nLab sequence algebra

Definition

For a commutative ring $R$, the sequence algebra is the commutative $R$-algebra $\mathbb{N} \to R$, with canonical injection $c \colon R \to (\mathbb{N} \to R)$ which sends each element $r$ in $R$ to a constant function $c(r)$. The commutative algebra operations are pointwise defined as:

$$c(r)(n) \coloneqq r$$

$$0(n) \coloneqq 0$$

$$(x + y)(n) \coloneqq x(n) + y(n)$$

$$(-x)(n) \coloneqq -x(n)$$

$$(r x)(n) \coloneqq r x(n)$$

$$(1)(n) \coloneqq 1$$

$$(x \cdot y)(n) \coloneqq x(n) \cdot y(n)$$

and thus the sequence set $\mathbb{N} \to R$ is a commutative $R$-algebra.

Polynomials

Given a commutative ring $R$, a sequential polynomial is a sequence $x \colon \mathbb{N} \to R$ where there exists a natural number $N$ such that for all $i \gt N$ we have $x(i) = 0$.

Operators

Left shift operator

Given a commutative ring $R$ and a sequence $x \colon \mathbb{N} \to R$ of terms in $R$, the left shift operator

$$T \colon (\mathbb{N} \to R) \to (\mathbb{N} \to R)$$

is defined as

$$T x(i) \coloneqq x(i + 1)$$

for $i \in \mathbb{N}$.

Series operator

Given a commutative ring $R$ and a sequence $x:\mathbb{N} \to R$ of terms in $R$, the series operator

$$\Sigma:(\mathbb{N} \to R) \to (\mathbb{N} \to R)$$

is inductively defined as

$$\Sigma x(0) \coloneqq x(0)$$

$$\Sigma x(i + 1) \coloneqq \Sigma x(i) + x(i + 1)$$

for $i \in \mathbb{N}$.

For any such sequence $x:\mathbb{N} \to R$, $\Sigma x$ is called a series.

Inverse series operator

Given a commutative ring $R$ and a sequence $x:\mathbb{N} \to R$ of terms in $R$, the inverse series operator

$$\Sigma^{-1}:(\mathbb{N} \to R) \to (\mathbb{N} \to R)$$

is inductively defined as

$$\Sigma^{-1} x(0) \coloneqq x(0)$$

$$\Sigma^{-1} x(i + 1) \coloneqq \Sigma^{-1} x(i) - x(i + 1)$$

for $i \in \mathbb{N}$.

For any such sequence $x:\mathbb{N} \to R$, $\Sigma^{-1}\Sigma x = x$ and $\Sigma\Sigma^{-1}x = x$.

Infinite product operator

Given a commutative ring $R$ and a sequence $x:\mathbb{N} \to R$ of terms in $R$, the infinite product operator

$$\Pi:(\mathbb{N} \to R) \to (\mathbb{N} \to R)$$

is inductively defined as

$$\Pi x(0) \coloneqq x(0)$$

$$\Pi x(i + 1) \coloneqq \Pi x(i) \cdot x(i + 1)$$

for $i \in \mathbb{N}$.

For any such sequence $x:\mathbb{N} \to R$, $\Pi x$ is called an infinite product.

Discrete convolution

Given a commutative ring $R$ and sequence $x:\mathbb{N} \to R$ and $y:\mathbb{N} \to R$, the discrete convolution

$$(-)*(-):(\mathbb{N} \to R) \times (\mathbb{N} \to R) \to (\mathbb{N} \to R)$$

is defined as

$$(x*y)(i) \coloneqq \sum_{n:[0,i]} x(i) y(i - n)$$

for $i \in \mathbb{N}$.

Given a commutative ring $R$ and sequence $x:\mathbb{N} \to R$ and $y:\mathbb{N} \to R$, the Cauchy product of the two series $\Sigma x \cdot \Sigma x$ is the discrete convolution of $x$ and $y$.

$$\Sigma x \cdot \Sigma y \coloneqq x * y$$

Sequential derivative

Given a commutative ring $R$ and a sequence $x:\mathbb{N} \to R$ of terms in $R$, the sequential derivative

$$\mathcal{D}:(\mathbb{N} \to R) \to (\mathbb{N} \to R)$$

is defined as

$$\mathcal{D} x(i) \coloneqq (i + 1) x(i + 1)$$

for $i \in \mathbb{N}$.

