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def finite_diff(expression, variable, increment=1):
"""
Takes as input a polynomial expression and the variable used to construct
it and returns the difference between function's value when the input is
incremented to 1 and the original function value. If you want an increment
other than one supply it as a third argument.
Examples
========
>>> from sympy.abc import x, y, z
>>> from sympy.series.kauers import finite_diff
>>> finite_diff(x**2, x)
2*x + 1
>>> finite_diff(y**3 + 2*y**2 + 3*y + 4, y)
3*y**2 + 7*y + 6
>>> finite_diff(x**2 + 3*x + 8, x, 2)
4*x + 10
>>> finite_diff(z**3 + 8*z, z, 3)
9*z**2 + 27*z + 51
"""
expression = expression.expand()
expression2 = expression.subs(variable, variable + increment)
expression2 = expression2.expand()
return expression2 - expression
def finite_diff_kauers(sum):
"""
Takes as input a Sum instance and returns the difference between the sum
with the upper index incremented by 1 and the original sum. For example,
if S(n) is a sum, then finite_diff_kauers will return S(n + 1) - S(n).
Examples
========
>>> from sympy.series.kauers import finite_diff_kauers
>>> from sympy import Sum
>>> from sympy.abc import x, y, m, n, k
>>> finite_diff_kauers(Sum(k, (k, 1, n)))
n + 1
>>> finite_diff_kauers(Sum(1/k, (k, 1, n)))
1/(n + 1)
>>> finite_diff_kauers(Sum((x*y**2), (x, 1, n), (y, 1, m)))
(m + 1)**2*(n + 1)
>>> finite_diff_kauers(Sum((x*y), (x, 1, m), (y, 1, n)))
(m + 1)*(n + 1)
"""
function = sum.function
for l in sum.limits:
function = function.subs(l[0], l[- 1] + 1)
return function
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