/*****************************************************************************
 *                                                                           *
 * ************************************************************************* *
 * ***                                                                   *** *
 * ***                           * Solve_rec *                           *** *
 * ***                                                                   *** *
 * ***           This file implements some funtions for solving          *** *
 * ***         recurrence equations with maxima. In particular it        *** *
 * ***           implements the Petkovsek's hyper algorithm and          *** *
 * ***       Abramov's algorithm for solving linear recurrences with     *** *
 * ***                      polynomial coefficients.                     *** *
 * ***                                                                   *** *
 * ***                                                                   *** *
 * ***   Version: 1.08                                                   *** *
 * ***   Author:  Andrej Vodopivec <andrejv@users.sourceforge.net>       *** *
 * ***                                                                   *** *
 * ************************************************************************* *
 *                                                                           *
 *                                                                           *
 * This package solves recurrences using:                                    *
 *   - characteristic equation method for solving linear recurrences with    *
 *     constant coefficients                                                 *
 *   - algorithm hyper for solving linear recurrences with polynomial        *
 *     coefficients (only homogeneous equations)                             *
 *   - algorithm ratsol for obtaining rational solutions for linear          *
 *     reccurences with polynomial coefficients (also inhomogeneous)         *
 *   - solves ricatti type recurrences                                       *
 *                                                                           *
 * Usage:                                                                    *
 *   - solve_rec(f[n]=f[n-1]+f[n-2], f[n]);                                  *
 *   - solve_rec(2*x*(x+1)*y[x] - (x^2+3*x-2)*y[x+1] + (x-1)*y[x+2],         *
 *               y[x], y[1]=1, y[3]=3);                                      *
 *                                                                           *
 * Top level functions:                                                      *
 *   - solve_rec                                                             *
 *   - solve_rec_poly                                                        *
 *   - solve_rec_rat                                                         *
 *   - solve_rec_hyper                                                       *
 *   - solve_rec_ic                                                          *
 *   - reduce_order                                                          *
 *                                                                           *
 *****************************************************************************/

eval_when(batch,
          ttyoff : true,
          nolabels : true)$

put('solve_rec, 1.08, 'version)$

define_variable(solve_rec_warn,        true,   bool)$

define_variable(simplify_products,     true,   bool)$
define_variable(normalize_products,    true,   bool)$
define_variable(distribute_products,   true,   bool)$
define_variable(simplify_products_deg,   10, fixnum)$
define_variable(product_use_gamma,     true,   bool)$

define_variable(use_hyper,             true,   bool)$
define_variable(use_ratsol,            true,   bool)$
define_variable(hyper_factor_solve,   false,   bool)$
define_variable(hyper_all_solutions,   true,   bool)$
define_variable(hyper_to_product,      true,   bool)$

define_variable(solve_rec_method, 'solve_rec_method, any)$
define_variable(shift_op, 'shift_op, any)$

define_variable(%n, '%n, any)$
define_variable(%f, '%f, any)$
define_variable(%m, '%m, any)$
define_variable(%n, '%n, any)$
define_variable(%l, '%l, any)$
define_variable(%j, '%j, any)$
define_variable(%k, '%k, any)$
define_variable(%x, '%x, any)$
define_variable(%u, '%u, any)$
define_variable(%z, '%z, any)$

sr_print([arg]) :=
  if solve_rec_warn then apply(print, arg)$

/**************************************************************************
 *
 * shift operator handling
 *
 *************************************************************************/

change_to_shift_op(expr, %f, %n) :=
  if atom(expr) then expr
  else if part(expr, 0)=%f then block(
    [k% : args(expr)],
    if numberp(first(k%)) then expr
    else funmake('shift_op, [%f, %n, k%[1]-%n])
  )
  else if member(part(expr, 0), ["+", "*", "/", "^", "-"]) then
    apply(part(expr, 0),
      map(lambda([%u], change_to_shift_op(%u, %f, %n)), args(expr)))
  else expr$

max_shift(expr) :=
  if atom(expr) then -inf
  else if part(expr, 0)=shift_op then part(expr, 3)
  else apply('max, map('max_shift, args(expr)))$

min_shift(expr) :=
  if atom(expr) then inf
  else if part(expr, 0)=shift_op then part(expr, 3)
  else apply('min, map('min_shift, args(expr)))$

differ_degree(expr) :=
  if atom(expr) then 0
  else if part(expr, 0)='shift_op then 1
  else if part(expr, 0)="+" or part(expr, 0)="-" then
    apply('max, map('differ_degree, args(expr)))
  else if part(expr, 0)="*" then apply("+", map('differ_degree, args(expr)))
  else if part(expr, 0)="^" then block(
    [dp : differ_degree(part(expr, 2)), db : differ_degree(part(expr, 1))],
    if dp#0 then inf
    else if db=0 then 0
    else db*part(expr, 2)
  )
  else if part(expr, 0)="/" then block(
    [dd : differ_degree(denom(expr)), nd : differ_degree(num(expr))],
    if dd#0 then inf
    else nd
  )
  else 0$

differ_order(expr) := max_shift(expr)-min_shift(expr)$

/**************************************************************************
 *
 * change difference equation to standard form with shift operator
 *
 *************************************************************************/

to_standard_form1(expr, %n, %m) :=
  if atom(expr) then (
    if expr=%n then %n-%m
    else expr
  )
  else if part(expr, 0)='shift_op then (
    if part(expr, 2)=%n then funmake('shift_op, [part(expr, 1), part(expr, 2),
                                                 part(expr, 3)-%m])
    else expr
  )
  else if member(part(expr, 0), ["+", "-", "*", "/", "^"]) then
  apply(part(expr, 0), map(lambda([u], to_standard_form1(u, %n, %m)),
                                  args(expr)))
  else subst(%n=%n-%m, expr)$


to_standard_form(expr, %f, %n) := block(
  [tmp : expand(lhs(expr)-rhs(expr)), m],
  tmp : change_to_shift_op(tmp, %f, %n),
  m : min_shift(tmp),
  to_standard_form1(tmp, %n, m)
)$