The sequential derivative is a derivation, where linearity is satisfied by addition and scalar multiplication of sequences and the Leibniz rule is satisfied for the Cauchy product of sequences. As a result, every sequence algebra is a differential algebra.

Right shift operators

Given a commutative ring $R$ and a sequence $x:\mathbb{N} \to R$, the set of right shifted sequences of $x$ is the fiber of the left shift operator at $x$:

$$\{y:\mathbb{N} \to R \vert T y = x\}$$

A right shift operator is an element of the above set.

Every sequence in $R$ has a right shift operator $T^{-1}_s:(\mathbb{N} \to R) \to (\mathbb{N} \to R)$ for every term $r \in R$, defined as

$$T^{-1}_r x(0) = r$$

$$T^{-1}_r x(i + i) \coloneqq x(i)$$

for $i \in \mathbb{N}$.

Sequential antiderivatives

Given a commutative ring $R$ and a sequence $x:\mathbb{N} \to R$, the set of sequential antiderivatives of $x$ is the fiber of the sequential derivative at $x$:

$$\{y:\mathbb{N} \to R \vert \mathcal{D} y = x\}$$

A sequential antiderivative is a element of the above set.

If $R$ is a commutative $\mathbb{Q}$-algebra, then every sequence in $R$ has an antiderivative operator $\mathcal{D}^{-1}_r:(\mathbb{N} \to R) \to (\mathbb{N} \to R)$ for every term $r \in R$, defined as

$$\mathcal{D}^{-1}_r x(0) = r$$

$$\mathcal{D}^{-1}_r x(i + i) \coloneqq \frac{1}{i} x(i)$$

for $i \in \mathbb{N}$.

Sequence functions

Given a commutative ring $R$, a sequence function is a function $f:R \to (\mathbb{N} \to R)$.

Functional shift operator

Given a commutative ring $R$ and a sequence function $f:R \to (\mathbb{N} \to R)$, the functional shift operator

$$\mathcal{T}:R \to (R \to (\mathbb{N} \to R)) \to (R \to (\mathbb{N} \to R))$$

is defined as

$$\mathcal{T}_c f(r)(i) \coloneqq f(r - c)(i)$$

for $c \in R$, and $r \in R$, and $i \in \mathbb{N}$.

Power sequence functions

Given a commutative ring $R$ and a sequence $x:\mathbb{N} \to R$ of terms in $R$, the power sequence function

$$\mathcal{P}:(\mathbb{N} \to R) \to (R \to (\mathbb{N} \to R))$$

is defined as

$$\mathcal{P}_x (r)(i) \coloneqq x(i) r^i$$

for $i \in \mathbb{N}$ and $r \in R$.

Functional derivative

Given a commutative ring $R$, the functional derivative of the power sequence function $f:R \to (\mathbb{N} \to R)$

$$\tilde{D}:(R \to (\mathbb{N} \to R)) \to (R \to (\mathbb{N} \to R))$$

is defined as

$$(\tilde{D} \mathcal{P}_x)(r)(i) \coloneqq (\mathcal{D} x(i)) r^i$$

Power series

For any such sequence $x:\mathbb{N} \to R$, $\Sigma\mathcal{P}_x(r)$ is called a formal power series or formal Maclaurin series. For any such sequence $x:\mathbb{N} \to R$ and element $c \in R$, $\Sigma \mathcal{T}_c \mathcal{P}_x (r)$ is called a formal Taylor series.

Analytic functions

Suppose $R$ is a commutative ring with a partial function

$$\lim_{n \to \infty} (-)(n) \colon (\mathbb{N} \to R) \to R$$

making $R$ into a sequentially Hausdorff space.

Then, a partial function $f \colon R \to R$ is an analytic function if there exists an element $c \in R$ and sequence $x \colon \mathbb{N} \to R$ such that

$$f(r) = \lim_{n \to \infty} \Sigma \mathcal{T}_c \mathcal{P}_x (r)(n)$$

for all elements $r \in \dom f$.

See also

• sequence

• sequence space

• function algebra

• differential algebra

• sequentially Hausdorff space

• analytic function