## 89 Package solve_rec ¶

### 89.1 Introduction to solve_rec ¶

`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)

Categories: Linear recurrences · Share packages · Package solve_rec ·

### 89.2 Functions and Variables for solve_rec ¶

Function: harmonic_number (x) ¶

  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

Function: harmonic_to_psi (x) ¶

  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

Function: reduce_order (rec, sol, var) ¶

  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

  Categories: Package solve_rec ·

Option variable: simplify_products ¶

  Default value: `true`

  If `simplify_products` is `true`, `solve_rec` will try to simplify products in result.

  See also: `solve_rec`.

  Categories: Package solve_rec ·

Function: simplify_sum (expr) ¶

  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

  Categories: Package solve_rec · Sums and products · Simplification functions ·

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`.

  Categories: Package solve_rec ·

Function: solve_rec_rat (eqn, var, [init]) ¶

  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`.

  Categories: Package solve_rec ·

Option variable: product_use_gamma ¶

  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`.

  Categories: Package solve_rec ·

Function: summand_to_rec
`summand_to_rec` (summand, k, n)
`summand_to_rec` (summand, [k, lo, hi], n) ¶

  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

  Categories: Package solve_rec ·

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