Hankel determinants of q-exponential polynomials

Some remarks and conjectures about Hankel determinants of polynomials which are related to Motzkin paths

This note collects some results and conjectures for the generating functions of the Hankel determinants of certain polynomials which are related to Motzkin paths.

Hankel determinants and shifted periodic continued fractions

Sulanke and Xin developed a continued fraction method that applies to evaluate Hankel determinants corresponding to quadratic generating functions. We use their method to give short proofs of Cigler's Hankel determinant conjectures, which were proved recently by Chang-Hu-Zhang using direct determinant computation. We find that shifted periodic continued fractions arise in our computation. We also discover and prove some new nice Hankel determinants relating to lattice paths with step set $\{(1,1),(q,0), (\ell-1,-1)\}$ for integer parameters $m,q,\ell$. Again shifted periodic continued fraction…

Shifted Hankel determinants of Catalan numbers and related results

In this (partly expository) paper we give a short overview about the close relationship between the sequence of Catalan numbers and Hankel determinants from the point of view of orthogonal polynomials and show that an analogous situation exists for more general sequences.

关于与类 Catalan 数相关序列的 Hankel 行列式的若干结果与猜想

Some results and conjectures about Hankel determinants of sequences which are related to Catalan-like numbers

本文研究类 Catalan 数对应容许矩阵各列序列的 Hankel 行列式，归纳计算机实验揭示的性质，并通过前置零行探讨后向移位情形。

On the automaticity of the Hankel determinants of a family of automatic sequences

Hankel determinants and automatic sequences are two classical subjects widely studied in Mathematics and Theoretical Computer Science. However, these two topics were considered totally independently, until in 1998, when Allouche, Peyrière, Wen and Wen proved that all the Hankel determinants of the Thue-Morse sequence are nonzero. This property allowed Bugeaud to prove that the irrationality exponents of the Thue-Morse-Mahler numbers are exactly 2. Since then, the Hankel determinants of several other automatic sequences, in particular, the paperfolding sequence, the Stern sequence, the period-d…

Some observations about Hankel determinants of the columns of Pascal triangle and related topics

This note collects some facts and conjectures about the Hankel determinants and their generating functions of the columns of Hoggatt triangles which apparently are related to combinatorial objects such as Young tableaux and Narayana numbers.

Hankel determinants of linear combinations of moments of orthogonal polynomials

We prove evaluations of Hankel determinants of linear combinations of moments of orthogonal polynomials (or, equivalently, of generating functions for Motzkin paths), thus generalising known results for Catalan numbers.

A q-analogue of Catalan Hankel determinants

In this paper we shall survey the various methods of evaluating Hankel determinants and as an illustration we evaluate some Hankel determinants of a q-analogue of Catalan numbers. Here we consider $\frac{(aq;q)_{n}}{(abq^{2};q)_{n}}$ as a q-analogue of Catalan numbers $C_{n}=\frac1{n+1}\binom{2n}{n}$, which is known as the moments of the little q-Jacobi polynomials. We also give several proofs of this q-analogue, in which we use lattice paths, the orthogonal polynomials, or the basic hypergeometric series. We also consider a q-analogue of Schröder Hankel determinants, and give a new proof of M…

Almost Product Evaluation of Hankel Determinants

An extensive literature exists describing various techniques for the evaluation of Hankel determinants. The prevailing methods such as Dodgson condensation, continued fraction expansion, LU decomposition, all produce product formulas when they are applicable. We mention the classic case of the Hankel determinants with binomial entries ${3k+2 \choose k}$ and those with entries ${3k \choose k}$; both of these classes of Hankel determinants have product form evaluations. The intermediate case, ${3k+1 \choose k}$ has not been evaluated. There is a good reason for this: these latter determinants do…

幂级数卷积的汉克尔行列式：Cigler结果的推广

Hankel Determinants for Convolution of Power Series: An Extension of Cigler's Results

该研究将Cigler关于卡特兰数卷积幂的汉克尔行列式恒等式，推广到满足特定二次方程的一般幂级数卷积场景，并给出Motzkin数卷积幂的汉克尔行列式恒等式作为应用。

Hankel determinants of middle binomial coefficients and conjectures for some polynomial extensions and modifications

The middle binomial coefficients can be interpreted as numbers of Motzkin paths which have no horizontal steps at positive heights. Assigning suitable weights gives some nice polynomial extensions. We determine the Hankel determinants and their generating functions for the middle binomial coefficients and derive many conjectures for their polynomial extensions. Finally, we explore experimentally some modifications of the middle binomial coefficients whose Hankel determinants show an interesting modular pattern and obtain some q-analogs.

