WOLFRAM

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

open allclose all

Basic Examples  (4)Summary of the most common use cases

Discrete ratio with respect to i:

Out[2]=2

Discrete ratio for a geometric progression corresponds to the ratio:

Out[1]=1

Enter using dratio, and subscripts using :

Out[1]=1

Discrete ratio is the inverse operator to Product:

Out[1]=1
Out[2]=2

Scope  (20)Survey of the scope of standard use cases

Basic Use  (4)

Compute the discrete ratio:

Out[1]=1

The second discrete ratio:

Out[2]=2

Explicit shift structure in the function will typically be canceled:

Out[1]=1
Out[2]=2

Compute the discrete ratio of step h:

Out[1]=1

The second discrete ratio of step h:

Out[2]=2

Compute the partial discrete ratio:

Out[1]=1

Mix any orders:

Out[2]=2

Or any steps:

Out[3]=3

Special Sequences  (14)

Polynomials have rational function ratios:

Out[6]=6

The root locations get shifted:

Out[8]=8

Rational functions have rational function ratios:

Out[1]=1

The root and pole locations get shifted:

Out[2]=2

Factorial functions have rational ratios including FactorialPower:

Out[1]=1
Out[2]=2
Out[3]=3

Pochhammer:

Out[4]=4
Out[5]=5
Out[6]=6

Factorial and Gamma:

Out[7]=7
Out[8]=8

Binomial:

Out[9]=9
Out[10]=10

Exponential sequences have constant ratios:

Out[1]=1

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

Out[2]=2
Out[3]=3

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

Out[1]=1

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

Out[2]=2
Out[3]=3

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

Out[1]=1

The roots are shifted in a geometric fashion:

Out[2]=2

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

Out[1]=1

The roots and poles are shifted in a geometric fashion:

Out[2]=2

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

Out[1]=1
Out[2]=2
Out[3]=3

QFactorial:

Out[4]=4

QBinomial:

Out[5]=5
Out[6]=6

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

Out[1]=1

Products of factorial functions have factorial ratios, including BarnesG:

Out[1]=1

Then the second ratio is rational:

Out[2]=2

Hyperfactorial is a product of ii:

Out[1]=1

A multivariate hypergeometric term is hypergeometric in each variable:

Out[1]=1

The binomial distribution is a multivariate hypergeometric term:

Out[2]=2
Out[3]=3

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

Out[1]=1

This gives a simple expression for the ratio:

Out[2]=2
Out[3]=3

Similarly for BetaRegularized:

Out[4]=4

The difference for MarcumQ is expressed in terms of BesselI:

Out[1]=1

Special Operators  (2)

DiscreteRatio is the inverse operator to Product:

Out[1]=1
Out[2]=2

Definite products:

Out[3]=3

Multivariate products:

Out[4]=4

Other special operators:

Out[1]=1
Out[2]=2
Out[3]=3

In this case the variable x is scoped:

Out[4]=4
Out[5]=5

Applications  (6)Sample problems that can be solved with this function

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

Out[1]=1

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

Out[1]=1

DiscreteRatio gives the interest rate the compounding sequence:

Out[2]=2

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

Out[1]=1

Synthesize tones directly from frequencies:

Out[3]=3

Compare to a note scale:

Out[4]=4

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

Compute the DiscreteRatio for this series:

Out[2]=2

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

Out[3]=3

Verify the result using SumConvergence:

Out[4]=4

Verify the answer for an indefinite product:

Out[1]=1

The DiscreteRatio of a product is equivalent to the factor:

Out[2]=2

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

Out[1]=1
Out[2]=2

Properties & Relations  (6)Properties of the function, and connections to other functions

DiscreteRatio is the inverse for indefinite Product:

Out[1]=1
Out[2]=2

DiscreteRatio distributes over products and integer powers:

Out[1]=1
Out[2]=2

DiscreteRatio is closely related to DifferenceDelta:

Out[1]=1
Out[2]=2

DiscreteRatio can be expressed in terms of DifferenceDelta:

Out[2]=2

Use Ratios to compute ratios of adjacent terms:

Out[1]=1
Out[2]=2

Second-order ratios:

Out[3]=3
Out[4]=4

Ratios of step 2:

Out[5]=5
Out[6]=6

Use PowerRange to generate a list with constant ratio:

Out[1]=1
Out[2]=2

This is the sequence 2k with constant ratio:

Out[3]=3

Neat Examples  (1)Surprising or curious use cases

Create a gallery of discrete ratios:

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

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.

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

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

BibTeX

@misc{reference.wolfram_2025_discreteratio, author="Wolfram Research", title="{DiscreteRatio}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/DiscreteRatio.html}", note=[Accessed: 14-May-2025 ]}

@misc{reference.wolfram_2025_discreteratio, author="Wolfram Research", title="{DiscreteRatio}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/DiscreteRatio.html}", note=[Accessed: 14-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_discreteratio, organization={Wolfram Research}, title={DiscreteRatio}, year={2008}, url={https://reference.wolfram.com/language/ref/DiscreteRatio.html}, note=[Accessed: 14-May-2025 ]}

@online{reference.wolfram_2025_discreteratio, organization={Wolfram Research}, title={DiscreteRatio}, year={2008}, url={https://reference.wolfram.com/language/ref/DiscreteRatio.html}, note=[Accessed: 14-May-2025 ]}