/**************************************************************************
 *
 * Solver functions
 *
 *************************************************************************/

solve_rec(eq, fn, [cond]) := block(
  [std_form, %f, %n, deg, ord, hyper_to_product : true, sol : false],
  if atom(fn) then return(false),
  if length(fn)#1 then return(false),
  %f : part(fn, 0),
  %n : part(fn, 1),
  if not(atom(%f)) or not(mapatom(%n)) then return(false),
  
  std_form : subst(cond, eq),

  std_form : to_standard_form(eq, %f, %n),
  std_form : expand(num(ratsimp(std_form))),
  ord : differ_order(std_form),

  if ord=1 then sol : solve_rec_order_1(std_form, %f, %n, cond)
  else sol : solve_rec_gen(std_form, %f, %n, cond),

  if sol=false then false
  else (
    if listp(sol) then sol
    else fn=sol
  )
)$

solve_rec_order_1(std_form, %f, %n, cond) := block(
  [deg : differ_degree(std_form), sol],

  if deg=1 then block(
    [c2, c1, c0],
    c2 : ratcoeff(std_form, funmake('shift_op, [%f, %n, 1])),
    c1 : ratcoeff(std_form, funmake('shift_op, [%f, %n, 0])),
    c0 : ratsimp(std_form - c2*funmake('shift_op, [%f, %n, 1])
                          - c1*funmake('shift_op, [%f, %n, 0])),
    solve_rec_lin_1(expand(-c1/c2), expand(-c0/c2), %f, %n, cond)
  )

  else if deg=2 then (
    return(ricatti_type(std_form, %f, %n, cond))
  )

  else return(false)
)$

solve_rec_gen(std_form, %f, %n, cond) := block(
  [deg : differ_degree(std_form), coeffs : [], ihom, cc : true, coeffs1,
   pc : true, sol : false, ord : differ_order(std_form), var, num_sol, i%],

  if deg#1 then return(false),

  coeffs : makelist(coeff(std_form, funmake('shift_op, [%f, %n, ord-i%])),
                    i%, 0, ord),
  coeffs1 : makelist(ratsimp(coeffs[i%]/coeffs[1]), i%, 1, ord+1),
  for i%:0 thru ord do (
    std_form : std_form - coeffs[i%+1]*funmake('shift_op, [%f, %n, ord-i%]),
    if not(freeof(%n, coeffs1[i%+1])) then cc : false,
    if not(rec_polyp(coeffs[i%+1], %n)) then pc : false
  ),

  ihom : expand(std_form),

  if cc=true then sol : solve_rec_lin_cc(coeffs1, ihom/coeffs[1], %f, %n, cond)

  else if pc=true then (

    if ihom=0 and use_hyper then (
      sol : hyper(reverse(coeffs), %n),
      if sol#false then block(
        [i%],
        sol : sort(remove_duplicates(sol)),
        sol : get_linearly_independent1(sol, %n, ord),
        num_sol : length(sol),
        sol : sum(%k[i%]*sol[i%]/abs(numfactor(sol[i%])), i%, 1, length(sol)),
        if num_sol<ord then (
          sr_print("WARNING: found some hypergeometrical solutions!"),
          if length(cond)>0 then
            sol : false
        )
        else if length(cond)>0 then
          sol : solve_rec_ic1(sol, %f, %n, cond,
                              makelist(%k[i%], i%, 1, length(cond)))
      )
    )

    else if rec_polyp(ihom, %n) and use_ratsol then (
      sol : solve_rec_rat1(reverse(coeffs), ihom, %n),
      if sol#false then (
        var : get_variables(sol),
        if length(var)<ord then (
          sr_print("WARNING: found some rational solutions!"),
          if length(cond)#0 then
            sol: false
        )
        else if length(cond)#0 then
          sol : solve_rec_ic1(sol, %f, %n, cond,
                              makelist(var[i%], i%, 1, length(cond)))
      )
    )

  ),
  sol
)$

/**************************************************************************
 *
 * Get a solution from general solution, which
 * satisfies conditions in cond.
 *
 *************************************************************************/

solve_rec_ic(eq, fn, cond) := block(
  [%f, %n, gen_sol, i, c, var],
  if part(eq, 0)="=" then (
    eq : rhs(eq)-lhs(eq),
    c : coeff(eq, fn),
    eq : eq - c*fn
  ),
  %f : part(fn, 0),
  %n : part(fn, 1),
  var : get_variables(eq),
  fn = solve_rec_ic1(eq, %f, %n, cond, var)
)$

get_variables(expr) := block(
  [var : [], v],
  for v in listofvars(expr) do
    if not(atom(v)) and part(v, 0)=%k then var : append(var, [v]),
  var
)$

solve_rec_ic1(gen_sol, %f, %n, cond, var) := block(
  [i, %d : length(cond), eqns : [], sol, used_vars : [], s, simpsum : true],

  if length(cond)<length(var) then return(false),

  for i:1 thru %d do block(
    [f_l : part(cond, i, 1), f_r : part(cond, i, 2), m],
    eqns : append(eqns, [subst(part(f_l, 1), %n, gen_sol)=f_r]),
    used_vars : append(used_vars, [var[i]])
  ),
  eqns : ev(eqns, product),
  sol : linsolve(eqns, used_vars),
  gen_sol : subst(sol, gen_sol),
  gen_sol
)$