On Hankel Determinants for Dyck Paths with Peaks Avoiding Multiple Classes of Heights

For any integer $m\ge 2$ and a set $V\subset \{1,\dots,m\}$, let $(m,V)$ denote the union of congruence classes of the elements in $V$ modulo $m$. We study the Hankel determinants for the number of Dyck paths with peaks avoiding the heights in the set $(m,V)$. For any set $V$ of even elements of an even modulo $m$, we give an explicit description of the sequence of Hankel determinants in terms of subsequences of arithmetic progression of integers. There are numerous instances for varied $(m,V)$ with periodic sequences of Hankel determinants. We present a sufficient condition for the set $(m,V)…

部分theta函数系数后向移位的Hankel行列式

Hankel determinants of backward shifts of the coefficients of a partial theta function

本文研究与部分theta函数系数后向移位的Hankel行列式相关的多项式，并在附录中给出该函数倒数的系数的简单公式。

关于卡特兰数卷积幂的汉克尔行列式的若干实验观察

Some experimental observations about Hankel determinants of convolution powers of Catalan numbers

本研究通过计算机实验提出了若干关于卡特兰数卷积幂汉克尔行列式的猜想，虽多数暂无证明，但仍予以公布以期学界后续证明。

卡特兰数的移位汉克尔行列式及相关结果 II：后向移位

Shifted Hankel determinants of Catalan numbers and related results II: Backward shifts

该研究通过前置零定义序列后向移位的汉克尔行列式，得到卡特兰数及相关数与多项式序列的相关结果，并对卡特兰数卷积幂序列提出猜想。

Hankel determinants for convolution powers of Catalan numbers

The Hankel determinants $\left(\frac{r}{2(i+j)+r}\binom{2(i+j)+r}{i+j}\right)_{0\leq i,j \leq n-1}$ of the convolution powers of Catalan numbers were considered by Cigler and by Cigler and Krattenthaler. We evaluate these determinants for $r\le 31$ by finding shifted periodic continued fractions, which arose in application of Sulanke and Xin's continued fraction method. These include some of the conjectures of Cigler as special cases. We also conjectured a polynomial characterization of these determinants. The same technique is used to evaluate the Hankel determinants $\left(\binom{2(i+j)+r}{i…

Catalan 数卷积幂的 Hankel 行列式再探

Hankel determinants of convolution powers of Catalan numbers revisited

本文利用推广的 Andrews-Wimp 定理，简洁证明了 Catalan 数卷积幂平移的 Hankel 行列式，并推广至 Narayana 多项式。

巴里四个汉克尔行列式猜想的证明

Proof of Barry's Four Hankel Determinant Conjectures

巴里对四个有理族中心变换的汉克尔变换提出四个猜想，本文在共同代数框架内，通过将汉克尔行列式解释为格拉姆行列式，利用移位首一 切比雪夫多项式基揭示矩阵结构，从而证明了这四个猜想。

r阶广义错排多项式的Hankel行列式

The Hankel determinants for the generalized derangement polynomials of order r

本文引入r阶广义错排多项式，建立其恒等式与递推关系，并用概率方法给出相关Hankel行列式的显式表达式，进而推导错排多项式与循环错排数的Hankel行列式。

Hankel determinants of a Sturmian sequence

Let $τ$ be the substitution $1\to 101$ and $0\to 1$ on the alphabet $\{0,1\}$. The fixed point of $τ$ leading by 1, denoted by $\mathbf{s}$, is a Sturmian sequence. We first give a characterization of $\mathbf{s}$ using $f$-representation. Then we show that the distribution of zeros in the determinants induces a partition of integer lattices in the first quadrant. Combining those properties, we give the explicit values of the Hankel determinants $H_{m,n}$ of $\mathbf{s}$ for all $m\ge 0$ and $n\ge 1$.

