Symmetric Identities Research Articles

A New Class of Hermite-Based Higher Order Central Fubini Polynomials

In this paper, we introduce a new class of Hermite-based higher-order central Fubini polynomials and numbers. We obtain some properties, identities, and recurrence relationships by using the generating function of these polynomials. We establish summation formulas of these polynomials by summation techniques series. Also, we derive symmetric identities of three-variable Hermite-based higher-order central Fubini polynomials by using generating functions.

Some Identities Involving Two-Variable Partially Degenerate Hermite Polynomials Induced from Differential Equations and Structure of Their Roots

In this paper, we introduce two-variable partially degenerate Hermite polynomials and get some new symmetric identities for two-variable partially degenerate Hermite polynomials. We study differential equations induced from the generating functions of two-variable partially degenerate Hermite polynomials to give identities for two-variable partially degenerate Hermite polynomials. Finally, we study the symmetric properties of the structure of the roots of the two-variable partially degenerate Hermite equations.

Differential Equations Associated with Two Variable Degenerate Hermite Polynomials

In this paper, we introduce the two variable degenerate Hermite polynomials and obtain some new symmetric identities for two variable degenerate Hermite polynomials. In order to give explicit identities for two variable degenerate Hermite polynomials, differential equations arising from the generating functions of degenerate Hermite polynomials are studied. Finally, we investigate the structure and symmetry of the zeros of the two variable degenerate Hermite equations.

Isometries of real subspaces of self-adjoint operators in Banach symmetric ideals

Let (mathcal C_E, cdot_mathcal C_E) be a Banach symmetric ideal of compact operators, acting in a complex separable infinite-dimensional Hilbert space mathcal H. Let mathcal C_Ehxin mathcal C_E : xx be the real Banach subspace of self-adjoint operators in (mathcal C_E, cdot_mathcal C_E). We show that in the case when (mathcal C_E, cdot_mathcal C_E) is a separable or perfect Banach symmetric ideal (mathcal C_E eq mathcal C_2) any skew-Hermitian operator H: mathcal C_Ehto mathcal C_Eh has the following form H(x)i(xa - ax) for same aain mathcal B(mathcal H) and for all xin mathcal C_Eh. Using this description of skew-Hermitian operators, we obtain the following general form of surjective linear isometries V:mathcal C_Eh to mathcal C_Eh. Let (mathcal C_E, cdot_mathcal C_E) be a separable or a perfect Banach symmetric ideal with not uniform norm, that is p_mathcal C_E 1 for any finite dimensional projection p inmathcal C_E with dim p(mathcal H)1, let mathcal C_E eq mathcal C_2, and let V: mathcal C_Eh to mathcal C_Eh be a surjective linear isometry. Then there exists unitary or anti-unitary operator u on mathcal H such that V(x)uxu orV(x)-uxu for all x in mathcal C_Eh.

Properties of Partially Degenerate Complex Appell Polynomials

Degenerate versions of polynomial sequences have been recently studied to obtain useful properties such as symmetric identities by introducing degenerate exponential-type generating functions. As part of our continued work in degenerate versions of generating functions, we subsequently present our study on degenerate complex Appell polynomials by considering a partially degenerate version of the generating functions of ordinary complex Appell polynomials in this paper. We only consider partially degenerate generating functions to retain the crucial properties of the Appell sequence, and we present useful identities and general properties by splitting complex values into their real and imaginary parts; moreover, we provide several explicit examples. Additionally, the differential equations satisfied by degenerate complex Bernoulli and Euler polynomials are derived by the quasi-monomiality principle using Appell-type polynomials.

Some Properties for Multiple Twisted (p, q)-L-Function and Carlitz’s Type Higher-Order Twisted (p, q)-Euler Polynomials

The main goal of this paper is to study some interesting identities for the multiple twisted ( p , q ) -L-function in a complex field. First, we construct new generating functions of the new Carlitz-type higher order twisted ( p , q ) -Euler numbers and polynomials. By applying the Mellin transformation to these generating functions, we obtain integral representations of the multiple twisted ( p , q ) -Euler zeta function and multiple twisted ( p , q ) -L-function, which interpolate the Carlitz-type higher order twisted ( p , q ) -Euler numbers and Carlitz-type higher order twisted ( p , q ) -Euler polynomials at non-positive integers, respectively. Second, we get some explicit formulas and properties, which are related to Carlitz-type higher order twisted ( p , q ) -Euler numbers and polynomials. Third, we give some new symmetric identities for the multiple twisted ( p , q ) -L-function. Furthermore, we also obtain symmetric identities for Carlitz-type higher order twisted ( p , q ) -Euler numbers and polynomials by using the symmetric property for the multiple twisted ( p , q ) -L-function.

