Rank-Metric Codes, Generalized Binomial Moments and their Zeta Functions

In certain problems in model building and Bayesian analysis, the results end up in forms connected with generalized zeta functions. This necessitates the introduction of an extended form of the generalized zeta function. Such an extended form of the zeta function is introduced in this paper. In model building situations and in various types of applications in physical, biological and social sciences and engineering, a basic model taken is the Gaussian model in the univariate, multivariate and matrix-variate situations. A real scalar variable logistic model behaves like a Gaussian model but with a thicker tail. Hence, for many of industrial applications, a logistic model is preferred to a Gaussian model. When we study the properties of a logistic model in the multivariate and matrix-variate cases, in the real and complex domains, invariably the problem ends up in the extended zeta function defined in this paper. Several such extended logistic models are considered. It is also found that certain Bayesian considerations also end up in the extended zeta function introduced in this paper. Several such Bayesian models in the multivariate and matrix-variate cases in the real and complex domains are discussed. It is stated in a recent paper that “Quantum Mechanics is just the Bayesian theory generalized to the complex Hilbert space”. Hence, the models developed in this paper are expected to have applications in quantum mechanics, communication theory, physics, statistics and related areas.

Minimal rank-metric codes or, equivalently, linear cutting blocking sets are characterized in terms of the second generalized rank weight, via their connection with evasiveness properties of the associated q-system. Using this result, we provide the first construction of a family of mathbb F_{q^m}-linear MRD codes of length 2m that are not obtained as a direct sum of two smaller MRD codes. In addition, such a family has better parameters, since its codes possess generalized rank weights strictly larger than those of the previously known MRD codes. This shows that not all the MRD codes have the same generalized rank weights, in contrast to what happens in the Hamming metric setting.

We generalize the Ihara-Selberg zeta function to hypergraphs in a natural way. Hashimoto's factorization results for biregular bipartite graphs apply, leading to exact factorizations. For $(d,r)$-regular hypergraphs, we show that a modified Riemann hypothesis is true if and only if the hypergraph is Ramanujan in the sense of Winnie Li and Patrick Solé. Finally, we give an example to show how the generalized zeta function can be applied to graphs to distinguish non-isomorphic graphs with the same Ihara-Selberg zeta function.

The familiar Fermi-Dirac and Bose-Einstein functions are of importance not only for their role in quantum statistics, but also for their several interesting mathematical properties in themselves. Here, in our present investigation, we have extended these functions by introducing an extra parameter in a way that gives new insights into these functions and their relationship to the family of zeta functions. These extensions are dual to each other in a sense that is explained in this paper. Some identities are proved here for each of these general functions and their relationship with the general Hurwitz-Lerch zeta function Φ(z, s, a) is exploited to derive some other (presumably new) identities.

For k∈ℤ, the generalized Arakawa–Kaneko zeta functions with a, b, c parameters are given by the Laplace-Mellin integral ξks,x;a,b,c=1/Γs∫0∞Lik1−ab−t/bt−a−tc−xtts−1dt, where ℜs>0 and x>0 if k≥1, and ℜs>0 and x>k+1 if k≤0. In this paper, an interpolation formula between these generalized zeta functions and the poly-Bernoulli polynomials with a,b,c parameters is obtained. Moreover, explicit, difference, and Raabe’s formulas for ξks,x;a,b,c are derived.

The Riemann Zeta Function and the Generalized Zeta Function

This article presents a systematic investigation of various integrals and computational representations for some families of generalized Hurwitz–Lerch Zeta functions which are introduced here. We first establish their relationship with the -function, which enables us to derive the Mellin–Barnes type integral representations for nearly all of the generalized and specialized Hurwitz–Lerch Zeta functions. The integral expressions studied in this paper provide extensions of the corresponding results given by many authors, including (for example) Garg et al. [A further study of general Hurwitz-Lerch zeta function, Algebras Groups Geom. 25 (2008), pp. 311–319] and Lin and Srivastava [Some families of the Hurwitz-Lerch Zeta functions and associated fractional derivative and other integral representations, Appl. Math. Comput. 154 (2004), pp. 725–733]. We also derive a further analytic continuation formula which provides an elegant extension of the well-known analytic continuation formula for the Gauss hypergeometric function. Fractional derivatives associated with the generalized Hurwitz–Lerch Zeta functions are obtained. The relationship between the generalized Hurwitz–Lerch Zeta function and the -function, which was given by Garg et al., is seen to be erroneous and we give its corrected version here. Finally, a unification and extension of the Hurwitz–Lerch Zeta function, introduced in this article, is presented and two of its interesting special cases associated with the Mittag–Leffler type functions due to Barnes [The asymptotic expansion of integral functions defined by Taylor series, Philos. Trans. Roy. Soc. London Ser. A 206 (1906), pp. 249–297] and the generalized M-series considered recently by Sharma and Jain [A note on a generalzed M-series as a special function of fractional calculus, Fract. Calc. Appl. Anal. 12 (2009), pp. 449–452.] are deduced.

We compare the two duality theories of rank-metric codes proposed by Delsarte and Gabidulin, proving that the former generalizes the latter. We also give an elementary proof of MacWilliams identities for the general case of Delsarte rank-metric codes. The identities which we derive are very easy to handle, and allow us to re-establish in a very concise way the main results of the theory of rank-metric codes first proved by Delsarte employing association schemes and regular semilattices. We also show that our identities imply as a corollary the original MacWilliams identities established by Delsarte. We describe how the minimum and maximum rank of a rank-metric code relate to the minimum and maximum rank of the dual code, giving some bounds and characterizing the codes attaining them. Then we study optimal anticodes in the rank metric, describing them in terms of optimal codes (namely, MRD codes). In particular, we prove that the dual of an optimal anticode is an optimal anticode. Finally, as an application of our results to a classical problem in enumerative combinatorics, we derive both a recursive and an explicit formula for the number of $$k \times m$$ matrices over a finite field with given rank and $$h$$ -trace.