/**************************************************************************
 *
 * First order linear equations
 *
 *************************************************************************/

solve_rec_lin_1(%a, %b, %f, %n, cond) := block(
  [gen_sol, simpsum:true, ap, %j, %l, lo, prodhack:true],

  if length(cond)=0 then (
    if %b#0 then
      lo : get_lower_bound(%a*%b, %n)
    else
      lo : get_lower_bound(%a, %n)
  )
  else lo : part(cond, 1, 1, 1),

  %b : subst(%j, %n, %b),
  %a : subst(%l, %n, %a),

  if freeof(%l, %a) then
    ap : %a^(%n-%j-1)
  else (
    ap : minfactorial(simplify_product(product(%a, %l, %j+1, %n-1))),
    ap : ap/numfactor(ap)
  ),
  if freeof(%l, %a) then
    gen_sol : %k[1]*%a^(%n-lo)
  else (
    gen_sol : minfactorial(simplify_product(product(%a, %l, lo, %n-1))),
    gen_sol : %k[1]*gen_sol/abs(numfactor(gen_sol))
  ),

  gen_sol : gen_sol + nusum(%b*ap, %j, lo, %n-1),
  solve_rec_method : "First order linear reccurence",

  if length(cond)>0 then
    solve_rec_ic1(gen_sol, %f, %n, cond, [%k[1]])
  else
    gen_sol
)$

get_lower_bound(expr, %n) := block(
  [ex : ratsimp(expr), sol, s, i, bound : 0, de, nu],

  de : denom(ex),
  nu : num(ex),

  sol : solve(de, %n),

  if length(sol)>0 then (
    for i in sol do (
      s : rhs(i),
      if integerp(s) and s>= bound then bound : s+1
    )
  ),

  sol : solve(nu, %n),
  if sol#all and length(sol)>0 then (
    for i in sol do (
      s : rhs(i),
      if integerp(s) and s>= bound then bound : s+1
    )
  ),

  bound
)$

/**************************************************************************
 *
 * Higher order linear equations with constant coefficients
 *
 *************************************************************************/

solve_rec_lin_cc(coeffs, ihom, %f, %n, cond) := block(
  [gen_sol, hi, i],

  gen_sol : solve_rec_char_gen(coeffs, %f, %n),

  if ihom#0 then block(
    [exps : solve_rec_get_exps(expand(ihom), %n), curr_ihom, pol],
/*
    exps : remove_duplicates(exps),
*/
    for ex in exps do block(
      [base, pow, mul, %emode : false],
      [pol, ex] : ex,

      if not(numberp(ex)) then
        if part(ex, 0)="^" then
          pow : part(ex, 2)
        else
          pow : part(ex, 2, 2),

      if hipow(pow, %n)>1 then curr_ihom : false
      else (
        mul : coeff(pow, %n),
        if numberp(ex) then base:1
        else
          if part(ex, 0)="^" then
            base : part(ex, 1)
          else
            base : 1/part(ex, 2, 1),
        base : base^mul,
        if not(freeof(%n, expand(pow-mul*%n))) then curr_ihom : false
        else
          curr_ihom:
            solve_rec_char_part(coeffs, base, pol, ex*pol, %n)),
      if curr_ihom#false then (
        gen_sol : gen_sol + curr_ihom,
        ihom : ratexpand(ihom-pol*ex)
      )
    )
  ),
  /*
    Try quite hard to see if ihom = 0, since the a^i/b^i => (a/b)^i
    step in solve_rec_get_exps might have outwitted the simplifier.
  */
  block ([ihom_zero: is (0 = radcan (ihom))],
    if not(ihom_zero) then
      if not (rec_polyp(ihom, %n)) then
        return(false)
      else
        ihom : solve_rec_char_part(coeffs, 1, ihom, ihom, %n),
    if ihom=false then
      return(false)
    else
      gen_sol: gen_sol + ihom,
    solve_rec_method: "Characteristic equation method",
    if length(cond)>0 then
      solve_rec_ic1(gen_sol, %f, %n, cond, makelist(%k[i], i, 1, length(cond)))
    else gen_sol));

solve_rec_char_gen(coeffs, %f, %n) := block(
  [deg : length(coeffs), %x, pol, sol, i, j:1, gen_sol : 0, %k, i%],

  pol : sum(coeffs[i%]*%x^(deg-i%), i%, 1, deg),
  sol : solve(pol, %x),

  for i:1 thru length(sol) do block(
    [k%],
    gen_sol : gen_sol + rhs(sol[i])^%n*
              sum(%k[j+k%]*%n^k%, k%, 0, multiplicities[i]-1),
    j : j+multiplicities[i]
  ),

