PhyslibAlpha.AlgebraicFramework.JordanOrderUnit.JB.GeneratedByOne.PosInvertibility

Let $E$ be a normed Jordan algebra and $a \in E$. Let $C_J(a)$ denote the norm-closed associative subalgebra generated by $1$ and $a$. For an element $z \in C_J(a)$, this function defines an element in $C_J(a)$ as the sum of the series: $$\sum_{n=0}^{\infty} \binom{1/2}{n} (-z)^n$$ where $\binom{1/2}{n}$ are the generalized binomial coefficients. This series represents the "half-binomial" candidate for $\sqrt{1-z}$ within the algebra.

In the ring of formal power series over the real numbers $\mathbb{R}$, the binomial series with exponent $1/2$, defined as $\sum_{n=0}^{\infty} \binom{1/2}{n} X^n$, satisfies the identity: $$\left( \sum_{n=0}^{\infty} \binom{1/2}{n} X^n \right)^2 = 1 + X$$ where $X$ is the indeterminate of the power series.

For any natural number $n$, let $C_n$ denote the $n$-th Catalan number and $\binom{2n}{n}$ denote the $n$-th central binomial coefficient. The following telescoping identity holds: $$\frac{C_n}{4^n} = 2 \left( \frac{\binom{2n}{n}}{4^n} - \frac{\binom{2(n+1)}{n+1}}{4^{n+1}} \right)$$

For any natural number $N$, the partial sum of the sequence $\frac{C_n}{4^n}$ from $n=0$ to $N-1$ is given by: $$\sum_{n=0}^{N-1} \frac{C_n}{4^n} = 2 \left( 1 - \frac{1}{4^N} \binom{2N}{N} \right),$$ where $C_n$ denotes the $n$-th Catalan number and $\binom{2N}{N}$ denotes the $N$-th central binomial coefficient.

The series $\sum_{n=0}^{\infty} \frac{C_n}{4^n}$ is summable, where $C_n$ denotes the $n$-th Catalan number.

For any real number $a$ and any natural number $n$, the generalized binomial coefficient $\binom{a}{n+1}$ satisfies the recurrence relation: $$\binom{a}{n+1} = \binom{a}{n} \frac{a - n}{n + 1}$$

For any natural number $n$, the $(n+1)$-th Catalan number $C_{n+1}$ and the $n$-th Catalan number $C_n$ satisfy the recurrence relation $$C_{n+1} = \frac{2(2n + 1)}{n + 2} C_n,$$ where the terms are considered as real numbers.

For any natural number $n$, the generalized binomial coefficient $\binom{1/2}{n+1}$ satisfies the identity $$\binom{1/2}{n+1} = \frac{(-1)^n C_n}{2 \cdot 4^n}$$ where $C_n$ denotes the $n$-th Catalan number. This result provides an exact signed relationship between the half-binomial coefficients and the Catalan sequence.

For any natural number $n$, the absolute value of the generalized binomial coefficient $\binom{1/2}{n+1}$ is given by $$\left| \binom{1/2}{n+1} \right| = \frac{C_n}{2 \cdot 4^n}$$ where $C_n$ denotes the $n$-th Catalan number.

The sequence of the absolute values of the generalized binomial coefficients $\binom{1/2}{n}$ is summable over the natural numbers. That is, the series $\sum_{n=0}^{\infty} \left| \binom{1/2}{n} \right|$ converges, where $\binom{1/2}{n}$ denotes the generalized binomial coefficient $\frac{(1/2)(1/2-1)\dots(1/2-n+1)}{n!}$ for $n > 0$ and $\binom{1/2}{0} = 1$.

Let $E$ be a nontrivial JB algebra. Let $a$ be an element of $E$, and let $C_J(a)$ be the norm-closed Jordan subalgebra generated by $1$ and $a$. For any element $z \in C_J(a)$ with $\|z\| \le 1$, the series $\sum_{n=0}^{\infty} \binom{1/2}{n} (-z)^n$ is absolutely summable. That is, the series of norms $$\sum_{n=0}^{\infty} \left\| \binom{1/2}{n} (-z)^n \right\|$$ converges, where $\binom{1/2}{n}$ denotes the generalized binomial coefficient.

Let $E$ be a nontrivial normed Jordan algebra and $a \in E$. Let $C_J(a)$ be the norm-closed Jordan subalgebra generated by the identity element $1$ and $a$. For any element $z \in C_J(a)$ with $\|z\| \leq 1$, the series $$\sum_{n=0}^{\infty} \binom{1/2}{n} (-z)^n$$ is summable in $C_J(a)$, where $\binom{1/2}{n}$ denotes the generalized binomial coefficient.

Let $E$ be a normed Jordan algebra and $a \in E$. Let $C_J(a)$ be the norm-closed associative Jordan subalgebra generated by the identity element $1$ and $a$. For any element $z \in C_J(a)$ and any natural number $n$, the $n$-th coefficient of the Cauchy product of the binomial series for $(1-z)^{1/2}$ satisfies: $$\sum_{i+j=n} \left( \binom{1/2}{i} (-z)^i \right) \cdot \left( \binom{1/2}{j} (-z)^j \right) = \begin{cases} 1 & \text{if } n = 0 \\ -z & \text{if } n = 1 \\ 0 & \text{if } n > 1 \end{cases}$$ where $\binom{1/2}{k}$ denotes the generalized binomial coefficient `Ring.choose (1/2 : ℝ) k`.

