DiscreteRatio
✖
DiscreteRatio
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 allBasic Examples (4)Summary of the most common use cases
Discrete ratio with respect to i:

https://wolfram.com/xid/0i1otqgma-du5kh1

Discrete ratio for a geometric progression corresponds to the ratio:

https://wolfram.com/xid/0i1otqgma-c7rabl

Enter using dratio
, and subscripts using
:

https://wolfram.com/xid/0i1otqgma-h8rz92

Discrete ratio is the inverse operator to Product:

https://wolfram.com/xid/0i1otqgma-babd1


https://wolfram.com/xid/0i1otqgma-m3bg

Scope (20)Survey of the scope of standard use cases
Basic Use (4)

https://wolfram.com/xid/0i1otqgma-h929r4


https://wolfram.com/xid/0i1otqgma-nvljol

Explicit shift structure in the function will typically be canceled:

https://wolfram.com/xid/0i1otqgma-hh8aii


https://wolfram.com/xid/0i1otqgma-na7z5f

Compute the discrete ratio of step h:

https://wolfram.com/xid/0i1otqgma-geh1vp

The second discrete ratio of step h:

https://wolfram.com/xid/0i1otqgma-f24c2y

Compute the partial discrete ratio:

https://wolfram.com/xid/0i1otqgma-b4qk0b


https://wolfram.com/xid/0i1otqgma-cyk09g


https://wolfram.com/xid/0i1otqgma-f3djro

Special Sequences (14)
Polynomials have rational function ratios:

https://wolfram.com/xid/0i1otqgma-bgmzmy

The root locations get shifted:

https://wolfram.com/xid/0i1otqgma-cnnnyz

Rational functions have rational function ratios:

https://wolfram.com/xid/0i1otqgma-bhr4d

The root and pole locations get shifted:

https://wolfram.com/xid/0i1otqgma-elasy2

Factorial functions have rational ratios including FactorialPower:

https://wolfram.com/xid/0i1otqgma-660wx


https://wolfram.com/xid/0i1otqgma-kjn657


https://wolfram.com/xid/0i1otqgma-g5c2fz


https://wolfram.com/xid/0i1otqgma-vd0os


https://wolfram.com/xid/0i1otqgma-f0ezso


https://wolfram.com/xid/0i1otqgma-gukyy


https://wolfram.com/xid/0i1otqgma-frh37i


https://wolfram.com/xid/0i1otqgma-jaf6az


https://wolfram.com/xid/0i1otqgma-dzmbug


https://wolfram.com/xid/0i1otqgma-w0xyy

Exponential sequences have constant ratios:

https://wolfram.com/xid/0i1otqgma-jjys72

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

https://wolfram.com/xid/0i1otqgma-qlza1v


https://wolfram.com/xid/0i1otqgma-c4yv6t

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

https://wolfram.com/xid/0i1otqgma-icgwi4

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

https://wolfram.com/xid/0i1otqgma-ilo2ki


https://wolfram.com/xid/0i1otqgma-n46tj

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

https://wolfram.com/xid/0i1otqgma-1jabl

The roots are shifted in a geometric fashion:

https://wolfram.com/xid/0i1otqgma-soe61

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

https://wolfram.com/xid/0i1otqgma-d2vbfc

The roots and poles are shifted in a geometric fashion:

https://wolfram.com/xid/0i1otqgma-xrs8

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

https://wolfram.com/xid/0i1otqgma-c5180g


https://wolfram.com/xid/0i1otqgma-fcywrw


https://wolfram.com/xid/0i1otqgma-mn51e


https://wolfram.com/xid/0i1otqgma-bdxqzi


https://wolfram.com/xid/0i1otqgma-ezyfzd


https://wolfram.com/xid/0i1otqgma-bvpnbh

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

https://wolfram.com/xid/0i1otqgma-fergjc

Products of factorial functions have factorial ratios, including BarnesG:

https://wolfram.com/xid/0i1otqgma-glhj8z

Then the second ratio is rational:

https://wolfram.com/xid/0i1otqgma-d191rr

Hyperfactorial is a product of ii:

https://wolfram.com/xid/0i1otqgma-ftzncj

A multivariate hypergeometric term is hypergeometric in each variable:

https://wolfram.com/xid/0i1otqgma-7zjki

The binomial distribution is a multivariate hypergeometric term:

