Thotsaporn Aek Thanatipanonda, Yi Zhang
Polynomial, C-finite, Holonomic are the most common ansatz to describe the pattern of the sequences. We propose a new ansatz called X-recursive that generalize those we mentioned. We also discuss its closure properties and compare this ansatz to another new similar ansatz from another paper.
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Rational reductions for holonomic sequences
该研究刻画了使完整序列与有理函数的乘积可求和的有理函数特征，给出分子次数上下界并说明分母可由零化子得到，提出的有理约化可用于生成新的多和等式与同余式。
Polynomial reduction for $q$-holonomic sequences
本文提出了一种针对 $q$-全纯序列的（Laurent）多项式约化方法，通过分解Laurent多项式为可求和部分和低次余项，并引入对称条件下的幂可分约化，应用于$q$-中心Delannoy数以推导新的$q$-恒等式和$q$-同余式。
Holonomic Sequences Arising from Laplace Transforms of Generalized Second Order Sequences
研究通过拉普拉斯变换得到的斐波那契 - 拉普拉斯和卢卡斯 - 拉普拉斯序列，关联序列族并证明其完整性，通过加权彩色平铺和广义莫茨金路径给出组合解释，推导连分数展开式，扩展了前人工作。
We show how the continuous Almkvist-Zeilberger algorithm can be used to efficiently discover and prove differential equations satisfied by generating functions of sequences defined as integrals of powers of C-finite polynomial sequences (like the Chebyshev polynomials) from which one can automatically derive linear recurrences with polynomial coefficients for the sequences themselves. We fully implement this with a Maple package, CfiniteIntergal.txt .
Various sequences that possess explicit analytic expressions can be analysed asymptotically through integral representations due to Lindelöf, which belong to an attractive but somewhat neglected chapter of complex analysis. One of the outcomes of such analyses concerns the non-existence of linear recurrences with polynomial coefficients annihilating these sequences, and, accordingly, the non-existence of linear differential equations with polynomial coefficients annihilating their generating functions. In particular, the corresponding generating functions are transcendental. Asymptotic estimat…
On the non-holonomic character of logarithms, powers, and the n-th prime function
我们建立了由整数的幂和“分数”幂形成的序列以及素数序列是非完整的，从而回答了Gerhold的三个公开问题[Electronic Journal of Combinatorics 11（2004），R87]。我们的证明依赖于基本的复分析，即连接的结构定理的奇异性的解决方案，线性微分方程和阿贝尔定理。简要讨论了基于奇异性的方法的范围，并证明了几个自然发生的序列是非完整的。
Polynomial reduction, designed first for hypergeometric terms, can be used to automatically prove and generate new hypergeometric identities from old ones. In this paper, we extend the reduction method to holonomic sequences. As applications, we describe an algorithmic way to prove and generate new multi-summation identities. Especially we present new families of $π$-series involving Domb numbers and Franel numbers.
Non-holonomicity of the sequence $\log 1, \log 2, \log 3, ...$
Gerhold提出并有条件地证明了（log n：n= 1，2，.）并不是一个完整的序列Flajolet，Gerhold和Salvy给出了一个证明使用分析机器。我们给出一个简单的证明。
Given a sequence 1, 1, 5, 23, 135, 925, 7285, 64755, 641075, 6993545, 83339745,..., how can we guess a formula for it? This article will quickly walk you through the concept of ansatz for classes of polynomial, $C$-finite, holonomic, and the most recent addition $C^2$-finite sequences. For each of these classes, we discuss in detail various aspects of the guess and check, generating functions, closure properties, and closed-form solutions. Every theorem is presented with an accessible proof, followed by several examples intended to motivate the development of the theories. Each example is acco…
We describe a method to compute the inverse Mellin transform of holonomic sequences, that is based on a method to compute the Mellin transform of holonomic functions. Both methods are implemented in the computer algebra package HarmonicSums.
The SL_3 colored Jones polynomial of the trefoil knot is a $q$-holonomic sequence of two variables with natural origin, namely quantum topology. The paper presents an explicit set of generators for the annihilator ideal of this $q$-holonomic sequence as a case study. On the one hand, our results are new and useful to quantum topology: this is the first example of a rank 2 Lie algebra computation concerning the colored Jones polynomial of a knot. On the other hand, this work illustrates the applicability and computational power of the employed computer algebra methods.
