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In this paper a general theory of semi-classical matrix orthogonal polynomials is developed. We define the semi-classical linear functionals by means of a distributional equation $D(u A) = u B,$ where $A$ and $B$ are matrix polynomials.…
The aim of this work is to characterize three fundamental normalization proprieties in lambda-calculus trough the Taylor expansion of $ \lambda$-terms. The general proof strategy consists in stating the dependence of ordinary reduction…
These notes aim to provide a classical approach to solving some conformable differential equations based on prior knowledge of how to solve ordinary differential equations. That is, using the methods of separation of variables, homogeneous…
We define the pattern fragment for higher-order unification problems in linear and affine type theory and give a deterministic unification algorithm that computes most general unifiers.
The Laguerre functions constitute one of the fundamental basis sets for calculations in atomic and molecular electron-structure theory, with applications in hadronic and nuclear theory as well. While similar in form to the Coulomb…
We present a novel lambda calculus that casts the categorical approach to the study of quantum protocols into the rich and well established tradition of type theory. Our construction extends the linear typed lambda calculus with a linear…
This paper contains a new elementary proof of the Fundamental Theorem of Calculus for the Lebesgue integral. The hardest part of our proof simply concerns the convergence in ${\rm L}^1$ of a certain sequence of step functions, and we prove…
This paper presents an up-to-date and refined version of the SCL calculus for first-order logic without equality. The refinement mainly consists of the following two parts: First, we incorporate a stronger notion of regularity into…
The relations between solutions of the three types of totally linear partial differential equations of first order are presented. The approach is based on factorization of a non-homogeneous first order differential operator to products…
This thesis is intended to provide an account of the theory and applications of Operational Methods that allow the "translation" of the theory of special functions and polynomials into a "different" mathematical language. The language we…
We define a new class of functions, connected to the classical Laguerre-P\'{o}lya class, which we call the shifted Laguerre-P\'{o}lya class. Recent work of Griffin, Ono, Rolen, and Zagier shows that the Riemann Xi function is in this class.…
We develop the operational semantics of an untyped probabilistic lambda-calculus with continuous distributions, as a foundation for universal probabilistic programming languages such as Church, Anglican, and Venture. Our first contribution…
We present a call-by-need $\lambda$-calculus that enables strong reduction (that is, reduction inside the body of abstractions) and guarantees that arguments are only evaluated if needed and at most once. This calculus uses explicit…
A solution is proposed for the problem of composition of ordinary generating functions. A new class of functions that provides a composition of ordinary generating functions is introduced; main theorems are presented; compositae are written…
This paper introduces a new functional expansion framework that extends classical ideas beyond the Taylor series. Unlike traditional Taylor expansions based on local polynomial approximations, the proposed approach arises from exact…
We will introduce a cyclic derivative for fully (stably) matricial functions and study its basic properties. In particular, we will show the Poincar\'{e} lemma for stably matricial functions of certain classes. We will also position…
Proof assistants and programming languages based on type theories usually come in two flavours: one is based on the standard natural deduction presentation of type theory and involves eliminators, while the other provides a syntax in…
A new general and unified method of summation, which is both regular and consistent, is invented. It is based on the idea concerning a way of integers reordering. The resulting theory includes a number of explicit and closed form summation…
Based on a generalized Newton's identity, we construct a family of symmetric functions which deform the modular Hall-Littlewood functions. We also give a determinant formula for the Macdonald functions.
Taylor's formula holds significant importance in function representation, such as solving differential difference equations, ordinary differential equations, partial differential equations, and further promotes applications in visual…