https://wolfram.com/xid/0i1otqgma-dgwqik


https://wolfram.com/xid/0i1otqgma-hdj5r

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

https://wolfram.com/xid/0i1otqgma-fvu3en

This gives a simple expression for the ratio:

https://wolfram.com/xid/0i1otqgma-bo7j21


https://wolfram.com/xid/0i1otqgma-fdvpgv

Similarly for BetaRegularized:

https://wolfram.com/xid/0i1otqgma-uhv1u

The difference for MarcumQ is expressed in terms of BesselI:

https://wolfram.com/xid/0i1otqgma-f3i8f7

Special Operators (2)
DiscreteRatio is the inverse operator to Product:

https://wolfram.com/xid/0i1otqgma-cd5x8


https://wolfram.com/xid/0i1otqgma-emsij1


https://wolfram.com/xid/0i1otqgma-ia7t0x


https://wolfram.com/xid/0i1otqgma-hwsf3b


https://wolfram.com/xid/0i1otqgma-1zzx8


https://wolfram.com/xid/0i1otqgma-edlgry


https://wolfram.com/xid/0i1otqgma-lw8b29

In this case the variable x is scoped:

https://wolfram.com/xid/0i1otqgma-8gj0


https://wolfram.com/xid/0i1otqgma-cevrn2

Applications (6)Sample problems that can be solved with this function
The defining property for a geometric sequence is that its DiscreteRatio is constant:

https://wolfram.com/xid/0i1otqgma-bwohw5

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

https://wolfram.com/xid/0i1otqgma-b7uun7

DiscreteRatio gives the interest rate the compounding sequence:

https://wolfram.com/xid/0i1otqgma-dgjcad

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

https://wolfram.com/xid/0i1otqgma-g6vc6i

Synthesize tones directly from frequencies:

https://wolfram.com/xid/0i1otqgma-g4vsbk

https://wolfram.com/xid/0i1otqgma-bmjowm


https://wolfram.com/xid/0i1otqgma-iy6sfz

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

https://wolfram.com/xid/0i1otqgma-e3ja4c
Compute the DiscreteRatio for this series:

https://wolfram.com/xid/0i1otqgma-or9ve

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

https://wolfram.com/xid/0i1otqgma-gw0z8n

Verify the result using SumConvergence:

https://wolfram.com/xid/0i1otqgma-dvwrjt

Verify the answer for an indefinite product:

https://wolfram.com/xid/0i1otqgma-9t1bd

The DiscreteRatio of a product is equivalent to the factor:

https://wolfram.com/xid/0i1otqgma-c5lisv

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

https://wolfram.com/xid/0i1otqgma-j6dec


https://wolfram.com/xid/0i1otqgma-djbb2e

Properties & Relations (6)Properties of the function, and connections to other functions
DiscreteRatio is the inverse for indefinite Product:

https://wolfram.com/xid/0i1otqgma-bsci41


https://wolfram.com/xid/0i1otqgma-no0l1

DiscreteRatio distributes over products and integer powers:

https://wolfram.com/xid/0i1otqgma-fb9c44


https://wolfram.com/xid/0i1otqgma-dbot9s

DiscreteRatio is closely related to DifferenceDelta:

https://wolfram.com/xid/0i1otqgma-8mjio


https://wolfram.com/xid/0i1otqgma-f4yru

DiscreteRatio can be expressed in terms of DifferenceDelta:

https://wolfram.com/xid/0i1otqgma-feo3ek

https://wolfram.com/xid/0i1otqgma-mx7iez

Use Ratios to compute ratios of adjacent terms:

https://wolfram.com/xid/0i1otqgma-fjoou


https://wolfram.com/xid/0i1otqgma-b1s72h


https://wolfram.com/xid/0i1otqgma-b984cb


https://wolfram.com/xid/0i1otqgma-p78gf


https://wolfram.com/xid/0i1otqgma-b0giqo


https://wolfram.com/xid/0i1otqgma-c5ux4r

Use PowerRange to generate a list with constant ratio:

https://wolfram.com/xid/0i1otqgma-ga3i59


https://wolfram.com/xid/0i1otqgma-fjg8nd

This is the sequence 2k with constant ratio:

https://wolfram.com/xid/0i1otqgma-e4kiuv

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
]}
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
]}