A sequence of rational functions in a variable $q$ is $q$-holonomic if it satisfies a linear recursion with coefficients polynomials in $q$ and $q^n$. We prove that the degree of a $q$-holonomic sequence is eventually a quadratic quasi-polynomial. Our proof uses differential Galois theory (adapting proofs regarding holonomic $D$-modules to the case of $q$-holonomic $D$-modules) combined with the Lech-Mahler-Skolem theorem from number theory. En route, we use the Newton polygon of a linear $q$-difference equation, and introduce the notion of regular-singular $q$-difference equation and a WKB ba…
G-functions and multisum versus holonomic sequences
本文件的目的有三个方面：（a）证明了平衡超几何项的多维和序列都有Gevrey型1有理指数渐近展开式，（b）构造了一类由计数组合学得到的$G$ -函数，（c）给出了Zeilberger问题的一个反例，该问题提出完整序列是否可以写成平衡超几何项的多重和.证明利用的概念$G$ -功能，介绍了西格尔，其分析/算术性质最近显示安德烈。
We describe how the extension of a solver for linear differential equations by Kovacic's algorithm helps to improve a method to compute the inverse Mellin transform of holonomic sequences. The method is implemented in the computer algebra package HarmonicSums.
A sequence $f_n(q)$ is $q$-holonomic if it satisfies a nontrivial linear recurrence with coefficients polynomials in $q$ and $q^n$. Our main theorems state that $q$-holonomicity is preserved under twisting, i.e., replacing $q$ by $ωq$ where $ω$ is a complex root of unity, and under the substitution $q \to q^α$ where $α$ is a rational number. Our proofs are constructive, work in the multivariate setting of $\partial$-finite sequences and are implemented in the Mathematica package HolonomicFunctions. Our results are illustrated by twisting natural $q$-holonomic sequences which appear in quantum …
We give recurrence relations for any family of generalized Appell polynomials unifying so some known recurrences of many classical sequences of polynomials. Our main tool to get our goal is the Riordan group. We use the product of Riordan matrices to interpret some relationships between different families of polynomials. Moreover using the Hadamard product of series we get a general recurrence relation for the polynomial sequences associated to the so called generalized umbral calculus.
Strongly regular graph, strongly polynomial sequences, and two-level polynomials
本文研究强正则图与强多项式图序列的特征多项式和色多项式的深度性质，证明强正则图特征多项式具有无限深度，而Paley图色多项式深度为二，并构造了使同态计数保持无限深度的目标序列。
Sequences of Genocchi numbers of the first and second kind are considered. For these numbers, an approach based on their representation using sequences of polynomials is developed. Based on this approach, for these numbers some identities generalizing the known identities are constructed.
We study properties of constant coefficient Laurent biorthogonal polynomials using Riordan arrays. We give details of related orthogonal polynomials, and we explore relationships between the moments of these orthogonal polynomials, the moments of the defining Laurent biorthogonal polynomials, and the expansions of $T$-fractions. Closed form expressions are given for the polynomials and their moments.
We derive identities for the determinants of matrices whose entries are (rising) powers of (products of) polynomials that satisfy a recurrence relation. In particular, these results cover the cases for Fibonacci polynomials, Lucas polynomials and certain orthogonal polynomials. These identities naturally generalize the determinant identities obtained by Alfred, Carlitz, Prodinger, Tangboonduangjit and Thanatipanonda.
Umbral presentations for polynomial sequences
本文以随机变量为出发点，阐述了Rota和Taylor为经典的投影演算所发展的形式。各种各样的例子，最终在几个描述的二项式类型的序列的阴影多项式。
A unified approach to polynomial sequences with only real zeros
利用交错零点法给出了多项式序列只有实零点的新的充分条件。作为应用，我们得到了几个著名的事实，包括现实的正交多项式，匹配多项式，Narayana多项式和欧拉多项式的零点。我们还解决了某些Astratures斯塔尔属多项式证明他们为某些类别的图，同时表明，他们是假的一般。
We give simple new proofs of some Hankel determinant evaluations by Omer Egecioglu and Aleksandar Cvetkovic, Predrag Rajkovic and Milos Ivkovic and prove analogous results for sums of moments of symmetric orthogonal polynomials.
