# DiscreteRatio

DiscreteRatio[f,i]

gives the discrete ratio .

DiscreteRatio[f,{i,n}]

gives the multiple discrete ratio.

DiscreteRatio[f,{i,n,h}]

gives the multiple discrete ratio with step h.

DiscreteRatio[f,i,j,]

computes the partial difference ratio with respect to i, j, .

# Details and Options

• DiscreteRatio[f,i] can be input as if. The character is entered dratio or as \[DiscreteRatio]. The variable i is entered as a subscript.
• All quantities that do not explicitly depend on the variables given are taken to have discrete ratio equal to one.
• A multiple discrete ratio is defined recursively in terms of lower discrete ratios.
• Discrete ratio is the inverse operator to indefinite product. »
• DiscreteRatio[f,,Assumptions->assum] uses the assumptions assum in the course of computing discrete ratios.

# Examples

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## Basic Examples(4)

Discrete ratio with respect to i:

Discrete ratio for a geometric progression corresponds to the ratio:

Enter using dratio, and subscripts using :

Discrete ratio is the inverse operator to Product:

## Scope(20)

### Basic Use(4)

Compute the discrete ratio:

The second discrete ratio:

Explicit shift structure in the function will typically be canceled:

Compute the discrete ratio of step h:

The second discrete ratio of step h:

Compute the partial discrete ratio:

Mix any orders:

Or any steps:

### Special Sequences(14)

Polynomials have rational function ratios:

The root locations get shifted:

Rational functions have rational function ratios:

The root and pole locations get shifted:

Factorial functions have rational ratios including FactorialPower:

Factorial and Gamma:

Exponential sequences have constant ratios:

The ratio of an exponential sequence corresponds to the DifferenceDelta of the exponent:

Hypergeometric terms are products of factorial, rational, and exponential functions:

Hypergeometric terms have rational ratios, so CatalanNumber is a hypergeometric term:

Q-polynomials (polynomials of exponentials) have q-rational ratios:

The roots are shifted in a geometric fashion:

Q-rational functions (rational functions of exponentials) have q-rational ratios:

The roots and poles are shifted in a geometric fashion:

Q-factorial functions have q-rational ratios including QPochhammer:

Q-hypergeometric terms are defined by having a q-rational discrete ratio:

Products of factorial functions have factorial ratios, including BarnesG:

Then the second ratio is rational:

Hyperfactorial is a product of ii:

A multivariate hypergeometric term is hypergeometric in each variable:

The binomial distribution is a multivariate hypergeometric term:

The difference of GammaRegularized with respect to is a hypergeometric term:

This gives a simple expression for the ratio:

Similarly for BetaRegularized:

The difference for MarcumQ is expressed in terms of BesselI:

### Special Operators(2)

DiscreteRatio is the inverse operator to Product:

Definite products:

Multivariate products:

Other special operators:

In this case the variable x is scoped:

## Applications(6)

The defining property for a geometric sequence is that its DiscreteRatio is constant:

Solve a compound interest problem with interest rate 1+r:

DiscreteRatio gives the interest rate the compounding sequence:

The frequencies used in an even-tempered scale form a geometric progression with ratio :

Synthesize tones directly from frequencies:

Compare to a note scale:

Use the ratio test to verify convergence of a series whose general term is given by:

Compute the DiscreteRatio for this series:

The series converges since the limit at infinity of the ratio is less than 1:

Verify the result using SumConvergence:

Verify the answer for an indefinite product:

The DiscreteRatio of a product is equivalent to the factor:

Verify the solution from RSolve using a higher-step shift ratio:

## Properties & Relations(6)

DiscreteRatio is the inverse for indefinite Product:

DiscreteRatio distributes over products and integer powers:

DiscreteRatio is closely related to DifferenceDelta:

DiscreteRatio can be expressed in terms of DifferenceDelta:

Use Ratios to compute ratios of adjacent terms:

Second-order ratios:

Ratios of step 2:

Use PowerRange to generate a list with constant ratio:

This is the sequence 2k with constant ratio:

## Neat Examples(1)

Create a gallery of discrete ratios:

Wolfram Research (2008), DiscreteRatio, Wolfram Language function, https://reference.wolfram.com/language/ref/DiscreteRatio.html.

#### Text

Wolfram Research (2008), DiscreteRatio, Wolfram Language function, https://reference.wolfram.com/language/ref/DiscreteRatio.html.

#### CMS

Wolfram Language. 2008. "DiscreteRatio." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/DiscreteRatio.html.

#### APA

Wolfram Language. (2008). DiscreteRatio. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/DiscreteRatio.html

#### BibTeX

@misc{reference.wolfram_2024_discreteratio, author="Wolfram Research", title="{DiscreteRatio}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/DiscreteRatio.html}", note=[Accessed: 20-May-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_discreteratio, organization={Wolfram Research}, title={DiscreteRatio}, year={2008}, url={https://reference.wolfram.com/language/ref/DiscreteRatio.html}, note=[Accessed: 20-May-2024 ]}