  gen_sol
)$

solve_rec_char_part(coeffs, po, ihom_pol, ihom, %n) := block(
  [pol : expand(ihom_pol), hi, u, p%, %u, eqns, i%, char, k, i,
   %d : length(coeffs)],
  hi : hipow(pol, %n),
  if rec_polyp(pol, %n)=false and hi>0 then return(false),
  p% : sum(%u[i%]*%n^i%, i%, 0, hi),
  char : sum(coeffs[length(coeffs)-i%]*%x^i%, i%, 0, length(coeffs)-1),
  while subst(po, %x, char)=0 and char#0 do (
    p% : p%*%n,
    char : diff(char, %x)
  ),
  p% : expand(p%*po^%n),
  eqns : 0,
  for i:1 thru %d do eqns : eqns + subst(%n+%d-i, %n, p%)*coeffs[i],
  eqns : expand(eqns) + ihom,
  eqns : expand(eqns/po^%n),
  eqns : makelist(ratcoeff(eqns, %n, i)=0, i, lopow(eqns,%n), hipow(eqns, %n)),
  if errcatch(eqns : solve(eqns, makelist(%u[i], i, 0, hi)))=[] then
    false
  else (
    if (listp(eqns[1])) then
      subst(eqns[1], p%) /*at(p, eqns[1])*/
    else
      subst(eqns, p%)
  )
)$

rec_polyp(p, n) :=
  if freeof(n, p) then true
  else block(
    [hi : hipow(p, n)],
    rec_polyp1(p, n, hi)
  )$

rec_polyp1(p%, %n, hi) := block(
  [c : ratcoeff(p%, %n, hi)],
  if not(freeof(%n, hi)) then false
  else if freeof(%n, c) then (
    p% : expand(p%-c*%n^hi),
    if hi=hipow(p%, %n) then false
    else (
      hi : hipow(p%,%n),
      if hi=0 then
        true
      else
        rec_polyp1(p%, %n, hi)
      )
  )
  else
    false
)$

/*
  If expr is 1 or of the form u^v then return a pair [u,v] such that
  u^v=expr. If not, raise an error.
*/
solve_rec_split_exponent (expr) :=
  if expr = 1 then [1, 0]
  else if atom (expr) then
    error ("Found atom ", expr, " when expecting an exponent")
  else if (op (expr) = "/" and first (expr) = 1) then
    block ([split: solve_rec_split_exponent (second (expr))],
      [1/first (split), second (split)])
  else if (op (expr) = "^") then
    [first (expr), second (expr)]
  else
    error ("Found ", expr, " when expecting an exponent")$

/*
  Try to return a list of pairs [u,v] such that the sum of u*v across
  the list is equal to expr. Moreover, v is always of the form
  a^(b*var). If a term can't be factored helpfully like this, it is
  returned in the form [u, 1].

  Examples:

     solve_rec_get_exps(u*v^i, i)      =>  [[u, v^i]]
     solve_rec_get_exps(u*v^i + w, i)  =>  [[w, 1], [u, v^i]]
     solve_rec_get_exps(u^i/v^i, i)    =>  [[1, (u/v)^i]]
*/
solve_rec_get_exps(expr, var) :=
  if atom(expr) then [[expr, 1]]

  else if op(expr)="+" then
    apply(append,
          map(lambda([u], solve_rec_get_exps(u, var)), args(expr)))

  /* Should only ever see unary minus, but test for length anyway */
  else if (op(expr) = "-" and length(expr) = 1) then
    map(lambda([u], [-u[1], u[2]]), solve_rec_get_exps(-expr, var))

  else if op(expr)="^" then
  block([bs: first (expr), pw: second (expr)],
    if freeof(var, pw) or not (freeof (var, bs)) then
      [[expr, 1]]
    else block([a0,a1],
      [a1,a0]: bothcoef(pw, var),
      if not freeof(var, a0) then [[expr, 1]]
      else [[bs^a0, bs^(a1*var)]]))

  else if op(expr) = "*" then
    [solve_rec_get_exps_mult (args (expr), var)]

  else if op(expr) = "/" then
    /*
      We want to call solve_rec_get_exps_mult with
      [first(expr),1/second(expr)], but have to avoid an infinite
      recursion here: if first(expr) = 1, do it by hand.
    */
    if first (expr) = 1 then
      map (lambda ([pair],
                   [1/first (pair),
                    if second (pair) = 1 then 1
                    else
                      first(second(pair))^(-second(second(pair)))]),
           solve_rec_get_exps (second (expr), var))
    else
      [solve_rec_get_exps_mult([first (expr), 1/second (expr)], var)]$

/*
  This is the hard case for solve_rec_get_exps. It deals with the
  multiplication case, taking the terms of the product as the "terms"
  variable.

  It will always return exactly one pair.
*/
solve_rec_get_exps_mult (terms, var) :=
  block([pair_lists:
           map (lambda ([term], solve_rec_get_exps (term, var)),
                terms)],
    /* If any of these isn't a singleton, we've failed already */
    if some (lambda ([list], length(list) # 1), pair_lists) then
      [xreduce("*", terms), 1]
    /* Split results into "easy" (of form [u,1]) and hard (not!) */
    else
    block ([easy: 1, hard: []],
      for pair in map (first, pair_lists) do
        block ([split: solve_rec_split_exponent (second (pair))],
          if second (split) = 0 then
            easy: first (pair) * easy
          else
            hard: cons ([first (pair), split], hard)),
      /* If everything was easy, we're good! */
      if length (hard) = 0 then [xreduce("*", terms), 1]
      /* If there was just one "hard" example, we can use that */
      else if length (hard) = 1 then
        [easy * first (hard [1]), apply("^", second (hard [1]))]
      /*
        We can deal with the case when all the hard examples are "the
        same" in the sense that second(second(x)) is constant across
        hard.
      */
      else
        block([e: second (second (first (hard))),
               others:
                 map(lambda ([x], second (second (x))), rest (hard))],
          if not every (lambda ([ee], is (ee = e)), others) then
            [xreduce("*", terms), 1]
          else
            [easy * xreduce ("*", map (first, hard)),
             xreduce ("*",
                      map (lambda ([x],
                                   first (second(x))), hard)) ^ e])))$

