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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, ....
  • 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.
EXAMPLESCLOSE ALL
Basic Examples   (5)
Shift ratio with respect to i:
Shift ratio of factorial related functions:
Enter DiscreteRatio using Esc dratio Esc, and subscript Ctrl+_:
Shift ratio is the inverse operator to Product:  »
Shift ratio is the inverse operator to Product:  »
Shift ratio with respect to i:
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Shift ratio of factorial related functions:
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Click for copyable input
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Enter DiscreteRatio using Esc dratio Esc, and subscript Ctrl+_:
In[1]:=
Click for copyable input
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Shift ratio is the inverse operator to Product:  »
In[1]:=
Click for copyable input
Out[1]=
In[2]:=
Click for copyable input
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Shift ratio is the inverse operator to Product:  »
In[1]:=
Click for copyable input
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In[2]:=
Click for copyable input
Out[2]=
Scope   (15)
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:
Polynomial functions:
An explicit shift structure between roots cancels:
FactorialPower and Pochhammer have an explicit shift structure:
Rational functions:
Explicit shift ratio cancels:
Exponential functions:
DiscreteRatio of exponentials is closely related to DifferenceDelta:
Polynomial exponents:
Hypergeometric terms are defined by having a rational discrete ratio:
CatalanNumber is a hypergeometric term:
Q-polynomial functions:
An explicit shift structure in the exponents and roots cancels:
QPochhammer has an explicit shift structure:
Q-rational functions:
An explicit shift structure in exponents and roots cancels:
QBinomial has explicit shift structure:
Q-hypergeometric terms are defined by having a q-rational discrete ratio:
BarnesG is a product of Gamma functions:
The second discrete will be rational:
Hyperfactorial is a product of ii:
A multivariate hypergeometric term is hypergeometric in each variable:
The binomial distribution is a multivariate hypergeometric term:
DiscreteRatio is the inverse operator to Product:
Definite products:
Other special operators:
In this case the variable x is scoped:
Applications   (5)
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 series with ratio :
Synthesize tones directly from frequencies:
Compare to a note scale:
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   (2)
DiscreteRatio distributes over products and integer powers:
DiscreteRatio is closely related to DifferenceDelta:
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