursive sequence, then we use this sequence and the combinatorial methods to perform a deep study on the computational problem concerning one kind sums, which includes the Chebyshev polynomials. This makes it possible to simplify a class of complex computations involving the second type Chebyshev polynomials into a very simple problem. Finally, we give a new and interesting identity for it.

An Identity Involving the Integral of the First-Kind Chebyshev Polynomials

We used the algebraic manipulations and the properties of Chebyshev polynomials to obtain an interesting identity involving the power sums of the integral of the first-kind Chebyshev polynomials and solved an open problem proposed by Wenpeng Zhang and Tingting Wang.

A new identity involving the classical Kloosterman sums and 2-dimensional Kloosterman sums

The main purpose of this paper is, using the properties of Gauss sums and the estimate for trigonometric sums, to study the relationships between the sixth power mean of the classical Kloosterman sums and the fourth power mean of the 2-dimensional Kloosterman sums, and give an interesting identity for them.

The hybrid mean value of Dedekind sums and two-term exponential sums

In this paper, we use the mean value theorem of Dirichlet L-functions, the properties of Gauss sums and Dedekind sums to study the hybrid mean value problem involving Dedekind sums and the two-term exponential sums, and give an interesting identity and asymptotic formula for it.

The Λ-Fleming-Viot Process and a Connection with Wright-Fisher Diffusion

Thed-dimensional Λ-Fleming-Viot generator acting on functionsg(x), withxbeing a vector ofdallele frequencies, can be written as a Wright-Fisher generator acting on functionsgwith a modified random linear argument ofxinduced by partitioning occurring in the Λ-Fleming-Viot process. The eigenvalues and right polynomial eigenvectors are easy to see from this representation. The two-dimensional process, which has a one-dimensional generator, is considered in detail. A nonlinear equation is found for the Green's function. In a model with genic selection a proof is given that there is a critical selection value such that if the selection coefficient is greater than or equal to the critical value then fixation, when the boundary 1 is hit, has probability 1 beginning from any nonzero frequency. This is an analytic proof different from the proofs of Der, Epstein and Plotkin (2011) and Foucart (2013). An application in the infinitely-many-alleles Λ-Fleming-Viot process is finding an interesting identity for the frequency spectrum of alleles that is based on size biasing. The moment dual process in the Fleming-Viot process is the usual Λ-coalescent tree back in time. The Wright-Fisher representation using a different set of polynomialsgn(x) as test functions produces a dual death process which has a similarity to the Kingman coalescent and decreases by units of one. The eigenvalues of the process are analogous to the Jacobi polynomials when expressed in terms ofgn(x), playing the role ofxn. Under the stationary distribution when there is mutation,is analogous to thenth moment in a beta distribution. There is ad-dimensional versiongn(X), and even an intriguing Ewens' sampling formula analogy whend→ ∞.