/**************************************************************************
 *
 * Special types of difference equations
 *
 *************************************************************************/

ricatti_type(eqn, %f, %n, cond) := block(
  [q0, p1, p2, p3, %z, eq],
  q0 : coeff(eqn, funmake('shift_op, [%f, %n, 0])),
  q0 : coeff(q0, funmake('shift_op, [%f, %n, 1])),
  if q0=0 then return(false),
  eqn : eqn - q0*funmake('shift_op, [%f, %n, 0])*funmake('shift_op, [%f, %n, 1]),
  eqn : expand(eqn/q0),
  p1 : coeff(eqn, funmake('shift_op, [%f, %n, 1])),
  p2 : coeff(eqn, funmake('shift_op, [%f, %n, 0])),
  p3 : expand(eqn - p1*funmake('shift_op, [%f, %n, 1])
                  - p2*funmake('shift_op, [%f, %n, 0])),
  if ratsimp(p1*p2-p3)=0 then (
    solve_rec_method: "Ricatty equation.",
    return([%f[%n]=-p1, %f[%n]=-subst(%n-1, %n, p2)])
  ),
  eq : %z[%n+2] + (p2-ratsubst(%n+1, %n, p1))*%z[%n+1] +
                         (p3 - p1*p2)*%z[%n],
  %z : solve_rec(ratsimp(eq), %z[%n]),
  if %z=false then return(false),
  %z : normalize_product(subst(%n+1, %n, rhs(%z))/rhs(%z)) - p1,
  %z : subst(1, %k[2], %z),
  solve_rec_method : "Ricatti equation.",
  if length(cond)>0 then
    solve_rec_ic1(%z, %f, %n, cond, [%k[1]])
  else
    %z
)$

/**************************************************************************
 *
 * solve_rec_poly and solve_rec_hyper implement Petkovsek's Hyper
 * algorithm for hypergeometric solutions of linear recurrences with
 * polynomial coeffs.
 *
 *************************************************************************/

/**************************************************************************
 *
 * Algorithm Poly
 *
 *************************************************************************/

solve_rec_poly(eqn, fn) := block(
  [std_form, %f, %n, deg, ord, coeffs, c, sol],
  if atom(fn) then return(false),
  if length(fn)#1 then return(false),
  %f : part(fn, 0),
  %n : part(fn, 1),
  std_form : to_standard_form(eqn, %f, %n),
  std_form : expand(num(ratsimp(std_form))),
  ord : differ_order(std_form),
  deg : differ_degree(std_form),
  if deg#1 then return(false),
  coeffs : makelist(coeff(std_form, funmake('shift_op, [%f, %n, i])), i, 0, ord),
  for i:1 thru length(coeffs) do
    std_form : std_form-coeffs[i]*funmake('shift_op, [%f, %n, i-1]),
  std_form : expand(std_form),
  sol : hyper_poly(coeffs, std_form, %n),
  if sol#false then fn = sol
  else false
)$

hyper_poly(coeffs, ihom, %n) := block(
  [m : apply('max, map(lambda([u], hipow(u,%n)), coeffs)), s : -1, i%, j%,
   d : length(coeffs), j, deg_poly : 0, i, sol, deg, %x, gen_p, %u, eqn,
   val, solutions, linsolvewarn:false, linsolve_params:false, %k, vars],
  while true do (
    s : s+1,
    j : 0,
    deg_poly : coeff(coeffs[1], %n, m-s) +
               sum(
                  binomial(%x, j%)*
                    sum(i%^j%*coeff(coeffs[i%+1], %n, m-s+j%), i%, 1, d-1),
                  j%, 0, s
               ),
    deg_poly : ratsimp(deg_poly),
    if deg_poly#0 then return(true)
    ),
  sol : solve(deg_poly, %x),
  deg : -1,
  for j in sol do
    if integerp(part(j,2)) and part(j,2)>deg then deg : part(j,2),
  if ihom#0 then deg : max(deg, hipow(ihom, %n) - m + s),
  if deg=-1 then return(false)
  else (
    gen_p : sum(%k[i%+1]*%n^i%, i%, 0, deg),
    eqn : sum(coeffs[j%]*subst(%n+j%-1, %n, gen_p), j%, 1, d),
    eqn : expand(eqn+ihom),
    eqn : makelist(coeff(eqn, %n, i)=0, i, 0, hipow(eqn, %n)),
    if errcatch(sol : linsolve(eqn, makelist(%k[i+1], i, 0, deg)))=[] then
      return(false),
    gen_p : expand(subst(sol, gen_p)),
    gen_p
  )
)$

/**************************************************************************
 *
 * Algorithm Hyper
 *
 *************************************************************************/

solve_rec_hyper(eqn, fn) := block(
  [std_form, %f, %n, deg, ord, coeffs, c],
  if atom(fn) then return(false),
  if length(fn)#1 then return(false),
  %f : part(fn, 0),
  %n : part(fn, 1),
  std_form : to_standard_form(eqn, %f, %n),
  std_form : expand(num(ratsimp(std_form))),
  ord : differ_order(std_form),
  deg : differ_degree(std_form),
  if deg#1 then return(false),
  coeffs : makelist(coeff(std_form, funmake('shift_op, [%f, %n, i])), i, 0, ord),
  for i:1 thru length(coeffs) do
    std_form : std_form-coeffs[i]*funmake('shift_op, [%f, %n, i-1]),
  std_form : expand(std_form),
  if std_form#0 then return(false),
  coeffs : hyper(coeffs, %n),
  if coeffs=false then false
  else (
    if hyper_to_product then
      coeffs : map(lambda([u], u/abs(numfactor(u))), coeffs),
    coeffs : remove_duplicates(coeffs),
    coeffs
  )
)$