Codimension and projective dimension up to symmetry

AbstractSymmetric ideals in increasingly larger polynomial rings that form an ascending chain are investigated. We focus on the asymptotic behavior of codimensions and projective dimensions of ideals in such a chain. If the ideals are graded it is known that the codimensions grow eventually linearly. Here this result is extended to chains of arbitrary symmetric ideals. Moreover, the slope of the linear function is explicitly determined. We conjecture that the projective dimensions also grow eventually linearly. As part of the evidence we establish two non‐trivial lower linear bounds of the projective dimensions for chains of monomial ideals. As an application, this yields Cohen–Macaulayness obstructions.

A SOCP relaxation based branch-and-bound method for generalized trust-region subproblem

<p style='text-indent:20px;'>This paper proposes a second-order cone programming (SOCP) relaxation for the generalized trust-region problem by exploiting the property that any symmetric matrix and identity matrix can be simultaneously diagonalizable. We show that our proposed SOCP relaxation can provide a lower bound as tight as that of the standard semidefinite programming (SDP) relaxation. Moreover, we provide a sufficient condition under which the proposed SOCP relaxation is exact. Since the standard SDP relaxation suffers from a much heavier computing burden, the proposed SOCP relaxation has a much higher efficiency in solving process. Then we design a branch-and-bound algorithm based on this SOCP relaxation to obtain the global optimal solution for a general problem. Three types of numerical experiments are carried out to demonstrate the effectiveness and efficiency of our proposed SOCP relaxation.

M-embedded symmetric operator spaces and the derivation problem

AbstractLet ℳ be a semifinite von Neumann algebra with a faithful semifinite normal trace τ. Assume that E(0, ∞) is an M-embedded fully symmetric function space having order continuous norm and is not a superset of the set of all bounded vanishing functions on (0, ∞). In this paper, we prove that the corresponding operator space E(ℳ, τ) is also M-embedded. It extends earlier results by Werner [48, Proposition 4∙1] from the particular case of symmetric ideals of bounded operators on a separable Hilbert space to the case of symmetric spaces (consisting of possibly unbounded operators) on an arbitrary semifinite von Neumann algebra. Several applications are given, e.g., the derivation problem for noncommutative Lorentz spaces ℒp,1(ℳ, τ), 1 &lt; p &lt; ∞, has a positive answer.

A Note on Type 2 w-Daehee Polynomials

In the paper, by virtue of the p-adic invariant integral on Z p , the authors consider a type 2 w-Daehee polynomials and present some properties and identities of these polynomials related with well-known special polynomials. In addition, we present some symmetric identities involving the higher order type 2 w-Daehee polynomials. These identities extend and generalize some known results.

M6A RNA Methylation Maintains Hematopoietic Stem Cell Identity and Symmetric Commitment

SUMMARYStem cells balance cellular fates through asymmetric and symmetric divisions in order to self-renew or to generate downstream progenitors. Symmetric commitment divisions in stem cells are required for rapid regeneration during tissue damage and stress. The control of symmetric commitment remains poorly defined. Using single-cell RNA sequencing (scRNA-seq) in combination with transcriptomic profiling of HSPCs (hematopoietic stem and progenitor cells) from control and m6A methyltransferase Mettl3 conditional knockout mice, we found that m6A-deficient hematopoietic stem cells (HSCs) fail to symmetrically differentiate. Dividing HSCs are expanded and are blocked in an intermediate state that molecularly and functionally resembles multipotent progenitors. Mechanistically, RNA methylation controls Myc mRNA abundance in differentiating HSCs. We identified MYC as a marker for HSC asymmetric and symmetric commitment. Overall, our results indicate that RNA methylation controls symmetric commitment and cell identity of HSCs and may provide a general mechanism for how stem cells regulate differentiation fate choice.

Some Symmetric Identities for Degenerate Carlitz-type (p, q)-Euler Numbers and Polynomials

In this paper we define the degenerate Carlitz-type ( p , q ) -Euler polynomials by generalizing the degenerate Euler numbers and polynomials, degenerate Carlitz-type q-Euler numbers and polynomials. We also give some theorems and exact formulas, which have a connection to degenerate Carlitz-type ( p , q ) -Euler numbers and polynomials.

On Symmetric Identities of Carlitz’s Type q-Daehee Polynomials

In this paper, we study Carlitz’s type q-Daehee polynomials and investigate the symmetric identities for them by using the p-adic q-integral on Zp under the symmetry group of degree n.

