DiscreteRatio

The ratio $f(n+1)/f(n)$ between consecutive terms of a sequence, simplified.

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DiscreteRatio(f, n)f(n+1)/f(n), simplified

Typed signatures

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

Domain: Combinatorics

Details

• Substitutes $n \to n+1$ into $f$ and simplifies the quotient; stays symbolic when the ratio does not collapse further (e.g. a ratio of two named sequences with no known closed form).

Examples

Basic #

$(n+1)!/n! = n + 1$

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In$\mathrm{DiscreteRatio}(n!, n)$Out$n+1$

a constant ratio for a geometric sequence

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In$\mathrm{DiscreteRatio}(2^{n}, n)$Out$2$

A geometric term has constant ratio $r$

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In$\mathrm{DiscreteRatio}(r^{k}, k)$Out$r$

Ratio $\frac{(x)_{k+1}}{(x)_k}$, which simplifies to $x+k$

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In$\mathrm{DiscreteRatio}(\mathrm{Pochhammer}(x, k), k)$Out$\frac{\mathrm{Pochhammer}(x, k+1)}{\mathrm{Pochhammer}(x, k)}$

Scope #

Ratio of consecutive squares, $\frac{(x+1)^2}{x^2}$

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In$\mathrm{DiscreteRatio}(x^2, x)$Out$\frac{(x+1)^2}{x^2}$

A product of linear factors shifts each factor

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In$\mathrm{DiscreteRatio}((x-r_1)(x-r_2), x)$Out$\frac{(-r_1+x+1)(-r_2+x+1)}{(x-r_1)(x-r_2)}$

A rational function gives a rational ratio

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In$\mathrm{DiscreteRatio}(\frac{1+x}{1+x^2}, x)$Out$\frac{x^3+2x^2+x+2}{(x+1)((x+1)^2+1)}$

A hypergeometric-style numerator over denominator

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In$\mathrm{DiscreteRatio}(\frac{(x-r_1)(x-r_2)}{(x-p_1)(x-p_2)}, x)$Out$\frac{(-r_1+x+1)(-r_2+x+1)(x-p_1)(x-p_2)}{(-p_1+x+1)(-p_2+x+1)(x-r_1)(x-r_2)}$

Ratio of falling factorials in the lower index

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In$\mathrm{DiscreteRatio}(\mathrm{FallingFactorial}(a, x), x)$Out$\frac{\mathrm{FallingFactorial}(a, x+1)}{\mathrm{FallingFactorial}(a, x)}$

Rising factorial ratio in its length

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In$\mathrm{DiscreteRatio}(\mathrm{Pochhammer}(a, x), x)$Out$\frac{\mathrm{Pochhammer}(a, x+1)}{\mathrm{Pochhammer}(a, x)}$

$\frac{(x+1)!}{x!}=x+1$ simplifies fully

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In$\mathrm{DiscreteRatio}(x!, x)$Out$x+1$

Gamma ratio left as $\Gamma(x+1)/\Gamma(x)$

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In$\mathrm{DiscreteRatio}(Γ(x), x)$Out$\frac{Γ(x+1)}{Γ(x)}$

Binomial ratio in the top argument

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In$\mathrm{DiscreteRatio}(\binom{x}{k}, x)$Out$\frac{\binom{x+1}{k}}{\binom{x}{k}}$

Binomial ratio in the bottom argument

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In$\mathrm{DiscreteRatio}(\binom{n}{x}, x)$Out$\frac{\binom{n}{x+1}}{\binom{n}{x}}$

Constant ratio $r$ again, with continuous-looking variable

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In$\mathrm{DiscreteRatio}(r^{x}, x)$Out$r$

A mixed hypergeometric term assembled from each factor's ratio

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In$\mathrm{DiscreteRatio}(\frac{\mathrm{FallingFactorial}(x, a_1)\binom{x}{a_2}r^{x}}{\mathrm{Pochhammer}(x, b_1)x!}, x)$Out$\frac{x!\mathrm{Pochhammer}(x, b_1)\mathrm{FallingFactorial}(x+1, a_1)\binom{x+1}{a_2}{{r}}^{x+1}}{(x+1)!\mathrm{Pochhammer}(x+1, b_1)\mathrm{FallingFactorial}(x, a_1)\binom{x}{a_2}{{r}}^{x}}$