$(α, β)$型Somos 4汉克尔行列式的一个充分条件

A sufficient condition for $(α, β)$ Somos $4$ Hankel determinants

本研究发现Xin提出的用于解决Somos 4汉克尔行列式猜想的递推系统，是$(α, β)$型Somos 4序列的充分条件，可统一证明Barry提出的4个相关猜想。

Proof of a Conjecture on Hankel Determinants for Dyck Paths with Restricted Peak Heights

For any integer $m\geq 2$ and $r \in \{1,\dots, m\}$, let $f_n^{m,r}$ denote the number of $n$-Dyck paths whose peak's heights are $im+r$ for some integer $i$. We find the generating function of $f_n^{m,r}$ satisfies a simple algebraic functional equation of degree $2$. The $r=m$ case is particularly nice and we give a combinatorial proof. By using the Sulanke and Xin's continued fraction method, we calculate the Hankel determinants for $f_n^{m,r}$. The special case $r=m$ of our result solves a conjecture proposed by Chien, Eu and Fu. We also enriched the class of eventually periodic Hankel de…

A prime sensitive Hankel determinant of Jacobi symbol enumerators

We show that the determinant of a Hankel matrix of odd dimension n whose entries are the enumerators of the Jacobi symbols which depend on the row and the column indices vanishes iff n is composite. If the dimension is a prime p, then the determinant evaluates to a polynomial of degree p-1 which is the product of a power of p and the generating polynomial of the partial sums of Legendre symbols. The sign of the determinant is determined by the quadratic character of -1 modulo p. The proof of the evaluation makes use of elementary properties of Legendre symbols, quadratic Gauss sums and orthogo…

Some observations on the Rueppel sequence and associated Hankel determinants

Starting with a definition based on the Catalan numbers, we carry out an empirical study of the Rueppel sequence. We use the Hankel transform as the main technique. By means of this transform we find links to such sequences as the Jacobi sequence and the paper-folding sequence. We also study several number arrays defined by this sequence, which are analogs of the important Catalan triangles of combinatorics. We identify a sequence related to the Golay-Rudin-Shapiro sequence that plays a fundamental role in linking the Hankel transforms of the paper with classical sequences. Examples are given …

卷积Catalan数的Hankel行列式与非相交格路径：Cigler猜想的一个双射证明

Hankel Determinants of convoluted Catalan numbers and nonintersecting lattice paths: A bijective proof of Cigler's Conjecture

本文利用反射原理、Lindström-Gessel-Viennot方法及非相交格路径的反射叠合构造，为Cigler关于卷积Catalan数Hankel行列式的猜想提供了一个双射证明。

A simple approach to some Hankel determinants

I give simple elementary proofs for some well-known Hankel determinants and their q-analogues.

Narayana多项式卷积幂的Hankel行列式

Hankel determinants for convolution powers of Narayana polynomials

本文通过符号计算程序验证并推广了Narayana多项式卷积幂的Hankel行列式猜想，采用猜测与证明策略，利用Hankel连分数技术，最终给出简洁闭合表达式。

A curious class of Hankel determinants

We consider Hankel determinants of the sequence of Catalan numbers modulo 2 (interpreted as integers 0 and 1) and more generally Hankel determinants where the sum over all permutations reduces to a single signed permutation.