Periods and Reciprocity I

Abstract Given $\boldsymbol{F}$ a number field with ring of integers ${\mathcal{O}}_{\boldsymbol{F}}$ and ${\mathfrak{p}},{\mathfrak{q}}$ two coprime and squarefree ideals of ${\mathcal{O}}_{\boldsymbol{F}}$, we prove a reciprocity relation for the 1st moment of the triple product $L$-functions $L(\pi \otimes \pi _1\otimes \pi _2,\frac{1}{2})$ twisted by $\lambda _{\pi }({\mathfrak{p}})$, where $\pi _1$ and $\pi _2$ are fixed unitary automorphic representations of $\textrm{PGL}_2({\mathbb{A}}_{\boldsymbol{F}})$ with $\pi _1$ cuspidal and $\pi$ runs through unitary automorphic representations of conductor dividing ${\mathfrak{q}}$. The method uses adelic integral representations of $L$-functions and the symmetric identity is established for a particular period. Finally, the integral period is connected to the moment via Parseval formula.

Proofs and reductions of various conjectured partition identities of Kanade and Russell

Abstract We prove seven of the Rogers–Ramanujan-type identities modulo 12 that were conjectured by Kanade and Russell. Included among these seven are the two original modulo 12 identities, in which the products have asymmetric congruence conditions, as well as the three symmetric identities related to the principally specialized characters of certain level 2 modules of A 9 ( 2 ) {A_{9}^{(2)}} . We also give reductions of four other conjectures in terms of single-sum basic hypergeometric series.

Some Symmetric Identities for the Multiple (p, q)-Hurwitz-Euler eta Function

The main purpose of this paper is to find some interesting symmetric identities for the ( p , q ) -Hurwitz-Euler eta function in a complex field. Firstly, we define the multiple ( p , q ) -Hurwitz-Euler eta function by generalizing the Carlitz’s form ( p , q ) -Euler numbers and polynomials. We find some formulas and properties involved in Carlitz’s form ( p , q ) -Euler numbers and polynomials with higher order. We find new symmetric identities for multiple ( p , q ) -Hurwitz-Euler eta functions. We also obtain symmetric identities for Carlitz’s form ( p , q ) -Euler numbers and polynomials with higher order by using symmetry about multiple ( p , q ) -Hurwitz-Euler eta functions. Finally, we study the distribution and symmetric properties of the zero of Carlitz’s form ( p , q ) -Euler numbers and polynomials with higher order.

Identities of Symmetry for Type 2 Bernoulli and Euler Polynomials

The main purpose of this paper is to give several identities of symmetry for type 2 Bernoulli and Euler polynomials by considering certain quotients of bosonic p-adic and fermionic p-adic integrals on Z p , where p is an odd prime number. Indeed, they are symmetric identities involving type 2 Bernoulli polynomials and power sums of consecutive odd positive integers, and the ones involving type 2 Euler polynomials and alternating power sums of odd positive integers. Furthermore, we consider two random variables created from random variables having Laplace distributions and show their moments are given in terms of the type 2 Bernoulli and Euler numbers.

Two Symmetric Identities Involving Complete and Elementary Symmetric Functions

We introduce new relationships between complete and elementary symmetric functions. As specializations of these relations, we obtain new identities of Guo and Yang type. An open problem involving the Gaussian polynomials is derived in this context: For $$n>0$$, the expression $$\begin{aligned} \frac{(-1)^{n-1}}{(q;q)_n} \sum _{k=0}^{\lfloor (n-1)/2 \rfloor } \cos \frac{(n-2k)\pi }{3}\cdot \begin{bmatrix}n\\k\end{bmatrix}_{q} q^{\left( {\begin{array}{c}k\\ 2\end{array}}\right) +\left( {\begin{array}{c}n-k\\ 2\end{array}}\right) } \end{aligned}$$has nonnegative coefficients.

A Note on the Truncated-Exponential Based Apostol-Type Polynomials

In this paper, we propose to investigate the truncated-exponential-based Apostol-type polynomials and derive their various properties. In particular, we establish the operational correspondence between this new family of polynomials and the familiar Apostol-type polynomials. We also obtain some implicit summation formulas and symmetric identities by using their generating functions. The results, which we have derived here, provide generalizations of the corresponding known formulas including identities involving generalized Hermite-Bernoulli polynomials.

Delannoy Numbers and Preferential Arrangements

A preferential arrangement on [ [ n ] ] = { 1 , 2 , … , n } is a ranking of the elements of [ [ n ] ] where ties are allowed. The number of preferential arrangements on [ [ n ] ] is denoted by r n . The Delannoy number D ( m , n ) is the number of lattice paths from ( 0 , 0 ) to ( m , n ) in which only east ( 1 , 0 ) , north ( 0 , 1 ) , and northeast ( 1 , 1 ) steps are allowed. We establish a symmetric identity among the numbers r n and D ( p , q ) by means of algebraic and combinatorial methods.