Sum Of Ideals Research Articles

Some mixed character sum identities of Katz II

A conjecture connected with quantum physics led N. Katz to discover some amazing mixed character sum identities over a field of q elements, where q is a power of a prime p >3. His proof required deep algebro-geometric techniques, and he expressed interest in finding a more straightforward direct proof. The first author recently gave such a proof of his identities when q equiv 1 pmod 4, and this paper provides such a proof for the remaining case q equiv 3 pmod 4. Our proofs are valid for all characteristics p>2. Along the way we prove some elegant new character sum identities.

The Characteristic Function for Complex Doubly Infinite Jacobi Matrices

We introduce a class of doubly infinite complex Jacobi matrices determined by a simple convergence condition imposed on the diagonal and off-diagonal sequences. For each Jacobi matrix belonging to this class, an analytic function, called a characteristic function, is associated with it. It is shown that the point spectrum of the corresponding Jacobi operator restricted to a suitable domain coincides with the zero set of the characteristic function. Also, coincidence regarding the order of a zero of the characteristic function and the algebraic multiplicity of the corresponding eigenvalue is proved. Further, formulas for the entries of eigenvectors, generalized eigenvectors, a summation identity for eigenvectors, and matrix elements of the resolvent operator are provided. The presented method is illustrated by several concrete examples.

Identities for trigonometric divisor sums

The goal of this paper is to prove several identities for sums of the type [Formula: see text] where [Formula: see text] is the number of divisors of [Formula: see text] and [Formula: see text] is the number of representations of [Formula: see text] as a sum of two squares. Some of our identities involve weighted divisor functions, and most of our identities are new.

Two triple binomial sum supercongruences

In a recent article, Apagodu and Zeilberger discuss some applications of an algorithm for finding and proving congruence identities (modulo primes) of indefinite sums of many combinatorial sequences. At the end, they propose some supercongruences as conjectures. Here we prove one of them, including a new companion enumerating Abelian squares, and we leave some remarks for the others.

Inequalities for Chebyshev Functional in Banach Algebras

By utilizing some identities for double sums, some new inequalities for the Chebyshev functional in Banach algebras are given. Some examples for the exponential and resolvent functions on Banach algebras are also provided.

A generalization of power and alternating power sums to any Appell polynomials

The classical power sum and alternating power sum identities can be stated as ?m,i=0 sn(i)= 1/n+1 (Bn+1(m+1)- Bn+1), ?m,i=0(-1)i sn (i) = 1/2 ((-1)m En(m+1) + En), where sn(x)=xn is the simplest possible Appell polynomial for the Sheffer pair (1,t). The impetus for this research starts from the question that what if we replace sn(x)=xn by any Appell polynomial. In this paper, we give a generalization of power and alternating power sums to any Appell polynomials.

Higher spin six vertex model and symmetric rational functions

We consider a fully inhomogeneous stochastic higher spin six vertex model in a quadrant. For this model we derive concise integral representations for multi-point q-moments of the height function and for the q-correlation functions. At least in the case of the step initial condition, our formulas degenerate in appropriate limits to many known formulas of such type for integrable probabilistic systems in the (1 $$+$$ 1)d KPZ universality class, including the stochastic six vertex model, ASEP, various q-TASEPs, and associated zero range processes. Our arguments are largely based on properties of a family of symmetric rational functions that can be defined as partition functions of the inhomogeneous higher spin six vertex model for suitable domains. In the homogeneous case, such functions were previously studied in Borodin (On a family of symmetric rational functions, 2014); they also generalize classical Hall–Littlewood and Schur polynomials. A key role is played by Cauchy-like summation identities for these functions, which are obtained as a direct corollary of the Yang–Baxter equation for the higher spin six vertex model.

Proof Without Words: Factorial Sums

SummaryWe provide a visual proof of a factorial sum identity implying the existence of the factorial base number system.

The multinomial convolution sums of certain divisor functions

In this article we first introduce some results about the binomial, trinomial, and quadrinomial convolution sums of certain divisor functions and then we provide an identity for the multinomial convolution sums of the divisor function σr♯(n):=∑d|nn/d≡1(4)dr−(−1)r∑d|nn/d≡−1(4)dr.

Identity families of multiple harmonic sums and multiple zeta star values

In this paper we present many new families of identities for multiple harmonic sums using binomial coefficients. Some of these generalize a few recent results of Hessami Pilehrood et al. As applications we prove several conjectures involving multiple zeta star values (MZSV): the Two-one formula conjectured by Ohno and Zudilin, and a few conjectures of Imatomi et al. involving 2-3-2-1 type MZSV, where 2 means some finite string of 2's

A p-adic interpretation of some integral identities for Hall–Littlewood polynomials

If one restricts an irreducible representation Vλ of GL2n to the orthogonal group (respectively the symplectic group), the trivial representation appears with multiplicity one if and only if all parts of λ are even (resp. the conjugate partition λ′ is even). One can rephrase this statement as an integral identity involving Schur functions, the corresponding characters. Rains and Vazirani considered q,t-generalizations of such integral identities, and proved them using affine Hecke algebra techniques. In a recent paper, we investigated the q=0 limit (Hall–Littlewood), and provided direct combinatorial arguments for these identities; this approach led to various generalizations and a finite-dimensional analog of a recent summation identity of Warnaar. In this paper, we reformulate some of these results using p-adic representation theory; this parallels the representation-theoretic interpretation in the Schur case. The nonzero values of the identities are interpreted as certain p-adic measure counts. This approach provides a p-adic interpretation of these identities (and a new identity), as well as independent proofs. As an application, we obtain a new Littlewood summation identity that generalizes a classical result due to Littlewood and Macdonald. Finally, our p-adic method also leads to a generalized integral identity in terms of Littlewood–Richardson coefficients and Hall polynomials.

