The Fréchet $(\mbox{resp.}, (\mbox{LB})\mbox{-})$ sequence spaces $ces(p+) := \bigcap_{r > p} ces(r), 1 \leq p < \infty $ (resp. $ ces (p\mbox{-}) := \bigcup_{ 1 < r < p} ces (r), 1 < p \leq \infty),$ are known to be very different to the classical sequence spaces $ \ell_ {p+} $ (resp., $ \ell_{p_{\mbox{-}}}).$ Both of these classes of non-normable spaces $ ces (p+), ces (p\mbox{-})$ are defined via the family of reflexive Banach sequence spaces $ ces (p), 1 < p < \infty .$ The \textit{dual}\/ Banach spaces $ d (q), 1 < q < \infty ,$ of the discrete Cesàro spaces $ ces (p), 1 < p < \infty , $ were studied by G. Bennett, A. Jagers and others. Our aim is to investigate in detail the corresponding sequence spaces $ d (p+) $ and $ d (p\mbox{-}),$ which have not been considered before. Some of their properties have similarities with those of $ ces (p+), ces (p\mbox{-})$ but, they also exhibit differences. For instance, $ ces (p+)$ is isomorphic to a power series Fréhet space of order 1 whereas $ d (p+) $ is isomorphic to such a space of infinite order. Every space $ ces (p+), ces (p\mbox{-}) $ admits an absolute basis but, none of the spaces $ d (p+), d (p\mbox{-})$ have any absolute basis.

For systems of differential equations with singular $\varphi$-Laplacians the simple geometric argument is used to obtain the existence results for the Dirichlet, Sturm -- Liouville, periodic or anti--periodic boundary value problems.

Let $\mu$ be a positive Borel measure defined on the boundary of the unit disk. In this paper, we give a new characterization for the Dirichlet type space $\mathcal{D}_{\mu}$. As an application, we also give a new characterization for the closure of the $\mathcal{D}_{\mu}$ space in the Bloch space.

In this paper, we introduce the concept of the $\mathbb{J}_{\alpha, \beta}$-classical orthogonal polynomials, where $\mathbb{J}_{\alpha, \beta}$ is the raising operator $\mathbb{J}_{\alpha, \beta}:=(x^2-1)\frac{d}{dx}+ \big((\alpha+\beta)x-\alpha+\beta\big)\mathbb{I}$, with $\alpha$ and $\beta$ nonzero complex numbers and $\mathbb{I}$ representing the identity operator. Then, we show that the Jacobi polynomials $P^{(\alpha, \beta)}_n(x),\ n\geq0$, where $\alpha, \beta\in\mathbb{C}\backslash\{0,-1,-2,\ldots\}$, $(\alpha+\beta\neq-m, \ m\geq0)$, are the only $\mathbb{J}_{\alpha, \beta}$-classical orthogonal polynomials. As an application, we give some new connection formulas satisfied by the polynomials solution of our problem.

The main aim of this paper is to give a Hopf algebroid approach to the Picard-Vessiot theory of linear differential matrix equations with coefficients in the polynomial complex algebra. To this end, we introduce a general construction of what we call here \emph{the finite dual} of a co-commutative (right) Hopf algebroid and then apply this construction to the first Weyl algebra viewed as the universal enveloping Hopf algebroid of the Lie algebroid of all vector fields on the affine complex line. In this way, for a fixed linear differential matrix equation of order $\geq 1$, we are able to recognize the associated algebraic Galois groupoid as a closed subgroupoid of the induced groupoid of the general linear group along the trivial map, and show that is a transitive groupoid (i.e., it has only one type of isotropy algebraic groups). The polynomial coordinate ring of the Galois groupoid turns out to be a Hopf sub-algebroid of the finite dual of the first Weyl algebra and its total isotropy Hopf algebra (the bundle of all isotropy algebraic groups) is recognized as the Picard-Vessiot extension of the polynomial complex algebra for the linear differential equation we started with.

We establish that a Tychonoff space $X$ has the property $(\kappa)$ if all countable subsets of $X$ are scattered. Therefore a countably compact space $X$ does not need to be scattered if $C_p(X)$ is $\kappa$-Fréchet-Urysohn. However, if a hereditarily Baire space is sequential and has the property $(\kappa)$, then it must be scattered. We also prove, for any space $X$, that $C_p(X)$ is $\kappa$-Fréchet-Urysohn if and only if so is $C_p(X, [0,1])$. A pseudocompact first countable space must be scattered if it has the property $(\kappa)$.

Binomial sums with Pell and Lucas polynomials as weight functions are explicitly evaluated. Their special cases result in several interesting identities concerning Fibonacci and Lucas numbers. Two of them confirm the summation formulae proposed recently by Ohtsuka and Tauraso (2020).

The Baumslag-Solitar group $BS(2,3)$, is a so-called non-Hopfian group, meaning that it has an epimorphism $\phi$ onto itself, that is not injective. In particular this is equivalent to saying that $BS(2,3)$ has a non-trivial quotient that is isomorphic to itself. As a consequence the Cayley graph of $BS(2,3)$ has a quotient that is isomorphic to itself up to change of generators. We describe this quotient on the graph-level and take a closer look at the most common epimorphism $\phi$. We show its kernel is a free group of infinite rank with an explicit set of generators. Finally we show how $\phi$ appears as a morphism on fundamental groups induced by some continuous map. This point of view was communicated to the author by Gilbert Levitt.