gives the discrete ratio .


gives the multiple discrete ratio.


gives the multiple discrete ratio with step h.


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.


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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 n 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.


Wolfram Research (2008), DiscreteRatio, Wolfram Language function, 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.


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


@misc{reference.wolfram_2024_discreteratio, author="Wolfram Research", title="{DiscreteRatio}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/DiscreteRatio.html}", note=[Accessed: 25-June-2024 ]}


@online{reference.wolfram_2024_discreteratio, organization={Wolfram Research}, title={DiscreteRatio}, year={2008}, url={https://reference.wolfram.com/language/ref/DiscreteRatio.html}, note=[Accessed: 25-June-2024 ]}