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Description
This ticket would provide an interface to Maxima's and Sympy's recurrence-solving functions.
Maxima example:
sage: maxima.load('solve_rec')
?\/Users\/was\/s\/local\/share\/maxima\/5\.13\.0\/share\/contrib\/solve_rec\/solve_rec\.mac
sage: print maxima('solve_rec(a[n]=a[n-1]+a[n-2]+n/2^n, a[n])')
n n n
(sqrt(5) - 1) %k (- 1) (sqrt(5) + 1) %k
1 n 2 2
a = ------------------------- - ---- + ------------------ - ----
n n n n n
2 5 2 2 5 2
Sympy example:
>>> from sympy import Function, rsolve
>>> from sympy.abc import n
>>> y = Function('y')
>>> f = (n - 1)*y(n + 2) - (n**2 + 3*n - 2)*y(n + 1) + 2*n*(n + 1)*y(n)
>>> rsolve(f, y(n))
2**n*C0 + C1*factorial(n)
>>> rsolve(f, y(n), { y(0):0, y(1):3 })
3*2**n - 3*factorial(n)
The Maxima help:
sage: maxima.solve_rec?
Maxima 5.13.0 http://maxima.sourceforge.net
Using Lisp CLISP 2.41 (2006-10-13)
Distributed under the GNU Public License. See the file COPYING.
Dedicated to the memory of William Schelter.
This is a development version of Maxima. The function bug_report()
provides bug reporting information.
(%i1)
-- Function: solve_rec (<eqn>, <var>, [<init>])
Solves for hypergeometrical solutions to linear recurrence <eqn>
with polynomials coefficient in variable <var>. Optional arguments
<init> are initial conditions.
`solve_rec' can solve linear recurrences with constant
coefficients, finds hypergeometrical solutions to homogeneous
linear recurrences with polynomial coefficients, rational
solutions to linear recurrences with polynomial coefficients and
can solve Ricatti type recurrences.
Note that the running time of the algorithm used to find
hypergeometrical solutions is exponential in the degree of the
leading and trailing coefficient.
To use this function first load the `solve_rec' package with
`load(solve_rec);'.
Example of linear recurrence with constant coefficients:
(%i2) solve_rec(a[n]=a[n-1]+a[n-2]+n/2^n, a[n]);
n n
(sqrt(5) - 1) %k (- 1)
1 n
(%o2) a = ------------------------- - ----
n n n
2 5 2
n
(sqrt(5) + 1) %k
2 2
+ ------------------ - ----
n n
2 5 2
Example of linear recurrence with polynomial coefficients:
(%i7) 2*x*(x+1)*y[x] - (x^2+3*x-2)*y[x+1] + (x-1)*y[x+2];
2
(%o7) (x - 1) y - (x + 3 x - 2) y + 2 x (x + 1) y
x + 2 x + 1 x
(%i8) solve_rec(%, y[x], y[1]=1, y[3]=3);
x
3 2 x!
(%o9) y = ---- - --
x 4 2
Example of Ricatti type recurrence:
(%i2) x*y[x+1]*y[x] - y[x+1]/(x+2) + y[x]/(x-1) = 0;
y y
x + 1 x
(%o2) x y y - ------ + ----- = 0
x x + 1 x + 2 x - 1
(%i3) solve_rec(%, y[x], y[3]=5)$
(%i4) ratsimp(minfactorial(factcomb(%)));
3
30 x - 30 x
(%o4) y = - -------------------------------------------------
x 6 5 4 3 2
5 x - 3 x - 25 x + 15 x + 20 x - 12 x - 1584
See also: `solve_rec_rat', `simplify_products', and
`product_use_gamma'.
There are also some inexact matches for `solve_rec'.
Try `?? solve_rec' to see them.
CC: @burcin @sagetrac-kevin-stueve @kcrisman @robert-marik @eviatarbach @rwst @sagetrac-ktkohl
Component: calculus
Issue created by migration from https://trac.sagemath.org/ticket/1291