We present a formula that expresses the Hankel determinants of a linear combination of length $d+1$ of moments of orthogonal polynomials in terms of a $dtimes d$ determinant of the orthogonal polynomials. This formula exists somehow hidden in the fol
klore of the theory of orthogonal polynomials but deserves to be better known, and be presented correctly and with full proof. We present four fundamentally different proofs, one that uses classical formulae from the theory of orthogonal polynomials, one that uses a vanishing argument and is due to Elouafi [J. Math. Anal. Appl. 431} (2015), 1253-1274] (but given in an incomplete form there), one that is inspired by random matrix theory and is due to Brezin and Hikami [Comm. Math. Phys. 214 (2000), 111-135], and one that uses (Dodgson) condensation. We give two applications of the formula. In the first application, we explain how to compute such Hankel determinants in a singular case. The second application concerns the linear recurrence of such Hankel determinants for a certain class of moments that covers numerous classical combinatorial sequences, including Catalan numbers, Motzkin numbers, central binomial coefficients, central trinomial coefficients, central Delannoy numbers, Schroder numbers, Riordan numbers, and Fine numbers.
The theme of this article is a reciprocity between bounded up-down paths and bounded alternating sequences. Roughly speaking, this ``reciprocity manifests itself by the fact that the extension of the sequence of numbers of paths of length $n$, consis
ting of diagonal up- and down-steps and being confined to a strip of bounded width, to negative $n$ produces numbers of alternating sequences of integers that are bounded from below and from above. We show that this reciprocity extends to families of non-intersecting bounded up-down paths and certain arrays of alternating sequences which we call alternating tableaux. We provide as well weight
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.
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 evaluation
s 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.
We generalise Dworks theory of $p$-adic formal congruences from the univariate to a multi-variate setting. We apply our results to prove integrality assertions on the Taylor coefficients of (multi-variable) mirror maps. More precisely, with $mathbf z
=(z_1,z_2,...,z_d)$, we show that the Taylor coefficients of the multi-variable series $q(mathbf z)=z_iexp(G(mathbf z)/F(mathbf z))$ are integers, where $F(mathbf z)$ and $G(mathbf z)+log(z_i) F(mathbf z)$, $i=1,2,...,d$, are specific solutions of certain GKZ systems. This result implies the integrality of the Taylor coefficients of numerous families of multi-variable mirror maps of Calabi-Yau complete intersections in weighted projective spaces, as well as of many one-variable mirror maps in the Tables of Calabi-Yau equations [arXiv:math/0507430] of Almkvist, van Enckevort, van Straten and Zudilin. In particular, our results prove a conjecture of Batyrev and van Straten in [Comm. Math. Phys. 168 (1995), 493-533] on the integrality of the Taylor coefficients of canonical coordinates for a large family of such coordinates in several variables.
Given a finite irreducible Coxeter group $W$, a positive integer $d$, and types $T_1,T_2,...,T_d$ (in the sense of the classification of finite Coxeter groups), we compute the number of decompositions $c=si_1si_2 cdotssi_d$ of a Coxeter element $c$ o
f $W$, such that $si_i$ is a Coxeter element in a subgroup of type $T_i$ in $W$, $i=1,2,...,d$, and such that the factorisation is minimal in the sense that the sum of the ranks of the $T_i$s, $i=1,2,...,d$, equals the rank of $W$. For the exceptional types, these decomposition numbers have been computed by the first author. The type $A_n$ decomposition numbers have been computed by Goulden and Jackson, albeit using a somewhat different language. We explain how to extract the type $B_n$ decomposition numbers from results of Bona, Bousquet, Labelle and Leroux on map enumeration. Our formula for the type $D_n$ decomposition numbers is new. These results are then used to determine, for a fixed positive integer $l$ and fixed integers $r_1le r_2le ...le r_l$, the number of multi-chains $pi_1le pi_2le ...le pi_l$ in Armstrongs generalised non-crossing partitions poset, where the poset rank of $pi_i$ equals $r_i$, and where the block structure of $pi_1$ is prescribed. We demonstrate that this result implies all known enumerative results on ordinary and generalised non-crossing partitions via appropriate summations. Surprisingly, this result on multi-chain enumeration is new even for the original non-crossing partitions of Kreweras. Moreover, the result allows one to solve the problem of rank-selected chain enumeration in the type $D_n$ generalised non-crossing partitions poset, which, in turn, leads to a proof of Armstrongs $F=M$ Conjecture in type $D_n$.
We show that the Taylor coefficients of the series ${bf q}(z)=zexp({bf G}(z)/{bf F}(z))$ are integers, where ${bf F}(z)$ and ${bf G}(z)+log(z) {bf F}(z)$ are specific solutions of certain hypergeometric differential equations with maximal unipotent m
onodromy at $z=0$. We also address the question of finding the largest integer $u$ such that the Taylor coefficients of $(z ^{-1}{bf q}(z))^{1/u}$ are still integers. As consequences, we are able to prove numerous integrality results for the Taylor coefficients of mirror maps of Calabi-Yau complete intersections in weighted projective spaces, which improve and refine previous results by Lian and Yau, and by Zudilin. In particular, we prove the general ``integrality conjecture of Zudilin about these mirror maps. A further outcome of the present study is the determination of the Dwork-Kontsevich sequence $(u_N)_{Nge1}$, where $u_N$ is the largest integer such that $q(z)^{1/u_N}$ is a series with integer coefficients, where $q(z)=exp(F(z)/G(z))$, $F(z)=sum_{m=0} ^{infty} (Nm)! z^m/m!^N$ and $G(z)=sum_{m=1} ^{infty} (H_{Nm}-H_m)(Nm)! z^m/m!^N$, with $H_n$ denoting the $n$-th harmonic number, conditional on the conjecture that there are no prime number $p$ and integer $N$ such that the $p$-adic valuation of $H_N-1$ is strictly greater than 3.
We investigate arithmetic properties of values of the entire function $$ F(z)=F_q(z;lambda)=sum_{n=0}^inftyfrac{z^n}{prod_{j=1}^n(q^j-lambda)}, qquad |q|>1, quad lambda otin q^{mathbb Z_{>0}}, $$ that includes as special cases the Tschakaloff functio
n ($lambda=0$) and the $q$-exponential function ($lambda=1$). In particular, we prove the non-quadraticity of the numbers $F_q(alpha;lambda)$ for integral $q$, rational $lambda$ and $alpha otin-lambda q^{mathbb Z_{>0}}$, $alpha e0$.
It is shown how Andrews multidimensional extension of Watsons transformation between a very-well-poised $_8phi_7$-series and a balanced $_4phi_3$-series can be used to give a straightforward proof of a conjecture of Zudilin and the second author on t
he arithmetic behaviour of the coefficients of certain linear forms of 1 and Catalans constant. This proof is considerably simpler and more stream-lined than the first proof, due to the second author.
