Symmetric Identities Research Articles

Pseudo Symmetric Ideals in Near Subtraction Semigroups

Pseudo Symmetric Ideals in Near Subtraction Semigroups

Proof Without Words: A Symmetric Difference Identity Involving Polygonal Numbers

Summary We illustrate a symmetric difference identity involving polygonal numbers.

Higher Order Bivariate Bell-Based Apostol-Frobenius-Type Poly-Genocchi Polynomials with Parameters a and b

In this paper, we unveil a novel category of Frobenius-Genocchi polynomials, grounded in the Bell numbers and Apostol-type functions. Our exploration delves into a comprehensive examination of these polynomials, elucidating various properties. Employing diverse analytical methods and leveraging generating functions for Bell-based Apostol-Frobenius-Type poly-Genocchi polynomials of higher order, we derive explicit and implicit summation formulas, complemented by their symmetric identities.

Products and powers of principal symmetric ideals

Principal symmetric ideals were recently introduced by Harada et al. in [The minimal free resolution of a general principal symmetric ideal, preprint (2023), arXiv:2308.03141], where their homological properties are elucidated. They are ideals generated by the orbit of a single polynomial under permutations of variables in a polynomial ring. In this paper, we determine when a product of two principal symmetric ideals is principal symmetric and when the powers of a principal symmetric ideal are again principal symmetric ideals. We characterize the ideals that have the latter property as being generated by polynomials invariant up to a scalar multiple under permutation of variables. Recognizing principal symmetric ideals is an open question for the purpose of which we produce certain obstructions. We also demonstrate that the Hilbert functions of symmetric monomial ideals are not all given by symmetric monomial ideals, in contrast to the non-symmetric case.

Symmetric Identities Involving the Extended Degenerate Central Fubini Polynomials Arising from the Fermionic p-Adic Integral on ℤp

Since the constructions of p-adic q-integrals, these integrals as well as particular cases have been used not only as integral representations of many special functions, polynomials, and numbers, but they also allow for deep examinations of many families of special numbers and polynomials, such as central Fubini, Bernoulli, central Bell, and Changhee numbers and polynomials. One of the key applications of these integrals is for obtaining the symmetric identities of certain special polynomials. In this study, we focus on a novel generalization of degenerate central Fubini polynomials. First, we introduce two variable degenerate w-torsion central Fubini polynomials by means of their exponential generating function. Then, we provide a fermionic p-adic integral representation of these polynomials. Through this representation, we investigate several symmetric identities for these polynomials using special p-adic integral techniques. Also, using series manipulation methods, we obtain an identity of symmetry for the two variable degenerate w-torsion central Fubini polynomials. Finally, we provide a representation of the degenerate differential operator on the two variable degenerate w-torsion central Fubini polynomials related to the degenerate central factorial polynomials of the second kind.

Several Symmetric Identities of the Generalized Degenerate Fubini Polynomials by the Fermionic p-Adic Integral on Zp

After constructions of p-adic q-integrals, in recent years, these integrals with some of their special cases have not only been utilized as integral representations of many special numbers, polynomials, and functions but have also given the chance for deep analysis of many families of special polynomials and numbers, such as Bernoulli, Fubini, Bell, and Changhee polynomials and numbers. One of the main applications of these integrals is to obtain symmetric identities for the special polynomials. In this study, we focus on a novel extension of the degenerate Fubini polynomials and on obtaining some symmetric identities for them. First, we introduce the two-variable degenerate w-torsion Fubini polynomials by means of their exponential generating function. Then, we provide a fermionic p-adic integral representation of these polynomials. By this representation, we derive some new symmetric identities for these polynomials, using some special p-adic integral techniques. Lastly, by using some series manipulation techniques, we obtain more identities of symmetry for the two variable degenerate w-torsion Fubini polynomials.

Symmetrical Hermite–Hadamard type inequalities stemming from multiplicative fractional integrals

We firstly study ∗integrability and commutativity for multiplicative fractional integrals with exponential kernels, proposed by Peng et al. (2022). Secondly, making use of such operators, we present a symmetrical multiplicative fractional integrals identity. Based on it, and the fact that the function T∗ is multiplicatively convex or the function (lnT∗)θ is convex for θ>1, especially pondering the case of 0<θ≤1, we establish the symmetrical Hermite–Hadamard type inequalities for multiplicative convexity. We also give some applications in special means under multiplicative calculus.

