DifferenceDelta

f(n+1) - f(n), simplified -- the forward difference of a sequence.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageDifferenceDelta!↗

DifferenceDelta(f, n)f(n+1) - f(n), simplified

SymPydifference_delta(f, n)

Typed signatures

Type	Package	Applies
(any, symbol) -> any	combinatorics	always

Domain: Collections

Details

• Substitutes n -> n+1 into f and simplifies the difference; stays symbolic when it does not collapse further.

Examples

Basic #

$(n+1)^2 - n^2 = 2n + 1$

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In$\mathrm{DifferenceDelta}(n^2, n)$Out$2n+1$

The forward difference of a linear function is its slope, everywhere

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In$\mathrm{DifferenceDelta}(3n+5, n)$Out$3$

Forward difference of a symbolic falling factorial

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In$\mathrm{DifferenceDelta}(\mathrm{FallingFactorial}(k, n), k)$Out$\mathrm{FallingFactorial}(k+1, n)-\mathrm{FallingFactorial}(k, n)$

Scope #

Naming the index $i$ should keep it a bound variable, not the imaginary unit; compute-engine lifts a variable i to the imaginary unit when it boxes, so the difference comes back over $i$ and $1 + i$

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$\Delta k^3=(k+1)^3-k^3$, left unexpanded

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In$\mathrm{DifferenceDelta}(k^3, k)$Out$(k+1)^3-k^3$

$\Delta k^{\underline3}=3k^{\underline2}$, here as $3k^2-3k$

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In$\mathrm{DifferenceDelta}(\mathrm{FallingFactorial}(k, 3), k)$Out$3k^2-3k$

A rational term gives a difference of two fractions

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In$\mathrm{DifferenceDelta}(\frac{k+1}{k+3}, k)$Out$\frac{k+2}{k+4}+\frac{-k-1}{k+3}$

A negative falling factorial index gives a rational difference

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In$\mathrm{DifferenceDelta}(\mathrm{FallingFactorial}(k, -3), k)$Out$\frac{1}{(-k-3)(-k-2)(-k-1)}-\frac{1}{(-k-4)(-k-3)(-k-2)}$

Works with the Hurwitz zeta in its second argument

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In$\mathrm{DifferenceDelta}(ζ(2, k), k)$Out$ζ(2, k+1)-ζ(2, k)$

Exponentials: $\Delta a^k=a^{k+1}-a^k$

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In$\mathrm{DifferenceDelta}(a^{k}, k)$Out$a^{k+1}-a^{k}$

$\Delta 2^k=2^{k+1}-2^k=2^k$, shown unsimplified

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

Polynomial times exponential uses the product shift

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In$\mathrm{DifferenceDelta}((k^2+k+1)×2^{k}, k)$Out$((k+1)^2+k+2)×2^{k+1}-(k^2+k+1)×2^{k}$

Rational times exponential, stays as a difference of terms

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In$\mathrm{DifferenceDelta}(\frac{(k+1)×2^{k}}{k+3}, k)$Out$\frac{(k+2)×2^{k+1}}{k+4}-\frac{(k+1)×2^{k}}{k+3}$

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In$\mathrm{DifferenceDelta}(\sin(ak+b), k)$Out$\sin(ak+a+b)-\sin(ak+b)$

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In$\mathrm{DifferenceDelta}(\sinh(ak+b), k)$Out$\sinh(a×(k+1)+b)-\sinh(ak+b)$

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In$\mathrm{DifferenceDelta}(\mathrm{Pochhammer}(k, n), k)$Out$\mathrm{Pochhammer}(k+1, n)-\mathrm{Pochhammer}(k, n)$

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In$\mathrm{DifferenceDelta}(k!, k)$Out$kk!$

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In$\mathrm{DifferenceDelta}(\binom{k+n}{n}, k)$Out$\binom{k+n+1}{n}-\binom{k+n}{n}$

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In$\mathrm{DifferenceDelta}(Γ(k), k)$Out$Γ(k+1)-Γ(k)$

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In$\mathrm{DifferenceDelta}(\mathrm{B}(k, n), k)$Out$\mathrm{B}(k+1, n)-\mathrm{B}(k, n)$

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In$\mathrm{DifferenceDelta}(\frac{\mathrm{FallingFactorial}(a, k)}{\mathrm{FallingFactorial}(b, k)}, k)$Out$\frac{\mathrm{FallingFactorial}(a, k+1)}{\mathrm{FallingFactorial}(b, k+1)}-\frac{\mathrm{FallingFactorial}(a, k)}{\mathrm{FallingFactorial}(b, k)}$

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In$\mathrm{DifferenceDelta}(\operatorname{GammaRegularized}(k, z), k)$Out$\operatorname{GammaRegularized}(k+1, z)-\operatorname{GammaRegularized}(k, z)$

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In$\mathrm{DifferenceDelta}(\operatorname{BetaRegularized}(z, k, j), k)$Out$\operatorname{BetaRegularized}(z, k+1, j)-\operatorname{BetaRegularized}(z, k, j)$

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In$\mathrm{DifferenceDelta}(\operatorname{BetaRegularized}(z, k, j), j)$Out$\operatorname{BetaRegularized}(z, k, j+1)-\operatorname{BetaRegularized}(z, k, j)$

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In$\mathrm{DifferenceDelta}(\mathrm{QFactorial}(k, q), k)$Out$\mathrm{QFactorial}(k+1, q)-\mathrm{QFactorial}(k, q)$

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In$\mathrm{DifferenceDelta}(\mathrm{QPochhammer}(5, q, k), k)$Out$\mathrm{QPochhammer}(5, q, k+1)-\mathrm{QPochhammer}(5, q, k)$

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In$\mathrm{DifferenceDelta}(\mathrm{QBinomial}(k, m, q), k)$Out$\mathrm{QBinomial}(k+1, m, q)-\mathrm{QBinomial}(k, m, q)$

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In$\mathrm{DifferenceDelta}(\mathrm{QBinomial}(n, k, q), k)$Out$\mathrm{QBinomial}(n, k+1, q)-\mathrm{QBinomial}(n, k, q)$

About

Examples	28
Checked against	Wolfram Language 26 agree1 differSymPy 23 agree
Crosswalk	Wolfram LanguageSymPy

Implementation

mappedsympypackages/oracle/src/mappings.ts

See also: DiscreteRatio