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.
Examplesopen allclose all
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:
Basic Use (4)
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:
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:
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:
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:
Use PowerRange to generate a list with constant ratio:
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