The binomial-Stirling–Eulerian polynomials

We introduce the binomial-Stirling–Eulerian polynomials, denoted Ãn(x,y|α), which encompass binomial coefficients, Eulerian numbers and two Stirling statistics: the left-to-right minima and the right-to-left minima. When α=1, these polynomials reduce to the binomial-Eulerian polynomials Ãn(x,y), originally named by Shareshian and Wachs and explored by Chung–Graham–Knuth and Postnikov–Reiner–Williams. We investigate the γ-positivity of Ãn(x,y|α) from two aspects: • firstly by employing the grammatical calculus introduced by Chen; • and secondly by constructing a new group action on permutations. These results extend the symmetric Eulerian identity found by Chung, Graham and Knuth, and the γ-positivity of Ãn(x,y) first demonstrated by Postnikov, Reiner and Williams.

Gröbner bases, symmetric matrices, and type C Kazhdan–Lusztig varieties

AbstractWe study a class of combinatorially defined polynomial ideals that are generated by minors of a generic symmetric matrix. Included within this class are the symmetric determinantal ideals, the symmetric ladder determinantal ideals, and the symmetric Schubert determinantal ideals of A. Fink, J. Rajchgot, and S. Sullivant. Each ideal in our class is a type C analog of a Kazhdan–Lusztig ideal of A. Woo and A. Yong; that is, it is the scheme‐theoretic defining ideal of the intersection of a type C Schubert variety with a type C opposite Schubert cell, appropriately coordinatized. The Kazhdan–Lusztig ideals that arise are exactly those where the opposite cell is 123‐avoiding. Our main results include Gröbner bases for these ideals, prime decompositions of their initial ideals (which are Stanley–Reisner ideals of subword complexes), and combinatorial formulas for their multigraded Hilbert series in terms of pipe dreams.

Symmetric Identities Involving Discrete Appell Sequences

Symmetric Identities Involving Discrete Appell Sequences

Exploring the Characteristics of Δh Bivariate Appell Polynomials: An In-Depth Investigation and Extension through Fractional Operators

The objective of this article is to introduce the ∆h bivariate Appell polynomials ∆hAs[r](λ,η;h) and their extended form via fractional operators. The study described in this paper follows the line of study in which the monomiality principle is used to develop new results. It is further discovered that these polynomials satisfy various well-known fundamental properties and explicit forms. The explicit series representation of ∆h bivariate Gould–Hopper polynomials is first obtained, and, using this outcome, the explicit series representation of the ∆h bivariate Appell polynomials is further given. The quasimonomial properties fulfilled by bivariate Appell polynomials ∆h are also proved by demonstrating that the ∆h bivariate Appell polynomials exhibit certain properties related to their behavior under multiplication and differentiation operators. The determinant form of ∆h bivariate Appell polynomials is provided, and symmetric identities for the ∆h bivariate Appell polynomials are also exhibited. By employing the concept of the forward difference operator, operational connections are established, and certain applications are derived. Different Appell polynomial members can be generated by using appropriate choices of functions in the generating expression obtained in this study for ∆h bivariate Appell polynomials. Additionally, generating relations for the ∆h bivariate Bernoulli and Euler polynomials, as well as for Genocchi polynomials, are established, and corresponding results are obtained for those polynomials.

Symmetric ideals of generalized summing multilinear operators

Let be classes of Banach spaces-valued sequences. An n-linear operator A between Banach spaces belongs to the ideal of -summing multilinear operators if belongs to Y whenever belongs to . In this paper we develop techniques to generate non trivial symmetric ideals of this type. Illustrative examples and additional applications of the techniques are provided.

Universality for cokernels of random matrix products

For random integer matrices M1,…,Mk∈Matn(Z) with independent entries, we study the distribution of the cokernel Cok(M1⋯Mk) of their product. We show that this distribution converges to a universal one as n→∞ for a general class of matrix entry distributions, and more generally show universal limits for the joint distribution of Cok(M1),Cok(M1M2),…,Cok(M1⋯Mk). Furthermore, we characterize the universal distributions arising as marginals of a natural generalization of the Cohen-Lenstra measure to sequences of abelian groups with maps between them, which weights sequences inversely proportionally to their number of automorphisms. The proofs develop an extension of the moment method of Wood to joint moments of multiple groups, and rely also on the connection to Hall-Littlewood polynomials and symmetric function identities. As a corollary we obtain an explicit universal distribution for coranks of random matrix products over Fp as the matrix size tends to infinity.