hyper(polys, %n) := block(
  [d : length(polys), coeffs : [], Al, A, Bl,  B, j, i, m, eqn, %z, i%,
   solutions : [], have_solution : false, forbidden_deg_diff : []],
  Al : expand(polys[1]/coeff(polys[1], %n, hipow(polys[1], %n))),
  Al : monic_factors(Al, %n),
  Bl : expand(subst(%n-d+2, %n, polys[d])),
  Bl : expand(Bl/coeff(Bl, %n, hipow(Bl, %n))),
  Bl : monic_factors(Bl, %n),
  for A in Al do (
    if have_solution=true and hyper_all_solutions=false then return(1),
    for B in Bl do (
      if have_solution=true and hyper_all_solutions=false then return(1),
      if not(member(hipow(A, %n)-hipow(B, %n), forbidden_deg_diff)) then (
        coeffs : makelist(expand(polys[i]*
                          my_prod(subst(%n+j-1, %n, A), j, 0, i-1)*
                          my_prod(subst(%n+j-1, %n, B), j, i, d-1)), i, 1, d),
        m : apply('max, map(lambda([u], hipow(u, %n)), coeffs)),
        eqn : sum(%z^(i%-1)*coeff(coeffs[i%], %n, m), i%, 1, d),
        eqn : solve(eqn=0, [%z]),
        eqn : map('rhs, eqn),
        if (length(eqn)=1 and eqn[1]=0) or length(eqn)=0 then
          if not(member(hipow(A, %n)-hipow(B, %n), forbidden_deg_diff)) then
            forbidden_deg_diff : append([hipow(A, %n)-hipow(B, %n)],
                                         forbidden_deg_diff),
        for %z in eqn do block(
          [poly_eqn, gc],
          if %z=0 then return(false),
          gc : expand(A*subst(%n+d-2, %n, B)),
          poly_eqn : makelist(%z^i*coeffs[i+1]/gc, i, 0, d-1),
          poly_eqn : hyper_poly(expand(ratsimp(poly_eqn)), 0, %n),
          if poly_eqn#false then block(
            [basis : [], vars : listofvars(poly_eqn), v, sol, lo],
            for v in vars do (
              if not(atom(v)) and part(v, 0)=%k then
                basis : append(basis, [coeff(poly_eqn, v)])
            ),
            for sol in basis do (
              sol : %z*A/B*subst(%n+1, %n, sol)/sol,
              if hyper_to_product then (
                sol : subst(%j, %n, sol),
                lo : get_lower_bound(sol, %j),
                sol : apply(nounify('product), [sol, %j, lo, %n-1]),
                sol : minfactorial(simplify_product(sol)),
                sol : sol/numfactor(sol)
              ),
              solutions : append(solutions, [sol]),
              have_solution : true
            )
          )
        )
      )
    )
  ),
  if length(solutions)=0 then false
  else (
    solve_rec_method : "Hyper algorithm",
    factor(solutions)
  )
)$

monic_factors(p, %n) :=
  if hyper_factor_solve then monic_factors_sol(p, %n)
  else monic_factors_fac(p, %n)$

monic_factors_fac(p, %n) := block(
  [zeros, factors, sol : [1], f, d, tmp, i, pol, fac, lcoeff, %%n, deg],
  factors : factor(p),
  if atom(factors) then return([1, factors]),
  if part(factors, 0)="/" then factors : part(factors, 1),
  if part(factors, 0)="-" then factors : factors*(-1),

  /* This is a dirty trick to get a product */
  if member(part(factors, 0), ["^", "+"]) then factors : %%n*factors,

  for f in factors do (
    if not(freeof(%n, f)) then (
      tmp : sol,
      if not(atom(f)) and part(f, 0)="^" then (
        deg : part(f, 2),
        f : part(f, 1)
      )
      else
        deg : 1,
      for i:1 thru deg do (
        for s in sol do (
          fac : expand(s*f^i),
          tmp : append(tmp, [expand(s*f^i/coeff(fac, %n, hipow(fac, %n)))])
        )
      ),
      sol : tmp
    )
  ),
  sol
)$

monic_factors_sol(p, %n) := block(
  [zeros : [], factors],
  factors : [1],
  zeros : solve(p, %n),
  zeros : map('rhs, zeros),
  for i:1 thru length(zeros) do block(
    [tmp : []],
    for j:0 thru multiplicities[i] do
      for z in factors do tmp : append(tmp, [expand(z*(%n-zeros[i])^j)]),
    factors : tmp
  ),
  factors
)$

/**************************************************************************
 *
 * solve_rec_rat implements Abramov's algorithm for rational solutions of
 * linear reccurences with polynomial coefficients.
 *
 *************************************************************************/