Catalan numbers as a ratio of consecutive terms

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In$\mathrm{DiscreteRatio}(\mathrm{CatalanNumber}(x), x)$Out$\frac{\mathrm{CatalanNumber}(x+1)}{\mathrm{CatalanNumber}(x)}$

A term not depending on the variable has ratio $1$

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In$\bigl\lbrack\mathrm{DiscreteRatio}(f, n), \mathrm{DiscreteRatio}(f, k)\bigr\rbrack$Out$\bigl\lbrack1, 1\bigr\rbrack$

Ratio in the shape parameter of regularized gamma

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In$\mathrm{DiscreteRatio}(\operatorname{GammaRegularized}(n, z), n)$Out$\frac{\operatorname{GammaRegularized}(n+1, z)}{\operatorname{GammaRegularized}(n, z)}$

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In$\bigl\lbrack\mathrm{DiscreteRatio}(\binom{n}{k}, n), \mathrm{DiscreteRatio}(\binom{n}{k}, k)\bigr\rbrack$Out$\bigl\lbrack\frac{\binom{n+1}{k}}{\binom{n}{k}}, \frac{\binom{n}{k+1}}{\binom{n}{k}}\bigr\rbrack$

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In$\mathrm{DiscreteRatio}(aq^{x}+bq^{2x}, x)$Out$\frac{aq^{x+1}+bq^{2(x+1)}}{aq^{x}+bq^{2x}}$

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In$\mathrm{DiscreteRatio}((q^{x}-r_1)(q^{x}-r_2), x)$Out$\frac{(q^{x+1}-r_1)(q^{x+1}-r_2)}{(q^{x}-r_1)(q^{x}-r_2)}$

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In$\mathrm{DiscreteRatio}(\frac{aq^{x}+bq^{2x}}{1+q^{x}}, x)$Out$\frac{(aq^{x+1}+bq^{2(x+1)})(q^{x}+1)}{(aq^{x}+bq^{2x})(q^{x+1}+1)}$

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In$\mathrm{DiscreteRatio}(\frac{(q^{x}-r_1)(q^{x}-r_2)}{(q^{x}-p_1)(q^{x}-p_2)}, x)$Out$\frac{(q^{x}-p_1)(q^{x}-p_2)(q^{x+1}-r_1)(q^{x+1}-r_2)}{(q^{x+1}-p_1)(q^{x+1}-p_2)(q^{x}-r_1)(q^{x}-r_2)}$

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In$\mathrm{DiscreteRatio}(\mathrm{QPochhammer}(a, q, x), x)$Out$\frac{\mathrm{QPochhammer}(a, q, x+1)}{\mathrm{QPochhammer}(a, q, x)}$

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In$\mathrm{DiscreteRatio}(\frac{\mathrm{QBinomial}(a, x, q)\mathrm{QFactorial}(x, q)}{\mathrm{QPochhammer}(b, q, x)}, x)$Out$\frac{\mathrm{QPochhammer}(b, q, x)\mathrm{QFactorial}(x+1, q)\mathrm{QBinomial}(a, x+1, q)}{\mathrm{QPochhammer}(b, q, x+1)\mathrm{QFactorial}(x, q)\mathrm{QBinomial}(a, x, q)}$

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In$\mathrm{DiscreteRatio}(\mathrm{BarnesG}(x), x)$Out$Γ(x)$

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In$\mathrm{DiscreteRatio}(\mathrm{Hyperfactorial}(x), x)$Out$(x+1)^{x+1}$

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In$\mathrm{DiscreteRatio}(\operatorname{BetaRegularized}(z, n, m), m)$Out$\frac{\operatorname{BetaRegularized}(z, n, m+1)}{\operatorname{BetaRegularized}(z, n, m)}$

Applications #

$\frac{c^{a(k+1)+b}}{c^{ak+b}}=c^a$

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In$\mathrm{DiscreteRatio}(c^{ak+b}, k)$Out$c^{a}$

About

Examples	30
Checked against	Wolfram Language 30 agree
Crosswalk	Wolfram Language

See also: GeneratingFunction, FindSequenceFunction