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Question:
is there an established name for the set $\Big\lbrace\ f {\Large\ \boldsymbol{|}}\ f\in C^\infty\quad {\Large\boldsymbol{\land}}\quad \exists\,{k\in\mathbb{N}^+}:\frac{d^{i+k}}{dx^{i+k}}f(x)=\frac{d^{i}}{dx^{i}}f(x)\ \forall i\in\mathbb{N}^+ \Big\rbrace{\Large\text{?}}$
E.g. $f(x)=e^x\implies k=1;\ f(x)=\cos(x)\,\lor\,f(x)=\sin(x)\implies k=4$.

Addendum:

As there doesn't seem to be an established already, I'm tempted to call that class of functions "derivative-periodic" or deriodic for short by means of inventing yet another portmanteau.

These deriodic functions can be viewed as generalizing the rationals to functions if the derivatives are interpreted as generalized digits.

These deriodic functions resemble good local approximations to smooth functions: $n$ deriodics suffice to approximate a function's value and the first $2n-1$ of its derivatives for a given $x_0$.

Question:
is there an established name for the set $\Big\lbrace\ f {\Large\ \boldsymbol{|}}\ f\in C^\infty\quad {\Large\boldsymbol{\land}}\quad \exists\,{k\in\mathbb{N}^+}:\frac{d^{i+k}}{dx^{i+k}}f(x)=\frac{d^{i}}{dx^{i}}f(x)\ \forall i\in\mathbb{N}^+ \Big\rbrace{\Large\text{?}}$
E.g. $f(x)=e^x\implies k=1;\ f(x)=\cos(x)\,\lor\,f(x)=\sin(x)\implies k=4$.

Addendum:

As there doesn't seem to be an established name already, I'm tempted to call that class of functions "derivative-periodic" or deriodic for short by means of inventing yet another portmanteau.

These deriodic functions can be viewed as generalizing the rationals to functions if the derivatives are interpreted as generalized digits.

These deriodic functions resemble good local approximations to smooth functions: $n$ deriodics suffice to approximate a function's value and the first $2n-1$ of its derivatives for a given $x_0$.

Question:
is there an established name for the set $\Big\lbrace\ f {\Large\ \boldsymbol{|}}\ f\in C^\infty\quad {\Large\boldsymbol{\land}}\quad \exists\,{k\in\mathbb{N}^+}:\frac{d^{i+k}}{dx^{i+k}}f(x)=\frac{d^{i}}{dx^{i}}f(x)\ \forall i\in\mathbb{N}^+ \Big\rbrace{\Large\text{?}}$
e.g. $f(x)=e^x\implies k=1;\ f(x)=\cos(x)\,\lor\,f(x)=\sin(x)\implies k=4$

Addendum:

as there doesn't seem to be an established already, I#m tempted to call that class of functions "derivative-periodic" or deriodic for short by means of inventing yet another Portmanteau.

These deriodic functions can be viewed as generalzing the rationals to functions if the derivatives are interpreted as generalized digits.

These deriodic functions resemble good local approximations to smooth functions: $n$ deriodics suffice to approximate a function'value and the first $2n-1$ of its derivatives for a given $x_0$.

Question:
is there an established name for the set $\Big\lbrace\ f {\Large\ \boldsymbol{|}}\ f\in C^\infty\quad {\Large\boldsymbol{\land}}\quad \exists\,{k\in\mathbb{N}^+}:\frac{d^{i+k}}{dx^{i+k}}f(x)=\frac{d^{i}}{dx^{i}}f(x)\ \forall i\in\mathbb{N}^+ \Big\rbrace{\Large\text{?}}$
E.g. $f(x)=e^x\implies k=1;\ f(x)=\cos(x)\,\lor\,f(x)=\sin(x)\implies k=4$.

Addendum:

As there doesn't seem to be an established already, I'm tempted to call that class of functions "derivative-periodic" or deriodic for short by means of inventing yet another portmanteau.

These deriodic functions can be viewed as generalizing the rationals to functions if the derivatives are interpreted as generalized digits.

These deriodic functions resemble good local approximations to smooth functions: $n$ deriodics suffice to approximate a function's value and the first $2n-1$ of its derivatives for a given $x_0$.

Question:
is there an established name for the set $\Big\lbrace\ f {\Large\ \boldsymbol{|}}\ f\in C^\infty\quad {\Large\boldsymbol{\land}}\quad \exists\,{k\in\mathbb{N}^+}:\frac{d^{i+k}}{dx^{i+k}}f(x)=\frac{d^{i}}{dx^{i}}f(x)\ \forall i\in\mathbb{N}^+ \Big\rbrace{\Large\text{?}}$
e.g. $f(x)=e^x\implies k=1;\ f(x)=\cos(x)\,\lor\,f(x)=\sin(x)\implies k=4$

Question:
is there an established name for the set $\Big\lbrace\ f {\Large\ \boldsymbol{|}}\ f\in C^\infty\quad {\Large\boldsymbol{\land}}\quad \exists\,{k\in\mathbb{N}^+}:\frac{d^{i+k}}{dx^{i+k}}f(x)=\frac{d^{i}}{dx^{i}}f(x)\ \forall i\in\mathbb{N}^+ \Big\rbrace{\Large\text{?}}$
e.g. $f(x)=e^x\implies k=1;\ f(x)=\cos(x)\,\lor\,f(x)=\sin(x)\implies k=4$

Addendum:

as there doesn't seem to be an established already, I#m tempted to call that class of functions "derivative-periodic" or deriodic for short by means of inventing yet another Portmanteau.

These deriodic functions can be viewed as generalzing the rationals to functions if the derivatives are interpreted as generalized digits.

These deriodic functions resemble good local approximations to smooth functions: $n$ deriodics suffice to approximate a function'value and the first $2n-1$ of its derivatives for a given $x_0$.