DiscreteShift

The forward-shift operator: DiscreteShift(f, n) is f with every free n replaced by n + 1.

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DiscreteShift(f, n)f with every free occurrence of the symbol n replaced by n + 1.

DiscreteShift(f, {n, h})f with every free occurrence of n replaced by n + h -- an h-step shift, given as the {n, h} pair.

Typed signatures

Type	Package	Applies
(expression, expression) -> expression	analytic	always

Domain: Elementary functions

Details

• A substitution, not an arithmetic evaluation -- f need not itself be a declared function; DiscreteShift(a(n), n) is a(n + 1) even though a has no definition.
• The h-step form takes n and h together as the second argument, {n, h} -- not a third argument.

Examples

Basic #

One-step forward shift

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In$\mathrm{DiscreteShift}(a(n), n)$Out$a(n+1)$

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In$\mathrm{DiscreteShift}(\sin((-\frac{2}{7}i)π), \bigl\lbrack i, 10\bigr\rbrack)$Out$\mathrm{DiscreteShift}(i\sinh(-\frac{2π}{7}), \bigl\lbrack i, 10\bigr\rbrack)$

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In$\mathrm{DiscreteShift}(\sin(\frac{1}{7}×2πk^3), \bigl\lbrack k, 10\bigr\rbrack)$Out$\sin(\frac{1}{7}×2π(k+10)^3)$

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In$\mathrm{DiscreteShift}(\mathrm{Fibonacci}(k), \bigl\lbrack k, 5\bigr\rbrack)$Out$\mathrm{Fibonacci}(k+5)$

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In$\mathrm{DiscreteShift}(k!, \bigl\lbrack k, 5\bigr\rbrack)$Out$(k+5)!$

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In$\mathrm{DiscreteShift}(f^{\prime}(x), k)$Out$f^{\prime}(x)$

Scope #

The h-step form -- {n, h}, not a third argument

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In$\mathrm{DiscreteShift}(a(n), \bigl\lbrack n, 2\bigr\rbrack)$Out$a(n+2)$

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In$\mathrm{DiscreteShift}(\lfloor k\rfloor, k)$Out$\lfloor k+1\rfloor$

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In$\mathrm{DiscreteShift}(\mathrm{round}(k), \bigl\lbrack k, 3\bigr\rbrack)$Out$\mathrm{round}(k+3)$

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In$\mathrm{DiscreteShift}(2^{k}, k)$Out$2^{k+1}$

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In$\mathrm{DiscreteShift}(k^4+11k^3+20k^2, \bigl\lbrack k, 3\bigr\rbrack)$Out$(k+3)^4+11(k+3)^3+20(k+3)^2$

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In$\mathrm{DiscreteShift}(\frac{k^2+5}{k-7}, k)$Out$\frac{(k+1)^2+5}{k-6}$

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In$\mathrm{DiscreteShift}(\mathrm{FractionalPart}(k), k)$Out$\mathrm{FractionalPart}(k+1)$

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In$\mathrm{DiscreteShift}(Γ(k), \bigl\lbrack k, 3\bigr\rbrack)$Out$Γ(k+3)$

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In$\mathrm{DiscreteShift}(\mathrm{CatalanNumber}(k), \bigl\lbrack k, 3\bigr\rbrack)$Out$\mathrm{CatalanNumber}(k+3)$

About

Examples	15
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