Riordan array representation of recursive polynomial sequences, orthogonal polynomial sequences, and $d$-orthogonal polynomial sequences
研究具多项式系数线性齐次递推的多项式序列的Riordan阵列表示，刻画二阶递推情形并推广到任意阶；构建任意正交多项式序列的生成矩阵框架，还将其拓展至$d$-正交多项式序列，揭示对应递推与矩序列等性质。
A strongly polynomial sequence of graphs $(G_n)$ is a sequence $(G_n)_{n\in\mathbb{N}}$ of finite graphs such that, for every graph $F$, the number of homomorphisms from $F$ to $G_n$ is a fixed polynomial function of $n$ (depending on $F$). For example, $(K_n)$ is strongly polynomial since the number of homomorphisms from $F$ to $K_n$ is the chromatic polynomial of $F$ evaluated at $n$. In earlier work of de la Harpe and Jaeger, and more recently of Averbouch, Garijo, Godlin, Goodall, Makowsky, Nešetřil, Tittmann, Zilber and others, various examples of strongly polynomial sequences and constru…
We study q-holonomic sequences that arise as the colored Jones polynomial of knots in 3-space. The minimal-order recurrence for such a sequence is called the (non-commutative) A-polynomial of a knot. Using the "method of guessing", we obtain this polynomial explicitly for the K_p = (-2, 3, 3+2p) pretzel knots for p = -5, ..., 5. This is a particularly interesting family since the pairs (K_p, -K_{-p}) are geometrically similar (in particular, scissors congruent) with similar character varieties. Our computation of the non-commutative A-polynomial (a) complements the computation of the A-polynom…
It was discovered recently by Griffin, Ono, Rolen and Zagier that the Jensen polynomials associated to many sequences have Hermite polynomial limits. We develop this theory in detail, based on the log-polynomial property which is a refinement of log-concavity and log-convexity. Applications to various partition sequences are given. An application to the sequence of factorials leads naturally to evaluating limits of generalized Laguerre polynomials.
The role of binomial type sequences in determination identities for Bell polynomials
本文讨论了涉及贝尔多项式的恒等式。利用二项式型序列的生成函数和逐次导数导出Bell多项式的几个恒等式。第一部分给出了Bell多项式与二项式型序列之间的一些关系，第二部分推广了文[4]中的结果。
Multiplicative generalised polynomial sequences
本文完全分类了由广义多项式公式定义的完全乘性序列，并在几乎处处非零的附加条件下，对一般乘性序列得出了类似的分类结果。
We give a simplified presentation of some results about recurrences of certain sequences of binomial sums in terms of (generalized) Fibonacci and Lucas polynomials.
On the structure of polynomial multicorrelation sequences arising from nilpotent actions
该研究证明幂零作用产生的多项式多重相关序列可分解为广义零序列与零序列之和，解决了Frantzikinakis的长期问题，其证明基于Walsh范数收敛定理的思想。
We evaluate the Hankel determinants of various sequences related to Bernoulli and Euler numbers and special values of the corresponding polynomials. Some of these results arise as special cases of Hankel determinants of certain sums and differences of Bernoulli and Euler polynomials, while others are consequences of a method that uses the derivatives of Bernoulli and Euler polynomials. We also obtain Hankel determinants for sequences of sums and differences of powers and for generalized Bernoulli polynomials belonging to certain Dirichlet characters with small conductors. Finally, we collect a…
We present an algorithm to find invariant poynomial transformations of integer sequences, using the classical invariant theory approach.
We derive the necessary and sufficient condition, for a given Polynomial Recurrence Sequence to converge to a given target rational K. By converge, we mean that the Nth term of the sequence, is equal to K, as N tends to positive infinity. The basic idea of our approach is to construct a univariate polynomial equation in x, whose coefficients correspond to the terms of the Sequence. The approach then obtains the condition by analyzing five cases that cover all possible real values of x. The condition can be evaluated within time that is a polynomial function of the size of the description of th…
In this paper, we introduce two types of general classes of even and odd $q$-Lidstone polynomial sequences. We prove essential properties related to them like the matrix and determinate form representation, the generating function, recurrence relation, and conjugate sequences. Some illustrative examples are included.
We introduce three spectral sequences which give some expressions of colored Jones polynomials. Each spectral sequence contains a Khovanov-type homology groups. Two of them are derived from a bicomplex of the colored Jones polynomial. The other is the spectral sequence that deduces a colored Rasmussen invariant of links. We also introduce three functors between categories of nanophrases, generalizations of links, and obtain their applications using colored Jones polynomials and their categorifications.
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