Further Double Sums of Dunkl and Gasper

By combining the series rearrangement with a reformulation of Beta function, we prove a transformation theorem that reduces a double sum with seven free parameters to a terminating hypergeometric \({_4F_3}\)-series. Seven double sum identities are derived as consequences with one of them extending the recent double sum of Dunkl and Gasper (The sums of a double hypergeometric series and of the first m + 1 terms of \({_3F_2(a,b,c;(a+b+1)/2,2c;1)}\) when c = −m is a negative integer. arXiv:1412.4022v2 [math.CA], 2014).

Sums of divisors functions and Bessel function series

On page 335 in his lost notebook, Ramanujan recorded without proofs two identities involving finite trigonometric sums and doubly infinite series of Bessel functions. These two identities are intimately connected with the classical circle and divisor problems, respectively. There are three possible interpretations for the double series of these identities. The first identity has been proved under all three interpretations, and the second under two of them. Furthermore, several analogues of them were established, and they were extended to Riesz sum identities as well. In this paper, we provide analogous Riesz sum identities for the weighted sums of divisors functions, and in particular two of them yield a generalization of the Riesz sum identity for r6(n), where r6(n) denotes the number of representations of n as a sum of six squares.

Möbius inversion formulas related to the Fourier expansions of two-dimensional Apostol–Bernoulli polynomials

The two-dimensional (2D) Apostol–Bernoulli and Apostol–Euler polynomials are defined via the generating functionstext+ytmλet−1=∑n=0∞Bn(x,y;λ)tnn!,2ext+ytmλet+1=∑n=0∞En(x,y;λ)tnn!. The Apostol–Bernoulli and Apostol–Euler polynomials are essentially the same as parametrized polynomial families, thus we may restrict to the latter.The Fourier coefficients of x↦λxBn(x,y;λ) on [0,1) satisfy an arithmetical–dynamical transformation formula which makes the Fourier series amenable to a technique of generalized Möbius inversion. This yields some interesting arithmetic summation identities, among them parametrized versions of the following well-known classical formula of Davenport:∑k=1∞μ(k)k{kx}=−sin⁡(2πx)π(x∈R), where μ(n) is the Möbius function and {x} denotes the fractional part of x. Davenport's formula is the limiting case α=0 of−4π4π2−α2sin⁡(2πx)=∑k=1∞μ(k)k⋅sin⁡(αk({kx}−12))2sin⁡(α2k), which is valid for −π<α≤π.

New Finite and Infinite Summation Identities Involving the Generalized Harmonic Numbers

We state and prove a general summation identity. The identity is then applied to derive various summation formulas involving the generalized harmonic numbers and related quantities. Interesting results, mostly new, are obtained for both finite and infinite sums. The high points of this paper are perhaps the discovery of several previously unknown infinite summation results involving {\em non-linear} generalized harmonic number terms and the derivation of interesting alternating summation formulas involving these numbers

Some identities on the generalized twisted q-Euler polynomials with weak weight

In this paper, we study the symmetry for generalized twistedq-Euler numbers E ( ) n;;q; and polynomials E ( ) n;;q; (x) with weak weight . We obtain some interesting identities of the power sums and generalized twisted q-Euler polynomials E ( ) n;;q; (x) with weak weight using the symmetric properties for the p-adic invariant q-integral on Zp. Mathematics Subject Classication: 11B68, 11S40, 11S80

On $q$-analogs of some families of multiple harmonic sums and multiple zeta star value identities

In recent years, there is an intensive research on the Q-linear relations be- tween multiple zeta (star) values. In this paper, we prove many families of identities involving the q-analog of these values, from which we can always recover the correspond- ing classical identities by taking q → 1. The key idea of our approach is to first establish some q-analogs of binomial identities for multiple harmonic sums which generalize the corresponding results in the classical non-q setting obtained by the third author recently. Then by taking the limit as the number of terms in the multiple harmonic sums goes to infinity we can obtain the corresponding q-MZSV identities.

Inverse hyperbolic summations and product identities for Fibonacci and Lucas numbers

In this article we derive summation and product identities for Fibonacci and Lucas numbers. The results have a general character and contain some of the existing results as special cases. Mathematics Subject Classification: 11B39, 11Y60

Eigenvalue problem of Sturm–Liouville systems with separated boundary conditions

Let \(\lambda _j\) be the jth eigenvalue of Sturm–Liouville systems with separated boundary conditions, we build up the Hill-type formula, which represent \(\prod \nolimits _{j}(1-\lambda _j^{-1})\) as a determinant of finite matrix. Consequently, we get the Krein-type trace formula based on the Hill-type formula, which express \(\sum \nolimits _{j}{1\over \lambda _j^m}\) as trace of finite matrices. The trace formula can be used to estimate the conjugate point alone a geodesic in Riemannian manifold and to get some infinite sum identities.

Some identities for derangement and Ward number sequences and related bijections

Abstract We establish an alternating sum identity for three classes of singleton-free set partitions wherein the number of elements minus the number of blocks is fixed: (i) permutations, that is, partitions into cycles, (ii) unrestricted partitions, and (iii) contents-ordered partitions. Both algebraic and combinatorial proofs are given, the latter making use of a sign-changing involution in each ease. As a consequence, combinatorial proofs are found of specific cases of recent identities of Gould et al. involving both kinds of Stirling numbers.