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Lin Jiu

Publications and source records attributed to Lin Jiu.

31 records · Page 2Linked to original sources
arXiv2017

Matrix Representation for Multiplicative Nested Sums

We study multiplicative nested sums, which are generalizations of harmonic sums, and provide a calculation through multiplication of index matrices. Special cases interpret the index matrices as stochastic transition matrices of random walks on a finite number of sites. Relations among multiplicative nested sums, which are generalizations of relations between harmonic series and multiple zeta functions, can be easily derived from identities of the index matrices. Combinatorial identities and their generalizations can also be derived from this computation.

math.CO↗
arXiv2017

On the Method of Brackets: Rules, Examples, Interpretations and Modifications

The method of brackets, consisting of a small number of heuristic rules, was created by Ivan Gonzalez for the evaluation of definite integrals appearing in the resolution of Feynman diagrams. The current work verifies each step of this method, proves the independence on different representations of the integrand, and also modifies the rule by further considering analytic continuation.

math.CA↗
arXiv2017

An Extension of the Method of Brackets. Part 1

The method of brackets is an efficient method for the evaluation of a large class of definite integrals on the half-line. It is based on a small collection of rules, some of which are heuristic. The extension discussed here is based on the concepts of null and divergent series. These are formal representations of functions, whose coefficients $a_{n}$ have meromorphic representations for $n \in \mathbb{C}$, but might vanish or blow up when $n \in \mathbb{N}$. These ideas are illustrated with the evaluation of a variety of entries from the classical table of integrals by Gradshteyn and Ryzhik.

math.CA↗
arXiv2016

On Binomial Identities in Arbitrary Bases

We extend the digital binomial identity as given by Nguyen el al. to an identity in an arbitrary base $b$, by introducing the $b-$ary binomial coefficients. We then study the properties of these coefficients such as orthogonality, a link to Lucas' theorem and the corresponding $b-$ary Pascal triangles.

math.CO↗
arXiv2014

Riemannian Holonomy Groups of Statistical Manifolds

Normal distribution manifolds play essential roles in the theory of information geometry, so do holonomy groups in classification of Riemannian manifolds. After some necessary preliminaries on information geometry and holonomy groups, it is presented that the corresponding Riemannian holonomy group of the $d$-dimensional normal distribution is $SO\left(\frac{d\left(d+3\right)}{2}\right)$, for all $d\in\mathbb{N}$. As a generalization on exponential family, a list of holonomy groups follows.

math.DG↗
arXiv2014

Principal bundles over statistical manifolds

In this paper, we introduce the concept of principal bundles on statistical manifolds. After necessary preliminaries on information geometry and principal bundles on manifolds, we study the $α$-structure of frame bundles over statistical manifolds with respect to $α$-connections, by giving geometric structures. The manifold of one-dimensional normal distributions appears in the end as an application and a concrete example.

math.DG↗
arXiv2014

The finite Fourier transform of classical polynomials

The finite Fourier transform of a family of orthogonal polynomials $A_{n}(x)$, is the usual transform of the polynomial extended by $0$ outside their natural domain. Explicit expressions are given for the Legendre, Jacobi, Gegenbauer and Chebyshev families.

math.FA↗
arXiv2014

Identities for generalized Euler polynomials

For $N \in \mathbb{N}$, let $T_{N}$ be the Chebyshev polynomial of the first kind. Expressions for the sequence of numbers $p_{\ell}^{(N)}$, defined as the coefficients in the expansion of $1/T_{N}(1/z)$, are provided. These coefficients give formulas for the classical Euler polynomials in terms of the so-called generalized Euler polynomials. The proofs are based on a probabilistic interpretation of the generalized Euler polynomials recently given by Klebanov et al. Asymptotics of $p_{\ell}^{(N)}$ are also provided.

math.PR↗
arXiv2013

Recursion Rules for the Hypergeometric Zeta Functions

The hypergeometric zeta function is defined in terms of the zeros of the Kummer function M(a, a + b; z). It is established that this function is an entire function of order 1. The classical factorization theorem of Hadamard gives an expression as an infinite product. This provides linear and quadratic recurrences for the hypergeometric zeta function. A family of associated polynomials is characterized as Appell polynomials and the underlying distribution is given explicitly in terms of the zeros of the associated hypergeometric function. These properties are also given a probabilistic interpretation in the framework of Beta distributions.

math.NT↗