Related papers: Differential Equations, Associators, and Recurrenc…

We consider the problem of obtaining higher order in regularization parameter $\epsilon$ analytical results for master integrals with elliptics. The two commonly employed methods are provided by the use of differential equations and direct…

We propose a framework where Fer and Wilcox expansions for the solution of differential equations are derived from two particular choices for the initial transformation that seeds the product expansion. In this scheme intermediate…

The correlators of large-scale fluctuations belong to the most important observables in modern cosmology. Recently, there have been considerable efforts in analytically understanding the cosmological correlators and the related wavefunction…

Exceptional orthogonal Hermite and Laguerre polynomials have been linked to the k-step extension of harmonic and singular oscillators. The exceptional polynomials allow the existence of different supercharges from the Darboux-Crum and…

We prove an asymptotic expansion for the eigenvalues and eigenfunctions of Schr\"{o}dinger-type operator with a confining potential and with principle part a periodic elliptic operator in divergence form. We compare the spectrum to the…

Computation of the spherical harmonic rotation coefficients or elements of Wigner's d-matrix is important in a number of quantum mechanics and mathematical physics applications. Particularly, this is important for the Fast Multipole Methods…

The renormalization group relations for the higher-order hadronic vacuum polarization function perturbative expansion coefficients are studied. The folded recurrent and unfolded explicit forms of such relations are obtained. The explicit…

In this paper we show that, as in the spacelike case, the inverse logarithmic expansion is applicable for all values of the argument of the analytic coupling constant. We present two different approaches, one of which is based primarily on…

We provide explicit series expansions for the exponential and logarithm functions attached to a rank r Drinfeld module that generalize well known formulas for the Carlitz exponential and logarithm. Using these results we obtain a procedure…

The main purpose of this article is to introduce a comprehensive, unified theory of the geometry of all connections. We show that one can study a connection via a certain, closely associated second-order differential equation. One of the…

Using recent results on string on $AdS_{3}\times N^d$, where N is a d-dimensional compact manifold, we re-examine the derivation of the non trivial extension of the (1+2) dimensional-Poincar\'e algebra obtained by Rausch de Traubenberg and…

In this work we develop an algebraic theory of linear recurrence equations and systems with constant coefficients and reflection. We obtain explicit solutions and the Green's functions associated to different problems under general linear…

We engineer compact contours on the moduli spaces of genus-zero Riemann surfaces that achieve analytic continuation from Euclidean to Lorentzian worldsheets. These generalized Pochhammer contours are based on the combinatorics of…

We introduce a simple, general, and convergent scheme to compute generalized eigenfunctions of self-adjoint operators with continuous spectra on rigged Hilbert spaces. Our approach does not require prior knowledge about the eigenfunctions,…

Trigonometric formulas are derived for certain families of associated Legendre functions of fractional degree and order, for use in approximation theory. These functions are algebraic, and when viewed as Gauss hypergeometric functions,…

We prove that a solution of an elliptic operator with periodic coefficients behaves on large scales like an analytic function, in the sense of approximation by polynomials with periodic corrections. Equivalently, the constants in the…

A generalization of the generating function for Gegenbauer polynomials is introduced whose coefficients are given in terms of associated Legendre functions of the second kind. We discuss how our expansion represents a generalization of…

This paper is intended to investigate Grassmann and Clifford algebras over Peano spaces, introducing their respective associated extended algebras, and to explore these concepts also from the counterspace viewpoint. The exterior…

In this paper, we consider the Dirichlet problem for a new class of augmented Hessian equations. Under sharp assumptions that the matrix function in the augmented Hessian is regular and there exists a smooth subsolution, we establish global…

Frobenius companion matrices arise when we write an $n$-th order linear ordinary differential equation as a system of first order differential equations. These matrices and their transpose have very nice properties. By using the powers of…