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C. Krattenthaler

Publications and source records attributed to C. Krattenthaler.

13 recordsLinked to original sources
arXiv2021

A determinant identity for moments of orthogonal polynomials that implies Uvarov's formula for the orthogonal polynomials of rationally related densities

Let $p_n(x)$, $n=0,1,\dots$, be the orthogonal polynomials with respect to a given density $dμ(x)$. Furthermore, let $dν(x)$ be a density which arises from $dμ(x)$ by multiplication by a rational function in $x$. We prove a formula that expresses the Hankel determinants of moments of $dν(x)$ in terms of a determinant involving the orthogonal polynomials $p_n(x)$ and associated functions $q_n(x)=\int p_n(u) \,dμ(u)/(x-u)$. Uvarov's formula for the orthogonal polynomials with respect to $dν(x)$ is a corollary of our theorem. Our result generalises a Hankel determinant formula for the case where the rational function is a polynomial that existed somehow hidden in the folklore of the theory of orthogonal polynomials but has been stated explicitly only relatively recently (see [arXiv:2101.04225]). Our theorem can be interpreted in a two-fold way: analytically or in the sense of formal series. We apply our theorem to derive several curious Hankel determinant evaluations.

math.CA↗
arXiv2014

Generalised Apéry numbers modulo $9$

We characterise the modular behaviour of (generalised) Apéry number modulo $9$, thereby in particular establishing two conjectures in "A method for determining the mod-$3^k$ behaviour of recursive sequences" [arXiv:1308.2856].

math.NT↗
arXiv2013

Some divisibility properties of binomial and q-binomial coefficients

We first prove that if $a$ has a prime factor not dividing $b$ then there are infinitely many positive integers $n$ such that $\binom {an+bn} {an}$ is not divisible by $bn+1$. This confirms a recent conjecture of Z.-W. Sun. Moreover, we provide some new divisibility properties of binomial coefficients: for example, we prove that $\binom {12n} {3n}$ and $\binom {12n} {4n}$ are divisible by $6n-1$, and that $\binom {330n} {88n}$ is divisible by $66n-1$, for all positive integers $n$. As we show, the latter results are in fact consequences of divisibility and positivity results for quotients of $q$-binomial coefficients by $q$-integers, generalizing the positivity of $q$-Catalan numbers. We also put forward several related conjectures.

math.NT↗
arXiv2011

Superconformal indices of three-dimensional theories related by mirror symmetry

Recently, Kim and Imamura and Yokoyama derived an exact formula for superconformal indices in three-dimensional field theories. Using their results, we prove analytically the equality of superconformal indices in some U(1)-gauge group theories related by the mirror symmetry. The proofs are based on the well known identities of the theory of $q$-special functions. We also suggest the general index formula taking into account the $U(1)_J$ global symmetry present for abelian theories.

hep-th↗
arXiv2005

Some composition determinants

We compute two parametric determinants in which rows and columns are indexed by compositions, where in one determinant the entries are products of binomial coefficients, while in the other the entries are products of powers. These results generalize previous determinant evaluations due to the first and third author [SIAM J. Matrix Anal. Appl. 23 (2001), 459--471] and ["A polynomial generalization of the power-compositions determinant," Linear Multilinear Algebra (to appear)], and they prove two conjectures of the second author ["Advanced determinant calculus: a complement," preliminary version].

math.CO↗
arXiv2004

Hyperg{é}om{é}trie et fonction z{ê}ta de Riemann

We prove the second author's "denominator conjecture" [40] concerning the common denominators of coefficients of certain linear forms in zeta values. These forms were recently constructed to obtain lower bounds for the dimension of the vector space over $\mathbb Q$ spanned by $1,ζ(m),ζ(m+2),...,ζ(m+2h)$, where $m$ and $h$ are integers such that $m\ge2$ and $h\ge0$. In particular, we immediately get the following results as corollaries: at least one of the eight numbers $ζ(5),ζ(7),...,ζ(19)$ is irrational, and there exists an odd integer $j$ between 5 and 165 such that 1, $ζ(3)$ and $ζ(j)$ are linearly independent over $\mathbb{Q}$. This strengthens some recent results in [41] and [8], respectively. We also prove a related conjecture, due to Vasilyev [49], and as well a conjecture, due to Zudilin [55], on certain rational approximations of $ζ(4)$. The proofs are based on a hypergeometric identity between a single sum and a multiple sum due to Andrews [3]. We hope that it will be possible to apply our construction to the more general linear forms constructed by Zudilin [56], with the ultimate goal of strengthening his result that one of the numbers $ζ(5),ζ(7),ζ(9),ζ(11)$ is irrational.

