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solve_rec is a package for solving linear recurrences with polynomial
coefficients.
A demo is available with demo("solve_rec");.
Example:
(%i1) load("solve_rec")$
(%i2) solve_rec((n+4)*s[n+2] + s[n+1] - (n+1)*s[n], s[n]);
n
%k (2 n + 3) (- 1) %k
1 2
(%o2) s = -------------------- + ---------------
n (n + 1) (n + 2) (n + 1) (n + 2)
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When x is positive integer n, harmonic_number is
the n’th harmonic number. More generally,
harmonic_number(x) = psi[0](x+1) + %gamma. (See polygamma).
(%i1) load("simplify_sum")$
(%i2) harmonic_number(5);
137
(%o2) ---
60
(%i3) sum(1/k, k, 1, 5);
137
(%o3) ---
60
(%i4) float(harmonic_number(sqrt(2))); (%o4) %gamma + 0.6601971549171388
(%i5) float(psi[0](1+sqrt(2)))+%gamma; (%o5) %gamma + 0.6601971549171388
Converts expressions with harmonic_number to the equivalent
expression involving psi[0] (see polygamma).
(%i1) load("simplify_sum")$
(%i2) harmonic_to_psi(harmonic_number(sqrt(2)));
(%o2) psi (sqrt(2) + 1) + %gamma
0
Reduces the order of linear recurrence rec when a particular solution sol is known. The reduced reccurence can be used to get other solutions.
Example:
(%i3) rec: x[n+2] = x[n+1] + x[n]/n;
x
n
(%o3) x = x + --
n + 2 n + 1 n
(%i4) solve_rec(rec, x[n]);
WARNING: found some hypergeometrical solutions!
(%o4) x = %k n
n 1
(%i5) reduce_order(rec, n, x[n]);
(%t5) x = n %z
n n
n - 1
====
\
(%t6) %z = > %u
n / %j
====
%j = 0
(%o6) (- n - 2) %u - %u
n + 1 n
(%i6) solve_rec((n+2)*%u[n+1] + %u[n], %u[n]);
n
%k (- 1)
1
(%o6) %u = ----------
n (n + 1)!
So the general solution is
n - 1
==== j
\ (- 1)
%k n > -------- + %k n
2 / (j + 1)! 1
====
j = 0
Default value: true
If simplify_products is true, solve_rec will try to
simplify products in result.
See also: solve_rec.
Tries to simplify all sums appearing in expr to a closed form.
To use this function first load the simplify_sum package with
load("simplify_sum").
Example:
(%i1) load("simplify_sum")$
(%i2) sum(binomial(n+k,k)/2^k, k, 1, n) + sum(binomial(2*n, 2*k), k, 1,n);
n n
==== ====
\ binomial(n + k, k) \
(%o2) > ------------------ + > binomial(2 n, 2 k)
/ k /
==== 2 ====
k = 1 k = 1
(%i3) simplify_sum(%);
2 n - 1 n
(%o3) 2 + 2 - 2
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.
Solves for rational solutions to linear recurrences. See solve_rec for description of arguments.
To use this function first load the solve_rec package with
load("solve_rec");.
Example:
(%i1) (x+4)*a[x+3] + (x+3)*a[x+2] - x*a[x+1] + (x^2-1)*a[x];
(%o1) (x + 4) a + (x + 3) a - x a
x + 3 x + 2 x + 1
2
+ (x - 1) a
x
(%i2) solve_rec_rat(% = (x+2)/(x+1), a[x]);
1
(%o2) a = ---------------
x (x - 1) (x + 1)
See also: solve_rec.
Default value: true
When simplifying products, solve_rec introduces gamma function
into the expression if product_use_gamma is true.
See also: simplify_products, solve_rec.
Returns the recurrence satisfied by the sum
hi
====
\
> summand
/
====
k = lo
where summand is hypergeometrical in k and n. If lo and hi
are omitted, they are assumed to be lo = -inf and hi = inf.
To use this function first load the simplify_sum package with
load("simplify_sum").
Example:
(%i1) load("simplify_sum")$
(%i2) summand: binom(n,k); (%o2) binomial(n, k)
(%i3) summand_to_rec(summand,k,n);
(%o3) 2 sm - sm = 0
n n + 1
(%i7) summand: binom(n, k)/(k+1);
binomial(n, k)
(%o7) --------------
k + 1
(%i8) summand_to_rec(summand, [k, 0, n], n);
(%o8) 2 (n + 1) sm - (n + 2) sm = - 1
n n + 1
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