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 Moztkin Hankel determinants using an addition formula for ${}_2F_{1}$.
We provide a context around a conjectured closed form for the Hankel transform of linear combinations of consecutive pairs of Catalan numbers. This generalizes the formula for the Hankel transforms of the shifted Catalan numbers and the known results for linear combinations of pairs of Catalan numbers. Many interesting number triangles emerge, some of which we analyze using the language of Riordan arrays.
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.
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…
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.
We prove that the inverse of the Hankel matrix of the reciprocals of the Catalan numbers has integer entries. We generalize the result to an infinite family of generalized Catalan numbers. The Hankel matrices that we consider are associated with orthogonal polynomials that are variants of Jacobi polynomials. Our proofs use these polynomials and computer algebra based on Wilf-Zeilberger theory.
We give conjectures on the form of families of integer sequences whose Hankel transforms are, respectively, $(α, β)$ Somos $4$ sequences, $(α, 0, γ)$ Somos $6$ sequences, and $(α, β, γ, δ)$ Somos $8$ sequences, for particular values of $α$, $β$, $γ$, $δ$ which we describe. The sequences involved can be described in terms of the application of certain stretched Riordan arrays to the Catalan numbers, accompanied by a (sequence) Hankel transform. The combination of Riordan array and the Catalan numbers results from the study of certain generalized Jacobi continued fractions, based on the Counting…
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 …
In this (mostly expository) paper I want to share some observations prompted by a class of matrices whose determinants are Catalan numbers. Considering different methods of proof we obtain some generalizations and q-analogues and connections with Hankel determinants. Finally we state some conjectures.
This note collects some results and conjectures for the generating functions of the Hankel determinants of certain polynomials which are related to Motzkin paths.
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.
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…
Following Benjamin et al., a matrix with entries being sums of two neighbouring Catalan numbers is considered. Its LU-decomposition is given, by guessing the results and later prove it by computer algebra, with lots of human help. Specializing a parameter, the determinant turns out to be a Fibonacci number with odd index, confirming earlier results, obtained back then by combinatorial methods.
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…
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.
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.
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)…
Proof of Barry's Four Hankel Determinant Conjectures
Center for Combinatorics, LPMC Nankai University（南开大学组合数学中心，LPMC）；School of Mathematics and Statistics, North China University of Water Resources and Electric Power（华北水利水电大学数学与统计学院）
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 $n\ge 1$.
In this paper we give some sufficient conditions for the nonnegativity of immanants of square submatrices of Catalan-Stieltjes matrices and their corresponding Hankel matrices. To obtain these sufficient conditions, we construct new planar networks with a recursive nature for Catalan-Stieltjes matrices. As applications, we provide a unified way to produce inequalities for many combinatorial polynomials, such as the Eulerian polynomials, Schröder polynomials and Narayana polynomials.
We show that the Catalan-Schroeder convolution recurrences and their higher order generalizations can be solved using Riordan arrays and the Catalan numbers. We investigate the Hankel transforms of many of the recurrence solutions, and indicate that Somos $4$ sequences often arise. We exhibit relations between recurrences, Riordan arrays, elliptic curves and Somos $4$ sequences. We furthermore indicate how one can associate a family of orthogonal polynomials to a point on an elliptic curve, whose moments are related to recurrence solutions.
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…
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.
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.
Smith normal form evaluations found by Bessenrodt and Stanley for some Hankel matrices of q-Catalan numbers are proven in two ways. One argument generalizes the Bessenrodt-Stanley results for the Smith normal form of a certain multivariate matrix that refines one studied by Berlekamp, Carlitz, Roselle, and Scoville. The second argument, which uses orthogonal polynomials, generalizes to a number of other Hankel matrices, Toeplitz matrices, and Gram matrices. It gives new results for q-Catalan numbers, q-Motzkin numbers, q-Schröder numbers, q-Stirling numbers, q-matching numbers, q-factorials, q…
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.
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…
The Generating Function of Ternary Trees and Continued Fractions
Michael Somos证明了Hankel行列式的元素$\frac 1{2n+1}\binom{3n}n$计数三元树与某些平面划分和交替符号矩阵的数量之间的关系。塔姆评价这些决定表明，生成函数为这些条目有一个连分数，这是一个特殊情况下高斯的连分数的商的超几何级数。我们给出了一个系统的应用程序的连分数方法的一些类似的汉克尔行列式。我们还描述了一个简单的方法来转换行列式使用生成函数的条目。在这种方式中，我们转换索莫斯的汉克尔行列式已知的决定因素，我们得到，高达3的幂，一个汉克尔行列式的交替符号矩阵的数量。我们得到了一个组合证明，在不相交的路径，涉及的三元树的数量和更一般的行列式恒等式涉及的$r$ -ary树的数量的行列式恒等式。
An $O(n\log^2n)$ Algorithm for Computing Hankel Determinants up to Order $n$
Center for Combinatorics, LPMCNankai University（南开大学组合数学中心）；Nankai University（南开大学）；School of Mathematical Sciences,Capital Normal University（首都师范大学数学科学学院）；Capital Normal University（首都师范大学）；School of Mathematics and StatisticsBeijing Institute of Technology（北京理工大学数学与统计学院）；Beijing Institute of Technology（北京理工大学）