/**************************************************************************
 *
 * Algorithm RatSol
 *
 *************************************************************************/


solve_rec_rat(eqn, fn) := block(
  [std_form, %f, %n, deg, ord, coeffs, c, hyper_to_product : false],
  if atom(fn) then return(false),
  if length(fn)#1 then return(false),
  %f : part(fn, 0),
  %n : part(fn, 1),
  std_form : to_standard_form(eqn, %f, %n),
  std_form : expand(num(ratsimp(std_form))),
  ord : differ_order(std_form),
  deg : differ_degree(std_form),
  if deg#1 then return(false),
  coeffs : makelist(coeff(std_form, funmake('shift_op, [%f, %n, i])), i, 0, ord),
  for i:1 thru length(coeffs) do
    std_form : std_form-coeffs[i]*funmake('shift_op, [%f, %n, i-1]),
  std_form : expand(std_form),
  coeffs : solve_rec_rat1(coeffs, std_form, %n),
  if coeffs=false then false
  else fn = factor(coeffs)
)$

solve_rec_rat1(coeffs, ihom, %n) := block(
  [A, B, ord : length(coeffs), res, %y, i, M : -1, d, u : 1, sol, eq, i%,
   gc, coeffs1, prodhack:true],
  coeffs1 : makelist(subst(%n-i+1, %n, coeffs[i]), i, 1, ord),
  gc : apply('ezgcd, coeffs1)[1],
  u : gc,
  A : expand(subst(%n-ord+1, %n, coeffs[ord]))/gc,
  B : expand(coeffs[1])/gc,
  res : resultant(A, expand(subst(%n+%y, %n, B)), %n),
  res : map('rhs, solve(res, %y)),
  for i in res do
    if integerp(i) then M : max(M, i),
  for i:M thru 0 step -1 do (
    d : gcd(A, subst(%n+i, %n, B)),
    A : ratsimp(A/d),
    B : ratsimp(B/subst(%n-i, %n, d)),
    for j:0 thru i do u : u*subst(%n-j, %n, d)
  ),
  eq : sum(coeffs[i%]/subst(%n+i%-1, %n, u)*%y[%n+i%-1], i%, 1, ord) + ihom,
  sol : solve_rec_poly(eq, %y[%n]),
  if sol=false then return(false),
  solve_rec_method : "RatSol algorithm",
  rhs(sol)/u
)$

/**************************************************************************
 *
 * Select linearly independent solutions
 *
 *************************************************************************/

casoratian_determinant(l, %n) := block(
  [ml, M, i, j, n : length(l)],
  ml : makelist(
           makelist(subst(%n+j-1, %n, l[i]), i, 1, n),
         j, 1, n),
  M : apply('matrix, ml),
  determinant(M)
)$

get_linearly_independent(sol, %n) := block(
  [lin_in : [sol[1]], s, tst, d, m, dd],
  for s:2 thru length(sol) do (
    d : casoratian_determinant(append(lin_in, [sol[s]]), %n),
    d : makefact(d),
    d : minfactorial(d),
    d : factcomb(d),
    d : fullratsimp(d),
    if d#0 then lin_in : append(lin_in, [sol[s]])
  ),
  lin_in
)$

get_linearly_independent1(sol, %n, ord) := block(
  [lin_in : [sol[1]], mat, %%i, %%j, mat1],
  mat : makelist(subst(%n+%%j-1, %n, sol[1]), %%j, 1, ord),
  mat : apply('matrix, [mat]),
  mat : transpose(mat),
  for s:2 thru length(sol) do (
    mat1 : addcol(mat,
      minfactorial(makefact(makelist(subst(%n+%%j-1, %n, sol[s]), %%j, 1, ord)))
      ),
    if rank(mat1)=length(lin_in)+1 then (
      lin_in : cons(sol[s], lin_in),
      mat : mat1
    )
  ),
  lin_in
)$

/**************************************************************************
 *
 * Reduce the order of equation
 *
 *************************************************************************/

reduce_order(eqn, sol, fn) := block(
  [%u, %z, eq, hi, lo, i, c, sumsplitfact : false, %j, %f, %n, e1, e2],
  %f : part(fn, 0),
  %n : part(fn, 1),
  eq : to_standard_form(rhs(eqn)-lhs(eqn), %f, %n),
  hi : max_shift(eq),
  lo : min_shift(eq),
  for i:lo thru hi do (
    eq : subst(subst(%n+i, %n, sol)*funmake('shift_op, [%z, %n, i]),
                     funmake('shift_op, [%f, %n, i]), eq)
  ),
  for i:hi thru lo+1 step -1 do (
    c : ratcoeff(eq, funmake('shift_op, [%z, %n, i])),
    eq : expand(ratsubst(%u[%n+i-1]+funmake('shift_op, [%z, %n, i-1]),
                         funmake('shift_op, [%z, %n, i]), eq))
  ),
  eq : factcomb(eq),
  c : ratcoeff(eq, funmake('shift_op, [%z, %n, 0])),
  if c#0 then return(false),
  eq : expand(eq - c*funmake('shift_op, [%z, %n, 0])),
  e1 : %f[%n] = sol*%z[%n],
  ldisplay(e1),
  e2 : %z[%n] = 'sum(%u[%j], %j, 0, %n-1),
  ldisplay(e2),
  num(ratsimp(eq))
)$

/**************************************************************************
 *
 * Simplify products - sometimes we can get rational solutions from
 * hypergeometrical if we can simplify products well.
 *
 * Maxima does not handle products well - this simplifies product in some
 * cases.
 *
 * - simplify_product simplifies products like
 *
 *     product(n/(n+3), n, 1, k) ==> 6/((k+1)*(k+2)(k+3))
 *
 *   to disable automatic simplification, set simplify_products to false
 *
 * - normalize product changes the upper bound for products like
 *
 *     product(n^2-1, n, 1, k-1) ==> product(n^2-1, n, 1, k)/(k^2-1)
 *
 *   to disable automatic normalization, set normalize_products to false
 *
 *************************************************************************/

simplify_product(prod) := block(
  [term, %n, lo, hi, %kk : 0, nu, de, comm : [], p% : 1,
   deg : simplify_products_deg, %j1],
  if simplify_products=false then return(prod),
  if atom(prod) then return(prod),
  if member(part(prod, 0), ["+", "-", "*", "/", "^"]) then
    return(apply(part(prod, 0), map('simplify_product, args(prod)))),
  if part(prod, 0)#nounify('product) then return(prod),