A Multilinear Operator for Almost Product Evaluation of Hankel Determinants

In a recent paper we have presented a method to evaluate certain Hankel determinants as almost products; i.e. as a sum of a small number of products. The technique to find the explicit form of the almost product relies on differential-convolution equations and trace calculations. In the trace calculations a number of intermediate nonlinear terms involving determinants occur, but only to cancel out in the end. In this paper, we introduce a class of multilinear operators γacting on tuples of matrices as an alternative to the trace method. These operators do not produce extraneous nonlinear terms…

Some determinants of path generating functions, II

We evaluate Hankel determinants of matrices in which the entries are generating functions for paths consisting of up-steps, down-steps and level steps with a fixed starting point but variable end point. By specialisation, these determinant evaluations have numerous corollaries. In particular, one consequence is that the Hankel determinant of Motzkin prefix numbers equals 1, regardless of the size of the Hankel matrix.

Riordan 阵列与 OEIS A005773 的移位 Hankel 行列式

Riordan Arrays and Shifted Hankel Determinants of OEIS A005773

本文通过边界分解统一两个 Riordan 阵列，得到 Motzkin 端点分布的周期三变换及按高度模 3 的路径公式，并导出第三、第四 Hankel 行列式的闭式解。

Hankel determinants of the Cantor sequence

In the paper, we give the recurrent equations of the Hankel determinants of the Cantor sequence, and show that the Hankel determinants as a double sequence is 3-automatic. With the help of the Hankel determinants, we prove that the irrationality exponent of the Cantor number, i.e. the transcendental number with Cantor sequence as its b-ary expansion, equals 2.

Hankel determinants of harmonic numbers and related topics

We give an overview of known results about Hilbert matrices from the point of view of orthogonal polynomials and compute Hankel determinants of harmonic numbers and related topics.

Hankel Continued fractions and Hankel determinants of the Euler numbers

The Euler numbers occur in the Taylor expansion of $\tan(x)+\sec(x)$. Since Stieltjes, continued fractions and Hankel determinants of the even Euler numbers, on the one hand, of the odd Euler numbers, on the other hand, have been widely studied separately. However, no Hankel determinants of the (mixed) Euler numbers have been obtained and explicitly calculated. The reason for that is that some Hankel determinants of the Euler numbers are null. This implies that the Jacobi continued fraction of the Euler numbers does not exist. In the present paper, this obstacle is bypassed by using the Hankel…

Hankel Determinant Calculus for the Thue-Morse and related sequences

The Hankel determinants of certain automatic sequences $f$ are evaluated, based on a calculation modulo a prime number. In most cases, the Hankel determinants of automatic sequences do not have any closed-form expressions; the traditional methods, such as $LU$-decompo\-si\-tion and Jacobi continued fraction, cannot be applied directly. Our method is based on a simple idea: the Hankel determinants of each sequence $g$ equal to $f$ modulo $p$ are equal to the Hankel determinants of $f$ modulo $p$. The clue then consists of finding a nice sequence $g$, whose Hankel determinants have closed-form e…

Catalan numbers, Hankel determinants and Fibonacci polynomials

This (partly expository) paper originated from the study of Hankel determinants of convolution powers of Catalan numbers and of Narayana polynomials. This led to some Hankel determinants of signed Catalan numbers whose values are multiples of Fibonacci numbers and to some Hankel determinants of signed central binomial coefficients whose values are multiples of Lucas numbers. Most proofs are computational but we also include a combinatorial one due to Christian Krattenthaler. Finally we formulate some conjectures.

Hankel Determinants of sequences related to Bernoulli and Euler Polynomials

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…

A special class of Hankel determinants

In this expository paper we compute Hankel determinants of some sequences whose generating functions are given by C-fractions and derive orthogonality properties for associated polynomials.

Hankel Determinants for Some Common Lattice Paths

For a single value of $\ell$, let $f(n,\ell)$ denote the number of lattice paths that use the steps $(1,1)$, $(1,-1)$, and $(\ell,0)$, that run from $(0,0)$ to $(n,0)$, and that never run below the horizontal axis. Equivalently, $f(n,\ell)$ satisfies the quadratic functional equation $F(x) = \sum_{n\ge 0}f(n,\ell) x^n = 1+x^{\ell}F(x)+x^2F(x)^2.$ Let $H_n$ denote the $n$ by $n$ Hankel matrix, defined so that $[H_n]_{i,j} = f(i+j-2,\ell)$. Here we investigate the values of such determinants where $\ell = 0,1,2,3$. For $\ell = 0,1,2$ we are able to employ the Gessel-Viennot-Lindström method. For…