Properties of Multivariable Hermite Polynomials in Correlation with Frobenius–Genocchi Polynomials

The evolution of a physical system occurs through a set of variables, and the problems can be treated based on an approach employing multivariable Hermite polynomials. These polynomials possess beneficial properties exhibited in functional and differential equations, recurring and explicit relations as well as symmetric identities, and summation formulae, among other examples. In view of these points, comprehensive schemes have been developed to apply the principle of monomiality from mathematical physics to various mathematical concepts of special functions, the development of which has encompassed generalizations, extensions, and combinations of other functions. Accordingly, this paper presents research on a novel family of multivariable Hermite polynomials associated with Frobenius–Genocchi polynomials, deriving the generating expression, operational rule, differential equation, and other defining characteristics for these polynomials. Additionally, the monomiality principle for these polynomials is verified, as well as establishing the series representations, summation formulae, operational and symmetric identities, and recurrence relations satisfied by these polynomials. This proposed scheme aims to provide deeper insights into the behavior of these polynomials and to uncover new connections between these polynomials, to enhance understanding of their properties.

Specialized symmetric functions and trace identities

Specialized symmetric functions and trace identities

A Combinatorial Proof of a Symmetric Group Character Involution

We give a short combinatorial proof based on the Murnaghan-Nakayama rule of the symmetric group character identity , where is the conjugate of the partition .

Properties of Multivariate Hermite Polynomials in Correlation with Frobenius–Euler Polynomials

A comprehensive framework has been developed to apply the monomiality principle from mathematical physics to various mathematical concepts from special functions. This paper presents research on a novel family of multivariate Hermite polynomials associated with Apostol-type Frobenius–Euler polynomials. The study derives the generating expression, operational rule, differential equation, and other defining characteristics for these polynomials. Additionally, the monomiality principle for these polynomials is verified. Moreover, the research establishes series representations, summation formulae, and operational and symmetric identities, as well as recurrence relations satisfied by these polynomials.

Lateral order on complex vector lattices and narrow operators

AbstractIn this paper, we continue investigation of the lateral order on vector lattices started in [25]. We consider the complexification of a real vector lattice E and introduce the lateral order on . Our first main result asserts that the set of all fragments of an element of the complexification of an uniformly complete vector lattice E is a Boolean algebra. Then, we study narrow operators defined on the complexification of a vector lattice E, extending the results of articles [22, 27, 28] to the setting of operators defined on complex vector lattices. We prove that every order‐to‐norm continuous linear operator from the complexification of an atomless Dedekind complete vector lattice E to a finite‐dimensional Banach space X is strictly narrow. Then, we prove that every C‐compact order‐to‐norm continuous linear operator from to a Banach space X is narrow. We also show that every regular order‐no‐norm continuous linear operator from to a complex Banach lattice is narrow. Finally, in the last part of the paper we investigate narrow operators taking values in symmetric ideals of compact operators.

Applications of Symmetric Identities for Apostol–Bernoulli and Apostol–Euler Functions

In this paper, we perform a further investigation on the Apostol–Bernoulli and Apostol–Euler functions introduced by Luo. By using the Fourier expansions of the Apostol–Bernoulli and Apostol–Euler polynomials, we establish some symmetric identities for the Apostol–Bernoulli and Apostol–Euler functions. As applications, some known results, for example, Raabe’s multiplication formula and Hermite’s identity, are deduced as special cases.

A family of Apostol–Euler polynomials associated with Bell polynomials

Abstract Many authors investigated the characteristics of the Bell, Euler, Bernoulli, and Genocchi polynomials because of their numerous uses in statistics, number theory, and other branches of science. A generating function for mixed-type Apostol–Euler polynomials of order η related with Bell polynomials is presented in this study. We also construct essential identities of Apostol–Euler polynomials of order η connected with Bell polynomials, such as the correlation formula, the implicit summation formula, the derivative formula, some association with Stirling numbers, and specific cases. Also we establish certain symmetric identities and their known consequences. In addition, we investigate several interesting relationships related to umbral calculus.