  /* Read product arguments. */
  %n : part(prod, 2),
  lo : part(prod, 3),
  hi : part(prod, 4),
  term : factor(part(prod, 1)),

  if lo#1 then return(simplify_product(changevar(prod, %j1=%n-lo+1, %j1, %n))),

  /* Check for simple cases.   */
  if term=%n then return(hi!/(lo-1)!),

  if atom(term) or freeof(%n, term) then return(term^(hi-lo+1)),

  /* Distribute over products.  */
  if part(term, 0)="*" then (
      p%*apply("*", map(lambda([u], simplify_product(apply(nounify('product),
                [u, %n, lo, hi]))),
          args(term)))
  )
  /* Take care of fractions. */
  else if part(term, 0)="/" then block(
    [nu : num(term), de : denom(term)],
    /* Check for cancelations. */
    for %kk:-deg thru deg do block(
      [g],
      g : gcd(expand(subst(%n+%kk, %n, nu)), de),
      if not(freeof(%n, g)) then (
        comm : append(comm, [[%kk, g, subst(%n-%kk, %n, g)]]),
        de : ratsimp(de/g),
        nu : ratsimp(nu/subst(%n-%kk, %n, g))
      )
    ),
    /* Cancel common terms. */
    for c in comm do block(
      [kk : c[1], d],
      if kk<0 then (
        p% : p%*my_prod(c[3], %n, hi+kk+1, hi),
        d : my_prod(c[2], %n, lo, lo-kk-1),
        p% : p%/d
      )
      else (
        p% : p%*1/my_prod(c[2], %n, hi-kk+1, hi),
        d : my_prod(c[3], %n, lo, lo+kk-1),
        p% : p%*d
      )
    ),
    /* Distribute over fractions. */
    nu : simplify_product(apply(nounify('product), [nu, %n, lo, hi])),
    de : simplify_product(apply(nounify('product), [de, %n, lo, hi])),
    p%*nu/de
  )
  else if part(term, 0)="^" and integerp(part(term, 2)) then (
    simplify_product(apply(nounify('product), [part(term, 1), %n, lo, hi]))^
      part(term, 2)
  )
  /* Assume we have a poly. */
  else block(
    [aa, bb, lcoeff, %m],
    bb : bothcoef(term, %n), aa : bb[1], bb : bb[2],

    /* Take care of linear products. */
    if freeof(%n, aa) and freeof(%n, bb) then (
      if aa=1 then (
        %m : term - %n,
        (hi+%m)!/(lo+%m-1)!
      )
      else if aa=-1 then (
        %m : term + %n,
        (%m-lo)!/(%m-hi-1)!
      )
      else if product_use_gamma and integerp(aa) then
        gamma(subst(hi+1, %n, expand(term/aa)))/
        gamma(subst(lo, %n, expand(term/aa)))*aa^(hi-lo+1)
      else
        prod
    )
    else if part(term, 0)="-" and length(args(term))=1 then
      (-1)^(hi-lo)*simplify_product(apply(nounify('product), [-term, %n, lo, hi]))
    /* Give up! */
    else
      p%*apply(nounify('product), [factor(term), %n, lo, hi])
  )
)$

normalize_product(%expr) :=
  if atom(%expr) or not(normalize_products) then %expr
  else if part(%expr, 0)=nounify('product) then block(
    [%term : part(%expr, 1),
     %var : part(%expr, 2),
     %lo : part(%expr, 3),
     %hi : part(%expr, 4),
     %vars],
    %vars : listofvars(%hi),
    /* We check if upper limit is const. + numb. - then we normalize to const. */
    if length(%vars)=1 and hipow(%hi, %vars[1])=1 then block(
      [%co : coeff(%hi, %vars[1]), %dd, %r],
      %dd : %hi - %co*%vars[1],
      if integerp(%dd) and %dd#0 then (
        if %dd>0 then
          apply(nounify('product), [%term, %var, %lo, %vars[1]]) *
          my_prod(subst(%vars[1]+%r, %var, %term), %r, 1, %dd)
        else if %dd<0 then
          apply(nounify('product), [%term, %var, %lo, %vars[1]]) /
          my_prod(subst(%vars[1]-%r, %var, %term), %r, 0, -%dd-1)
      )
      else
        %expr
    )
    else
      %expr
  )
  else if member(part(%expr, 0), ["+", "-", "*", "/", "^", "="]) then
    apply(part(%expr, 0), map(' normalize_product, args(%expr)))
  else
    %expr$

my_prod(term, n, lo, hi) := block(
  [pr : 1],
  for i:lo thru hi do pr : pr*subst(i, n, term),
  pr
)$

/*************************************************************************
 *
 * Zeilberger package redefines union - we need a workaround.
 *
 *************************************************************************/

remove_duplicates(lst) := remove_duplicates1(lst, [])$

remove_duplicates1(lst1, lst2) :=
  if length(lst1)=0 then lst2
  else if member(first(lst1), lst2) then remove_duplicates1(rest(lst1), lst2)
  else remove_duplicates1(rest(lst1), cons(first(lst1), lst2))$

eval_when(batch,
          ttyoff : false,
          nolabels : false)$