Motzkin数的卷积幂的Hankel行列式

Hankel determinants for convolution powers of Motzkin numbers

本文通过移位周期连分数求出了$r\leq 27$的Motzkin数卷积幂的Hankel行列式，并提出了其多项式刻画猜想。

The Generating Function of Ternary Trees and Continued Fractions

Michael Somos conjectured a relation between Hankel determinants whose entries $\frac 1{2n+1}\binom{3n}n$ count ternary trees and the number of certain plane partitions and alternating sign matrices. Tamm evaluated these determinants by showing that the generating function for these entries has a continued fraction that is a special case of Gauss's continued fraction for a quotient of hypergeometric series. We give a systematic application of the continued fraction method to a number of similar Hankel determinants. We also describe a simple method for transforming determinants using the genera…

一种计算直至 n 阶 Hankel 行列式的 \(O(n\log^2n)\) 算法

An $O(n\log^2n)$ Algorithm for Computing Hankel Determinants up to Order $n$

研究计算 Hankel 行列式的算法，通过探究 Hankel 连分数与广义 Sturm 序列的关系，开发出用 \(O(n \log^2 n)\) 次算术运算计算 Hankel 行列式的新算法，并探究了广义 Sturm 序列与 Hankel 矩阵符号的联系。

A combinatorial proof of the non-vanishing of Hankel determinants of the Thue--Morse sequence

In 1998, Allouche, Peyrière, Wen and Wen established that the Hankel determinants associated with the Thue--Morse sequence on $\{-1, 1\}$ are always nonzero. Their proof depends on a set of sixteen recurrence relations. We present an alternative, purely combinatorial proof of the same result. We also re-prove a recent result of Coons on the non-vanishing of the Hankel determinants associated to two other classical integer sequences.

与二项式变换和 $q$-二项式变换相关的 Hankel 行列式中的首项系数

Leading coefficient in the Hankel determinants related to binomial and $q$-binomial transforms

本文将二项式变换下 Hankel 行列式不变的经典结论推广至 $q$-二项式变换，研究其首项系数行为，并利用特殊二项式变换分析偶数下标 Bernoulli 多项式情形。

Hankel determinants of sums of consecutive weighted Schröder numbers

For a real number $t$, let $r_\ell(t)$ be the total weight of all $t$-large Schröder paths of length $\ell$, and $s_\ell(t)$ be the total weight of all $t$-small Schröder paths of length $\ell$. For constants $α, β$, in this article we derive recurrence formulae for the determinats of the Hankel matrices $\det_{1\le i,j\le n} (αr_{i+j-2}(t) +βr_{i+j-1}(t))$, $\det_{1\le i,j\le n} (αr_{i+j-1}(t) +βr_{i+j}(t))$, $\det_{1\le i,j\le n} (αs_{i+j-2}(t) +βs_{i+j-1}(t))$, and $\det_{1\le i,j\le n} (αs_{i+j-1}(t) +βs_{i+j}(t))$ combinatorially via suitable lattice path models.

类卡特兰序列的汉克尔行列式

Hankel determinants of Catalan-like sequences

本文计算了源于非相交Motzkin meander加权计数的类卡特兰序列的移位汉克尔行列式，其中部分为新发现或已被提出，还应用其验证了Cigler与Krattenthaler的猜想二项式行列式恒等式。

On $t$-extensions of the Hankel determinants of certain automatic sequences

In 1998, Allouche, Peyrière, Wen and Wen considered the Thue--Morse sequence, and proved that all the Hankel determinants of the period-doubling sequence are odd integral numbers. We speak of $t$-extension when the entries along the diagonal in the Hankel determinant are all multiplied by~$t$. Then we prove that the $t$-extension of each Hankel determinant of the period-doubling sequence is a polynomial in $t$, whose leading coefficient is the {\it only one} to be an odd integral number. Our proof makes use of the combinatorial set-up developed by Bugeaud and Han, which appears to be very suit…