Let $E$ be a nontrivial normed Jordan algebra and $a \in E$. Let $C_J(a)$ be the norm-closed Jordan subalgebra generated by the identity element $1$ and $a$. For any element $z \in C_J(a)$ such that $\|z\| \leq 1$, the binomial square-root series $S(z)$ defined as $$S(z) = \sum_{n=0}^{\infty} \binom{1/2}{n} (-z)^n$$ satisfies the identity $S(z)^2 = 1 - z$, where $\binom{1/2}{n}$ are the generalized binomial coefficients.

Let $E$ be a nontrivial Jordan-Banach (JB) algebra. For any element $a \in E$, let $C_J(a)$ denote the norm-closed associative Jordan subalgebra generated by the identity element $1$ and $a$. If $y$ is a non-negative element in $C_J(a)$ (written $y \geq 0$), then there exists an element $s \in C_J(a)$ such that $s^2 = y$.

Let $E$ be a JB algebra and $a \in E$. Let $C_J(a)$ denote the norm-closed Jordan subalgebra generated by the identity $1$ and $a$ (also denoted as $\overline{J[a]}$). For any element $y \in C_J(a)$, $y \ge 0$ if and only if there exists an element $s \in C_J(a)$ such that $s^2 = y$.

Let $E$ be a nontrivial Jordan-Banach (JB) algebra and $a \in E$. Let $C_J(a)$ denote the norm-closed associative Jordan subalgebra generated by the identity $1$ and $a$. If $y \in C_J(a)$ is an invertible element (a unit) that is non-negative ($y \geq 0$), then $y$ lies in the order interior; specifically, there exists a real number $\epsilon > 0$ such that $\epsilon \cdot 1 \leq y$.

Let $E$ be a nontrivial Jordan-Banach (JB) algebra and $a \in E$. Let $C_J(a)$ denote the norm-closed associative Jordan subalgebra generated by the identity $1$ and $a$ (also denoted as $\overline{J[a]}$). For any element $x \in C_J(a)$, it is not the case that both $\|x\| \cdot 1 - x$ and $\|x\| \cdot 1 + x$ are invertible (units) in $C_J(a)$.

Let $C_J(a)$ be the norm-closed Jordan subalgebra of a JB algebra generated by the identity $1$ and an element $a$ (denoted $\overline{J[a]}$). Let $\{s_n\}_{n \in \mathbb{N}}$ be a sequence in $C_J(a)$, and let $r, y$ be elements in $C_J(a)$. If the sequence $s_n$ converges to $r$ and the sequence of squares $s_n^2$ converges to $y$ as $n \to \infty$, then $r^2 = y$.

Let $E$ be a normed Jordan algebra and $a \in E$. Let $C_J(a)$ be the norm-closed Jordan subalgebra generated by the identity element $1$ and $a$. For any element $y \in C_J(a)$, if $\|1 - y\| < 1$, then $y$ is invertible (is a unit) in $C_J(a)$.

Let $E$ be a JB algebra with an order unit, and let $C_J(a)$ be the norm-closed Jordan subalgebra generated by the identity element $1$ and an element $a \in E$. For any element $y \in C_J(a)$, if there exists a real number $\epsilon > 0$ such that $y \geq \epsilon \cdot 1$ (where $1$ is the identity element), then $y$ is a unit (invertible) in $C_J(a)$.

Let $E$ be a nontrivial normed Jordan algebra with an order unit structure. For an element $a \in E$, let $C_J(a)$ denote the norm-closed Jordan subalgebra generated by $1$ and $a$. For any element $y \in C_J(a)$ satisfying $y \ge 0$, $y$ is invertible in $C_J(a)$ if and only if there exists a real number $\epsilon > 0$ such that $y \ge \epsilon \cdot 1$.

Let $E$ be a JB algebra and let $C_J(a)$ be the norm-closed Jordan subalgebra generated by the identity element $1$ and an element $a \in E$. For any $x \in C_J(a)$ and real numbers $c, d \in \mathbb{R}$ such that $d \neq 0$, the element $(x - c \cdot 1)^2 + d^2 \cdot 1$ is a unit in $C_J(a)$. This result identifies that irreducible real quadratic factors evaluate to units within the generated subalgebra.

Let $E$ be a JB algebra and $C_J(a)$ be the norm-closed Jordan subalgebra generated by the identity $1$ and an element $a \in E$. For any element $x \in C_J(a)$ and real numbers $c, d \in \mathbb{R}$ such that $d \neq 0$, the evaluation of the polynomial $P(X) = (X - c)^2 + d^2$ at $x$ is a unit (invertible) in $C_J(a)$.

Let $E$ be a JB algebra and $a \in E$. Let $C_J(a)$ (also denoted as $\overline{J[a]}$) be the norm-closed Jordan subalgebra generated by the identity $1$ and the element $a$. For any element $s \in C_J(a)$, if $s^2$ is an invertible element (a unit) in $C_J(a)$, then there exists a non-negative element $t \in C_J(a)$ (i.e., $t \geq 0$) such that $t \cdot s^2 = 1$.