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:

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 :

Pochhammer :

Factorial and Gamma :

Binomial :

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 :

QFactorial :

QBinomial :

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 :

DiscreteRatio is the inverse operator to Product :

Definite products:

Multivariate products:

Other special operators:

In this case the variable x is scoped: