Related papers: On linearization coefficients of $q$-Laguerre poly…
A composition of birational maps given by Laurent polynomials need not be given by Laurent polynomials; however, sometimes---quite unexpectedly---it does. We suggest a unified treatment of this phenomenon, which covers a large class of…
To a $2\times2$ matrix $G$ with complex entries, we relate the sequence of Laurent polynomial $L_n(z,G)=\tr \big(G\big[\begin{smallmatrix}z&0\\ 0&z^{-1}\end{smallmatrix}\big]G^{\ast}\big)^n$. It turns out that for each \(n\), the family…
We introduce orthogonal polynomials $M_j^{\mu,\ell}(x)$ as eigenfunctions of a certain self-adjoint fourth order differential operator depending on two parameters $\mu\in\mathbb{C}$ and $\ell\in\mathbb{N}_0$. These polynomials arise as…
Let $w_{\alpha}(t)=t^{\alpha}\,e^{-t}$, $\alpha>-1$, be the Laguerre weight function, and $|\cdot|_{w_\alpha}$ denote the associated $L_2$-norm, i.e., $$ | f|_{w_\alpha}:=\Big(\int_{0}^{\infty}w_{\alpha}(t)| f(t)|^2\,dt\Big)^{1/2}. $$…
Loop corrections induce a dependence on the momentum squared of the coefficients of the Standard Model Lagrangian, making highly non-trivial (or even impossible) the diagonalization of its quadratic part. Fortunately, the introduction of…
Many classes of orthogonal polynomials satisfy a specific linearization property giving rise to a polynomial hypergroup structure, which offers an elegant and fruitful link to Fourier analysis, harmonic analysis and functional analysis.…
Pluri-Lagrangian systems are variational systems with the multi-dimensional consistency property. This notion has its roots in the theory of pluriharmonic functions, in the Z-invariant models of statistical mechanics, in the theory of…
We introduce a notion of $q$-deformed rational numbers and $q$-deformed continued fractions. A $q$-deformed rational is encoded by a triangulation of a polygon and can be computed recursively. The recursive formula is analogous to the…
In this paper asymptotic formulas are given for the Lebesgue constants generated by three special approximation processes related to the $\ell_1$-partial sums of Fourier series. In particular, we consider the Lagrange interpolation…
The Lucas sequence is a sequence of polynomials in s, and t defined recursively by {0}=0, {1}=1, and {n}=s{n-1}+t{n-2} for n >= 2. On specialization of s and t one can recover the Fibonacci numbers, the nonnegative integers, and the…
A version of Liouville's theorem is proved for solutions of some degenerate elliptic equations defined in $\mathbb{R}^n\backslash K$, where $K$ is a compact set, provided the structure of this equation and the dimension $n$ are related.…
Several years ago the second author playing with different "recognizers of real constants", e.g., the LLL algorithm, the Plouffe inverter, etc. found empirically the following formula. Let $p_n/q_n$ denote the $n$th convergent of the…
In the seminal book M\'echanique analitique, Lagrange, 1788, the notion of a Lagrange multiplier was first introduced in order to study a smooth minimization problem subject to equality constraints. The idea is that, under some regularity…
In this paper we introduce the polynomials $\{d_n^{(r)}(x)\}$ and $\{D_n^{(r)}(x)\}$ given by $d_n^{(r)}(x)=\sum_{k=0}^n\binom{x+r+k}k\binom{x-r}{n-k} \ (n\ge 0)$, $D_0^{(r)}(x)=1,\ D_1^{(r)}(x)=x$ and…
Let $w_\alpha(t) := t^{\alpha}\,e^{-t}$, where $\alpha > -1$, be the Laguerre weight function, and let $\|\cdot\|_{w_\alpha}$ be the associated $L_2$-norm, $$ \|f\|_{w_\alpha} = \left\{\int_{0}^{\infty} |f(x)|^2…
We consider a stationary variational inequality with gradient constraint and obstacle. We prove that this problem can be described by an equation using a Lagrange multiplier and a characteristic function. The Lagrange multiplier contains…
We find all polynomials solutions $P_n(x)$ of the abstract "hypergeometric" equation $L P_n(x) = \lambda_n P_n(x)$, where $L$ is a linear operator sending any polynomial of degree $n$ to a polynomial of the same degree with the property…
We prove positivity results about linearization and connection coefficients for Bessel polynomials. The proof is based on a recursion formula and explicit formulas for the coefficients in special cases. The result implies that the…
In this paper we prove graded versions of the Darboux Theorem and Weinstein's Lagrangian tubular neighbourhood Theorem in order to study the deformation theory of Lagrangian $NQ$-submanifolds of degree $n$ symplectic $NQ$-manifolds. Using…
For a linearly recurrent vector sequence P[n+1] = A(n) * P[n], consider the problem of calculating either the n-th term P[n] or L<=n arbitrary terms P[n_1],...,P[n_L], both for the case of constant coefficients A(n)=A and for a matrix A(n)…