Related papers: Taylor expansion for an operator function
In this paper we study the development in Taylor series of the function $f(x)=x^x$. First section establishes a recursive relationship among successive derivatives of the function by using the coefficients defined therein. From recursion…
Using the reflection formula of the Gamma function, we derive a new formula for the Taylor coefficients of the reciprocal Gamma function. The new formula provides effective asymptotic values for the coefficients even for very small values…
We study the existence of formal Taylor expansions for functions defined on fields of generalised series. We prove a general result for the existence and convergence of those expansions for fields equipped with a derivation and an…
We give a simple way to extend index-theoretical statements from partial differential operators with smooth coefficients to operators with coefficients of finite Sobolev order.
The paper contains an interesting generalization of the classical Taylor expansion formula and four applications
In this work at first the relation the Mittag-Lefler function to the exponential is given. The results are applied to the construction of the solution of Cauchy problem for ordinary linear operator differential equations with constant…
We consider series expansions in bases of classical orthogonal polynomials. When such a series solves a linear differential equation with polynomial coefficients, its coefficients satisfy a linear recurrence equation. We interpret this…
By employing the $q$-difference operator, various classes of $q$-extensions of starlike functions have emerged from many different viewpoints and perspectives. Ruscheweyh's work unified these $q$-extensions with convolution operations.…
The polynomial relationship between elementary symmetric functions (Cauchy enumeration formula) is formulated via a ``raising operator" and Fock space construction. A simple graphical proof of this relation is proposed. The new operator…
In this paper we establish a multivariable non-commutative generalization of L\"owner's classical theorem from 1934 characterizing operator monotone functions as real functions admitting analytic continuation mapping the upper complex…
We propose a new integral based on Taylor measures, study its properties extensively, and we illustrate that it includes many concepts from mathematics as special cases. In particular, the new integral emerges as a generalization of the…
The paper establishes conditions under which there are exact linear representations of nonlinear partial differential equations (Cauchy problems). By introducing a certain linear operator $A$, it is shown that under these conditions there…
We give necessary and sufficient condition so that we have d-hypercyclicity for operators who map a holomorphic function to a partial sum of the Taylor expansion. This problem is connected with doubly universal Taylors series and this is an…
The slice Dirac operator over octonions is a slice counterpart of the Dirac operator over quaternions. It involves a new theory of stem functions, which is the extension from the commutative $ O(1) $ case to the non-commutative $ O(3) $…
The addition relation for the Riemann theta functions and for its limits, which lead to the appearance of exponential functions in soliton type equations is discussed. The presented form of addition property resolves itself to the…
We consider the integral and derivative operators of tempered fractional calculus, and examine their analytic properties. We discover connections with the classical Riemann-Liouville fractional calculus and demonstrate how the operators may…
We introduce a notion of a noncommutative function defined on a domain of $d$-tuples of bounded operators on an infinite dimensional Hilbert space. Inverse and implicit function theorems in this setting are established. When these…
We consider linear operators defined on a subspace of a complex Banach space into its topological antidual acting positively in a natural sense. The goal of this paper is to investigate of this kind of operators. The main theorem is a…
This article presents a natural extension of the tensor algebra. In addition to "left multiplications" by vectors, we can consider "derivations" by covectors as basic operators on this extended algebra. These two types of operators satisfy…
We develop a general theory of operator realizations, or ``linear representations" of analytic functions in several non-commuting variables about a matrix-centre. In particular we show that a non-commutative function has a matrix-centre…