In view of the central symmetry of the set of Riemann zeros, \(\rho \leftrightarrow 1 - \rho\) which crucially reflects the functional equation (3.26), there is no clear reason to pick either the ρ themselves (with \(\rho \equiv \frac{1}{2} \pm i\tau_{k}, {\rm Re} \tau_{k} > 0\)) or the \(\rho(1 - \rho) = \frac{1}{4} + \tau_{k}^{2}\) as the basic set upon which to build zeta functions over the Riemann zeros. This is a nonlinear remapping freedom as discussed in Sect. 1.1, which leads to generalized zeta functions of several kinds. This chapter gives an introductory overview of the various possibilities. In decreasing order of analytical tractability, and in our older notation to be soon discarded, we may define: Functions of first kind: \(\sum_{\rho}(x - \rho)^{-s}, \qquad {\rm Re} s > 1;\) Functions of second kind: \(\sum\limits_{k = 1}^{\infty}(\tau_{k}^{2} + v)^{-\sigma},\qquad \mathfrak{R}\sigma > \frac{1}{2};\) Functions of third kind: \(\sum\limits_{k = 1}^{\infty}(\tau_{k}^{2} + \tau)^{-s},\qquad {\rm Re} s > 1.\) In all three kinds the argument (in exponent) is the principal variable, the other one a shift parameter. Each parametric family starts as an analytic function of its argument in its indicated half-plane of convergence, due to (4.7) or (4.26), but it will extend to a meromorphic function in the entire complex plane (of its argument), with a computable singular part. The domain of each shift parameter is left as large as possible: simply, none of the quantities to be raised to a power should lie on the cut \(\mathbb{R}_{-}\); this will hardly matter with the fairly small parameter values that we will emphasize.KeywordsMeromorphic FunctionZeta FunctionCentral SymmetryShift ParameterRiemann HypothesisThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Our previous work presented explicit formulas for the generalized zeta function and the generalized Ihara zeta function corresponding to the Grover walk and the positive-support version of the Grover walk on the regular graph via the Konno–Sato theorem, respectively. This paper extends these walks to a class of walks including random walks, correlated random walks, quantum walks, and open quantum random walks on the torus by the Fourier analysis.

A formal weight enumerator is a homogeneous polynomial in two variables which behaves like the Hamming weight enumerator of a self-dual linear code except that the coefficients are not necessarily nonnegative integers. The notion of formal weight enumerator was first introduced by Ozeki in connection with modular forms, and a systematic investigation of formal weight enumerators has been conducted by Chinen in connection with zeta functions and Riemann hypothesis for linear codes. In this paper, we establish a relation between formal weight enumerators and Chebyshev polynomials. Specifically, the condition for the existence of formal weight enumerators with prescribed parameters $$(n,\varepsilon ,q)$$ is given in terms of Chebyshev polynomials. According to the parity of n and the sign $$\varepsilon$$ , the four kinds of Chebyshev polynomials appear in the statement of the result. Further, we obtain explicit expressions of formal weight enumerators in the case where n is odd or $$\varepsilon =-1$$ using Dickson polynomials, which generalize Chebyshev polynomials. We also state a conjecture from a viewpoint of binomial moments, and see that the results in this paper partially support the conjecture.

We consider the rank polynomial of a matroid and some well-known applications to graphs and linear codes. We compare rank polynomials with two-variable zeta functions for algebraic curves. This leads us to normalize the rank polynomial and to extend it to a rational rank function. As applications to linear codes we mention: A formulation of Greene’s theorem similar to an identity for zeta functions of curves first found by Deninger, the definition of a class of generating functions for support weight enumerators, and a relation for algebraic-geometric codes between the matroid of a code and the two-variable zeta function of a curve.KeywordsZeta FunctionLinear CodeSupport WeightDivisor ClassWeight EnumeratorThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A Proof that extends hurwitz formula into the critical strip

The analytic continuation invoked in the theory of generalized zeta functions associated with infinite-dimensional operators is shown to be equivalent in structure to the basic analytic methods deployed in dimensional regularization.

The effective Lagrangian and vacuum energy-momentum tensor $〈{T}^{\ensuremath{\mu}\ensuremath{\nu}}〉$ due to a scalar field in a de Sitter-space background are calculated using the dimensional-regularization method. For generality the scalar field equation is chosen in the form $({\ensuremath{\square}}^{2}+\ensuremath{\xi}R+{m}^{2})\ensuremath{\phi}=0$. If $\ensuremath{\xi}=\frac{1}{6}$ and $m=0$, the renormalized $〈{T}^{\ensuremath{\mu}\ensuremath{\nu}}〉$ equals ${g}^{\ensuremath{\mu}\ensuremath{\nu}}{(960{\ensuremath{\pi}}^{2}{a}^{4})}^{\ensuremath{-}1}$, where $a$ is the radius of de Sitter space. More formally, a general zeta-function method is developed. It yields the renormalized effective Lagrangian as the derivative of the zeta function on the curved space. This method is shown to be virtually identical to a method of dimensional regularization applicable to any Riemann space.