Hankel-type determinants for some combinatorial sequences

In this paper we confirm several conjectures of Z.-W. Sun on Hankel-type determinants for some combinatorial sequences including Franel numbers, Domb numbers and Apéry numbers. For any nonnegative integer $n$, define \begin{gather*}f_n:=\sum_{k=0}^n\binom nk^3,\ D_n:=\sum_{k=0}^n\binom nk^2\binom{2k}k\binom{2(n-k)}{n-k}, b_n:=\sum_{k=0}^n\binom nk^2\binom{n+k}k,\ A_n:=\sum_{k=0}^n\binom nk^2\binom{n+k}k^2. \end{gather*} For $n=0,1,2,\ldots$, we show that $6^{-n}|f_{i+j}|_{0\leq i,j\leq n}$ and $12^{-n}|D_{i+j}|_{0\le i,j\le n}$ are positive odd integers, and $10^{-n}|b_{i+j}|_{0\leq i,j\leq n}…

Some nice Hankel determinants

I study Hankel determinants of a class of sequences which can be interpreted as generalizations of the Catalan numbers and the central binomial coefficients. They follow a modular pattern with a frequent appearance of zeroes, so that the theory of orthogonal polynomials is not applicable. Even so our theorems and conjectures show some similarity with the relations between Catalan numbers and Fibonacci polynomials or between central binomial coefficients and Lucas polynomials.

An interesting class of Hankel determinants

For small $r$ the Hankel determinants $d_r(n)$ of the sequence $\left({2n+r\choose n}\right)_{n\ge 0}$ are easy to guess and show an interesting modular pattern. For arbitrary $r$ and $n$ no closed formulae are known, but for each positive integer $r$ the special values $d_r(rn)$, $d_r(rn+1)$, and $d_r(rn+\lfloor\frac{r+1}{2}\rfloor)$ have nice values which will be proved in this paper.

Pfaffian decomposition and a Pfaffian analogue of q-Catalan Hankel determinants

Motivated by the Hankel determinant evaluation of moment sequences, we study a kind of Pfaffian analogue evaluation. We prove an LU-decomposition analogue for skew-symmetric matrices, called Pfaffian decomposition. We then apply this formula to evaluate Pfaffians related to some moment sequences of classical orthogonal polynomials. In particular we obtain a product formula for a kind of q-Catalan Pfaffians. We also establish a connection between our Pfaffian formulas and certain weighted enumeration of shifted reverse plane partitions.

$q$-变形实数的连分数、$\{-1,0,1\}$-Hankel行列式及Somos-Gale-Robinson序列

Continued fractions for $q$-deformed real numbers, $\{-1,0,1\}$-Hankel determinants, and Somos-Gale-Robinson sequences

本文研究$q$-变形实数的连分数展开，发现新例子，并找到使前几个Hankel行列式仅含$-1,0,1$的无穷幂级数族，这些序列满足Somos和Gale-Robinson递推。

Hankel continued fraction and its applications

The Hankel determinants of a given power series $f$ can be evaluated by using the Jacobi continued fraction expansion of $f$. However the existence of the Jacobi continued fraction needs that all Hankel determinants of $f$ are nonzero. We introduce {\it Hankel continued fraction}, whose existene and unicity are guaranteed without any condition for the power series $f$. The Hankel determinants can also be evaluated by using the Hankel continued fraction. It is well known that the continued fraction expansion of a quadratic irrational number is ultimately periodic. We prove a similar result for …

Hankel Determinants of Certain Sequences Of Bernoulli Polynomials: A Direct Proof of an Inverse Matrix Entry from Statistics

We calculate the Hankel determinants of sequences of Bernoulli polynomials. This corresponding Hankel matrix comes from statistically estimating the variance in nonparametric regression. Besides its entries' natural and deep connection with Bernoulli polynomials, a special case of the matrix can be constructed from a corresponding Vandermonde matrix. As a result, instead of asymptotic analysis, we give a direct proof of calculating an entry of its inverse.