We establish two truncations of Gauss’ square exponent theorem and a finite extension of Euler’s identity. For instance, we prove that for any positive integer $n$, $$\sum _{k=0}^n{(-1)}^k\left[\begin{array}{c}2n-k\\ k\end{array}\right]{(q;q^2)}_{n-k}{q}^{\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)}=\sum _{k=-n}^n{(-1)}^k{q}^{k^2},$$ where $$\left[\begin{array}{c}n\\ m\end{array}\right]=\prod _{k=1}^m\frac{1-q^{n-k+1}}{1-q^k}\text{and}{(a;q)}_n=\prod _{k=0}^{n-1}(1-a{q}^k).$$

Let $N\subseteq M$ be an inclusion of $I{I}_1$ factors with finite Jones index. Then $M=(N^{\text{'}}\cap M)\oplus [N,M]$ as a vector space. Here $[N,M]$ denotes the vector space spanned by the commutators of the form $[a,b]$ where $a\in N,b\in M$.

Let $p$ be a prime number, ${\mathbb{Q}}_p$ the field of $p$-adic numbers and ${ℂ}_p$ the completion of the algebraic closure of ${\mathbb{Q}}_p$. In this paper we obtain a representation theorem for rigid analytic functions on ${𝐏}^1\left({ℂ}_p\right)\setminus C(t,ϵ)$ which are equivariant with respect to the Galois group $G=Ga{l}_{cont}({ℂ}_p/{\mathbb{Q}}_p)$, where $t$ is a lipschitzian element of ${ℂ}_p$ and $C(t,ϵ)$ denotes the $ϵ$-neighborhood of the $G$-orbit of $t$.

We investigate in a geometrical way the point sets of  $ℝ$  obtained by the  $\beta$-numeration that are the  $\beta$-integers  ${\mathbb{Z}}_{\beta }\subset \mathbb{Z}\left[\beta \right]$  where  $\beta$  is a Perron number. We show that there exist two canonical cut-and-project schemes associated with the  $\beta$-numeration, allowing to lift up the  $\beta$-integers to some points of the lattice  ${\mathbb{Z}}^m$  ($m=$  degree of  $\beta$) lying about the dominant eigenspace of the companion matrix of  $\beta$ . When  $\beta$  is in particular a Pisot number, this framework gives another proof of the fact...

The present paper shows that for any $s$ sequences of real numbers, each with infinitely many distinct elements, ${\lambda }_n^j$, j=1,...,s, the rational combinations of $x_1^{{\lambda }_{m_1}^1}{x}_2^{{\lambda }_{m_2}^2}...x_s^{{\lambda }_{m_s}^s}$ are always dense in $C_{I^s}$.

In this paper, we give an integral representation for the boundary values of derivatives of functions of the de Branges–Rovnyak spaces $\mathscr{H}\left(b\right)$, where $b$ is in the unit ball of $H^{\infty }\left({ℂ}_{+}\right)$. In particular, we generalize a result of Ahern–Clark obtained for functions of the model spaces $K_b$, where $b$ is an inner function. Using hypergeometric series, we obtain a nontrivial formula of combinatorics for sums of binomial coefficients. Then we apply this formula to show the norm convergence of reproducing kernel...