Concrete Mathematics

Hypergeometric terms

The center stage, in recurrence solving and summations, play hypergeometric terms. Formally these are sequences annihilated by first order linear recurrence operators. In simple words if we are given term then it is hypergeometric if its consecutive term ratio is a rational function in .

To check if a sequence is of this type you can use method which is available in Basic class. Here is simple example involving a polynomial:

>>> from sympy import *
>>> n, k = symbols('nk')
>>> (n**2 + 1).is_hypergeometric(n)
True

Of course polynomials are hypergeometric but are there any more complicated sequences of this type? Here are some trivial examples:

>>> factorial(n).is_hypergeometric(n)
True
>>> binomial(n, k).is_hypergeometric(n)
True
>>> rf(n, k).is_hypergeometric(n)
True
>>> ff(n, k).is_hypergeometric(n)
True
>>> gamma(n).is_hypergeometric(n)
True
>>> (2**n).is_hypergeometric(n)
True

We see that all species used in summations and other parts of concrete mathematics are hypergeometric. Note also that binomial coefficients and both rising and falling factorials are hypergeometric in both their arguments:

>>> binomial(n, k).is_hypergeometric(k)
True
>>> rf(n, k).is_hypergeometric(k)
True
>>> ff(n, k).is_hypergeometric(k)
True

To say more, all previously shown examples are valid for integer linear arguments:

>>> factorial(2*n).is_hypergeometric(n)
True
>>> binomial(3*n+1, k).is_hypergeometric(n)
True
>>> rf(n+1, k-1).is_hypergeometric(n)
True
>>> ff(n-1, k+1).is_hypergeometric(n)
True
>>> gamma(5*n).is_hypergeometric(n)
True
>>> (2**(n-7)).is_hypergeometric(n)
True

However nonlinear arguments make those sequences fail to be hypergeometric:

>>> factorial(n**2).is_hypergeometric(n)
False
>>> (2**(n**3 + 1)).is_hypergeometric(n)
False

If not only the knowledge of being hypergeometric or not is needed, you can use hypersimp() function. It will try to simplify combinatorial expression and if the term given is hypergeometric it will return a quotient of polynomials of minimal degree. Otherwise is will return to say that sequence is not hypergeometric:

>>> hypersimp(factorial(2*n), n)
2 + 6*n + 4*n**2
>>> hypersimp(factorial(n**2), n)