Deformation Expression for Elements of Algebras (V) - Diagonal matrix calculus and $*$-special functions -

TL;DR

This paper introduces *-special functions, extending classical special functions by replacing exponential functions with *-exponentials in integral definitions, aiming to develop a *-version framework for special functions.

Contribution

It proposes a novel approach to define *-special functions using integral formulas with *-exponentials, expanding the theory of special functions in the context of deformation quantization.

Findings

Defined *-special functions via integral formulas with *-exponentials

Established foundational properties of *-special functions

Extended classical special functions into the *-deformation setting

Abstract

In this note, we are interested in the *-version of various special functions. Noting that many special functions are defined by integrals involving the exponential functions, we define *-special functions by similar integral formula replacing exponential functions by *-exponential functions.