In this paper, we consider the Prabhakar fractional logistic differential equation. By using appropriate limit relations, we recover some other logistic differential equations, giving representations of each solution in terms of a formal power series. Some numerical approximations are implemented by using truncated series.
By means of the Lagrange expansion formula, we establish a general pair of nonlinear inverse series relations, which are expressed via partial Bell polynomials with the connection coefficients involve an arbitrary formal power series. As applications, two examples are presented with one of them recovering the difficult theorems discovered recently by Birmajer, Gil and Weiner (2012 and 2019).
Abstract
In the field of formal power series over a finite field, we prove a result which enables us to construct explicit examples of
$U_{m}$
-numbers by using continued fraction expansions of algebraic formal power series of degree
$m>1$
.
The aim of paper is to find the condition under which a Fréchet-valued function [Formula: see text] admitting meromorphic extension along some pencil of complex lines can be meromorphically extended to a neighborhood of [Formula: see text] Some auxiliary results concerning the domains of existence for Fréchet-valued meromorphic functions, Rothstein’s theorem, Levi extension theorem for meromorphic functions with values in a locally complete space, convergence of formal power series of Fréchet-valued homogeneous polynomials are also proved in this work.
Suppose
p
n
be sequence of positive reals. By
H
w
p
n
, we represent the space of all formal power series
∑
n
=
0
∞
a
n
z
n
equipped with
∑
n
=
0
∞
λ
a
n
/
n
+
1
p
n
<
∞
, for some
λ
>
0
.
Various topological and geometric behavior of
H
w
p
n
and the prequasi ideal constructs by
s
-numbers and
H
w
p
n
have been considered. The upper bounds for
s
-numbers of infinite series of the weighted
n
-th power forward shift operator on
H
w
p
n
with applications to some entire functions are granted. Moreover, we investigate an extrapolation of Caristi’s fixed point theorem in
H
w
p
n
.