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We discuss polynomials orthogonal with respect to a semi-classical generalised higher order Freud weight \[\omega(x;t,\lambda)=|x|^{2\lambda+1}\exp\left(tx^2-x^{2m}\right),\qquad x\in\mathbb{R},\] with parameters $\lambda > -1$,…

Suppose $M$ and $N$ are positive integers and let $k = \gcd(M, N)$, $m = M/k$, and $n=N/k$. We define a symmetric function $L_{M,N}$ as a weighted sum over certain tuples of lattice paths. We show that $L_{M,N}$ satisfies a generalization…

We find solutions for a linear deformation of the symmetric three-term recursion relation. The orthogonal polynomials of the first and second kind associated with the deformed relation are obtained. The new density (weight) function is…

In this paper, we investigate permutation rotation-symmetric (shift-invariant) vectorial Boolean functions on $n$ bits that are liftings from Boolean functions on $k$ bits, for $k\leq n$. These functions generalize the well-known map used…

We expand the quantum mechanical wavefunction in a complete set of square integrable orthonormal basis such that the matrix representation of the Hamiltonian operator is tridiagonal and symmetric. Consequently, the matrix wave equation…

We show that there are two or more procedures to generalize the known four-dimensional transformation, aiming to generate cylindrically rotating charged exact solutions, to higher dimensional spacetimes . In the one procedure, presented in…

Weighted degrees of quasihomogeneous Hamiltonian functions of the Painlev\'{e} equations are investigated. A tuple of positive integers, called a regular weight, satisfying certain conditions related to singularity theory is classified.…

This paper studies three types of functions arising separately in the analysis of algorithms that we analyze exactly using similar Mellin transform techniques. The first is the solution to a Multidimensional Divide-and-Conquer (MDC)…

In this paper, we study variants of weight enumerators of linear codes over $\mathbb{F}_q$. We generalize the concept of average complete joint weight enumerators of two linear codes over $\mathbb{F}_q$. We also give its MacWilliams type…

The theory of bi-orthogonal polynomials on the unit circle is developed for a general class of weights leading to systems of recurrence relations and derivatives of the polynomials and their associated functions, and to…

Model merging aims to combine multiple fine-tuned models into a single set of weights that performs well across all source tasks. While prior work has shown that merging can approximate the performance of individual fine-tuned models for…

In this work we show how to get advantage from the Riemann--Hilbert analysis in order to obtain information about the matrix orthogonal polynomials and functions of second kind associated with a weight matrix. We deduce properties for the…

Let $K$ be a field of characteristic zero and suppose that $f:\mathbb{N}\to K$ satisfies a recurrence of the form $$f(n)\ =\ \sum_{i=1}^d P_i(n) f(n-i),$$ for $n$ sufficiently large, where $P_1(z),...,P_d(z)$ are polynomials in $K[z]$.…

We propose a variant of elliptic multiple polylogarithms that have at most logarithmic singularities in all variables and satisfy a differential equation without homogeneous term. We investigate several non-trivial elliptic two-loop Feynman…

In several applications, such as \tsc{weno} interpolation and reconstruction [Shu C.W.: SIAM Rev. 51 (2009) 82--126], we are interested in the analytical expression of the weight-functions which allow the representation of the approximating…

The flux reconstruction (FR) approach offers a flexible framework for describing a range of high-order numerical schemes; including nodal discontinuous Galerkin and spectral difference schemes. This is accomplished through the use of…

The long-standing problem of representing the general massive one-loop Feynman integral as a meromorphic function of the space-time dimension $d$ has been solved for the basis of scalar one- to four-point functions with indices one. In 2003…

We consider polynomials of the form $\operatorname{h}_m(y_1^{[\varkappa_1]},\ldots,y_n^{[\varkappa_n]})$, where $\operatorname{h}_m$ is the complete homogeneous polynomial of degree $m$ and $y_j^{[\varkappa_j]}$ denotes $y_j$ repeated…

Consider a nested, non-homogeneous recursion R(n) defined by R(n) = \sum_{i=1}^k R(n-s_i-\sum_{j=1}^{p_i} R(n-a_ij)) + nu, with c initial conditions R(1) = xi_1 > 0,R(2)=xi_2 > 0, ..., R(c)=xi_c > 0, where the parameters are integers…

In this manuscript, we analyze the expansions of functions in orthogonal polynomials associated with a general weight function in a multidimensional setting. Such orthogonal polynomials can be obtained by Gram-Schmidt orthogonalization.…