math.NT↗
arXiv2004

Analytic proof of the partition identity $A_{5,3,3}(n) = B^0_{5,3,3}(n)$

In this paper we give an analytic proof of the identity $A_{5,3,3}(n) =B^0_{5,3,3}(n)$, where $A_{5,3,3}(n)$ counts the number of partitions of $n$ subject to certain restrictions on their parts, and $B^0_{5,3,3}(n)$ counts the number of partitions of $n$ subject to certain other restrictions on their parts, both too long to be stated in the abstract. Our proof establishes actually a refinement of that partition identity. The original identity was first discovered by the first author jointly with M. Ruby Salestina and S. R. Sudarshan in ["A new theorem on partitions," Proc. Int. Conference on Special Functions, IMSC, Chennai, India, September 23-27, 2002; to appear], where it was also given a combinatorial proof, thus responding a question of Andrews.

math.CO↗
arXiv2003

Proof of two conjectures of Zuber on fully packed loop configurations

Two conjectures of Zuber [``On the counting of fully packed loops configurations. Some new conjectures,'' preprint] on the enumeration of configurations in the fully packed loop model on the square grid with periodic boundary conditions, which have a prescribed linkage pattern, are proved. Following an idea of de Gier [``Loops, matchings and alternating-sign matrices,'' Discrete Math., to appear], the proofs are based on bijections between such fully packed loop configurations and rhombus tilings, and the hook-content formula for semistandard tableaux.

math.CO↗
arXiv2003

Séries hypergéométriques basiques, $q$-analogues des valeurs de la fonction zêta et séries d'Eisenstein

Nous étudions la nature arithmétique de $q$-analogues des valeurs $ζ(s)$ de la fonction zêta de Riemann, notamment des valeurs des fonctions $ζ_q(s)= \sum_{k=1} ^{\infty}q^k \sum_{d\mid k} ^{}d^{s-1}$, $s=1,2,...$, o{ù} $q$ est un nombre complexe, $| q|<1$ (ces fonctions sont intimenent liées au monde automorphe). Le théorème principal de cet article montre que, si $1/q$ est un nombre entier différent de $\pm1$ et si $M$ est un nombre impair suffisamment grand, alors la dimension de l'espace vectoriel engendré sur $\mathbb Q$ par $1,ζ_q(3), ζ_q(5),..., ζ_q(M)$ est au moins $c_1\cdot\sqrt{M}$, avec $c_1=0,3358$. Ce résultat peut être considéré comme un $q$-analogue du résultat de \cite{ri, br}, qui affirme que la dimension de l'espace vectoriel engendré sur $\mathbb Q$ par $1,ζ(3),ζ(5),...,ζ(M)$ est au moins $c_2\cdot\log{M}$, avec $c_2=0,5906$. Pour les mêmes valeurs de $q$, une minoration similaire pour les valeurs $ζ_q(s)$ aux entiers $s$ pairs nous permet de redémontrer un cas particulier d'un r{é}sultat de Bertrand \cite{ber} qui affirme la transcendance sur $\mathbb{Q}$ de l'une des deux séries d'Eisenstein $E_4(q)$ et $E_6(q)$ pour tout nombre complexe $q$ tel que $0<| q| <1$.

math.NT↗
arXiv1999

Enumeration of lozenge tilings of hexagons with a central triangular hole

We deal with unweighted and weighted enumerations of lozenge tilings of a hexagon with side lengths $a,b+m,c,a+m,b,c+m$, where an equilateral triangle of side length $m$ has been removed from the center. We give closed formulas for the plain enumeration and for a certain $(-1)$-enumeration of these lozenge tilings. In the case that $a=b=c$, we also provide closed formulas for certain weighted enumerations of those lozenge tilings that are cyclically symmetric. For $m=0$, the latter formulas specialize to statements about weighted enumerations of cyclically symmetric plane partitions. One such specialization gives a proof of a conjecture of Stembridge on a certain weighted count of cyclically symmetric plane partitions. The tools employed in our proofs are nonstandard applications of the theory of nonintersecting lattice paths and determinant evaluations. In particular, we evaluate the determinants $\det_{0\le i,j\le n-1}\big(\om δ_{ij}+\binom {m+i+j}j\big)$, where $\om$ is any 6th root of unity. These determinant evaluations are variations of a famous result due to Andrews (Invent. Math. 53 (1979), 193--225), which corresponds to $\om=1$.

math.CO↗
arXiv1997

The number of centered lozenge tilings of a symmetric hexagon

Propp conjectured that the number of lozenge tilings of a semiregular hexagon of sides $2n-1$, $2n-1$ and $2n$ which contain the central unit rhombus is precisely one third of the total number of lozenge tilings. Motivated by this, we consider the more general situation of a semiregular hexagon of sides $a$, $a$ and $b$. We prove explicit formulas for the number of lozenge tilings of these hexagons containing the central unit rhombus, and obtain Propp's conjecture as a corollary of our results.

math.CO↗