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<a name="Functions-and-Variables-for-solve_005frec"></a>
<div class="header">
<p>
Previous: <a href="maxima_365.html#Introduction-to-solve_005frec" accesskey="p" rel="previous">Introduction to solve_rec</a>, Up: <a href="maxima_364.html#solve_005frec_002dpkg" accesskey="u" rel="up">solve_rec-pkg</a> [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_423.html#Function-and-Variable-Index" title="Index" rel="index">Index</a>]</p>
</div>
<a name="Functions-and-Variables-for-solve_005frec-1"></a>
<h3 class="section">88.2 Functions and Variables for solve_rec</h3>
<a name="harmonic_005fnumber"></a><a name="Item_003a-solve_005frec_002fdeffn_002fharmonic_005fnumber"></a><dl>
<dt><a name="index-harmonic_005fnumber"></a>Function: <strong>harmonic_number</strong> <em>(<var>x</var>)</em></dt>
<dd><p>When <var>x</var> is positive integer <em>n</em>, <code>harmonic_number</code> is
the <em>n</em>’th harmonic number. More generally,
<code>harmonic_number(x) = psi[0](x+1) + %gamma</code>. (See <a href="maxima_87.html#polygamma">polygamma</a>).
</p>
<div class="example">
<pre class="example">(%i1) load("simplify_sum")$
</pre><pre class="example">(%i2) harmonic_number(5);
137
(%o2) ---
60
</pre><pre class="example">(%i3) sum(1/k, k, 1, 5);
137
(%o3) ---
60
</pre><pre class="example">(%i4) float(harmonic_number(sqrt(2)));
(%o4) %gamma + 0.6601971549171388
</pre><pre class="example">(%i5) float(psi[0](1+sqrt(2)))+%gamma;
(%o5) %gamma + 0.6601971549171388
</pre></div>
</dd></dl>
<a name="harmonic_005fto_005fpsi"></a><a name="Item_003a-solve_005frec_002fdeffn_002fharmonic_005fto_005fpsi"></a><dl>
<dt><a name="index-harmonic_005fto_005fpsi"></a>Function: <strong>harmonic_to_psi</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Converts expressions with <code>harmonic_number</code> to the equivalent
expression involving <code>psi[0]</code> (see <a href="maxima_87.html#polygamma">polygamma</a>).
</p>
<div class="example">
<pre class="example">(%i1) load("simplify_sum")$
</pre><pre class="example">(%i2) harmonic_to_psi(harmonic_number(sqrt(2)));
(%o2) psi (sqrt(2) + 1) + %gamma
0
</pre></div>
</dd></dl>
<a name="reduce_005forder"></a><a name="Item_003a-solve_005frec_002fdeffn_002freduce_005forder"></a><dl>
<dt><a name="index-reduce_005forder"></a>Function: <strong>reduce_order</strong> <em>(<var>rec</var>, <var>sol</var>, <var>var</var>)</em></dt>
<dd>
<p>Reduces the order of linear recurrence <var>rec</var> when a particular solution
<var>sol</var> is known. The reduced reccurence can be used to get other solutions.
</p>
<p>Example:
</p>
<div class="example">
<pre class="example">(%i3) rec: x[n+2] = x[n+1] + x[n]/n;
x
n
(%o3) x = x + --
n + 2 n + 1 n
</pre><pre class="example">(%i4) solve_rec(rec, x[n]);
WARNING: found some hypergeometrical solutions!
(%o4) x = %k n
n 1
</pre><pre class="example">(%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
</pre><pre class="example">(%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
</pre></div>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-solve_005frec">Package solve_rec</a>
·</div>
</dd></dl>
<a name="simplify_005fproducts"></a><a name="Item_003a-solve_005frec_002fdefvr_002fsimplify_005fproducts"></a><dl>
<dt><a name="index-simplify_005fproducts"></a>Option variable: <strong>simplify_products</strong></dt>
<dd><p>Default value: <code>true</code>
</p>
<p>If <code>simplify_products</code> is <code>true</code>, <code>solve_rec</code> will try to
simplify products in result.
</p>
<p>See also: <code><a href="#solve_005frec">solve_rec</a></code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-solve_005frec">Package solve_rec</a>
·</div>
</dd></dl>
<a name="simplify_005fsum"></a><a name="Item_003a-solve_005frec_002fdeffn_002fsimplify_005fsum"></a><dl>
<dt><a name="index-simplify_005fsum"></a>Function: <strong>simplify_sum</strong> <em>(<var>expr</var>)</em></dt>
<dd>
<p>Tries to simplify all sums appearing in <var>expr</var> to a closed form.
</p>
<p>To use this function first load the <code>simplify_sum</code> package with
<code>load("simplify_sum")</code>.
</p>
<p>Example:
</p>
<div class="example">
<pre class="example">(%i1) load("simplify_sum")$
</pre><pre class="example">(%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
</pre><pre class="example">(%i3) simplify_sum(%);
2 n - 1 n
(%o3) 2 + 2 - 2
</pre></div>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-solve_005frec">Package solve_rec</a>
·<a href="maxima_424.html#Category_003a-Sums-and-products">Sums and products</a>
·<a href="maxima_424.html#Category_003a-Simplification-functions">Simplification functions</a>
·</div>
</dd></dl>
<a name="solve_005frec"></a><a name="Item_003a-solve_005frec_002fdeffn_002fsolve_005frec"></a><dl>
<dt><a name="index-solve_005frec"></a>Function: <strong>solve_rec</strong> <em>(<var>eqn</var>, <var>var</var>, [<var>init</var>])</em></dt>
<dd><p>Solves for hypergeometrical solutions to linear recurrence <var>eqn</var> with
polynomials coefficient in variable <var>var</var>. Optional arguments <var>init</var>
are initial conditions.
</p>
<p><code>solve_rec</code> 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.
</p>
<p>Note that the running time of the algorithm used to find hypergeometrical
solutions is exponential in the degree of the leading and trailing
coefficient.
</p>
<p>To use this function first load the <code>solve_rec</code> package with
<code>load("solve_rec");</code>.
</p>
<p>Example of linear recurrence with constant coefficients:
</p>
<div class="example">
<pre class="example">(%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
</pre></div>
<p>Example of linear recurrence with polynomial coefficients:
</p>
<div class="example">
<pre class="example">(%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
</pre><pre class="example">(%i8) solve_rec(%, y[x], y[1]=1, y[3]=3);
x
3 2 x!
(%o9) y = ---- - --
x 4 2
</pre></div>
<p>Example of Ricatti type recurrence:
</p>
<div class="example">
<pre class="example">(%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
</pre><pre class="example">(%i3) solve_rec(%, y[x], y[3]=5)$
</pre><pre class="example">(%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
</pre></div>
<p>See also: <code><a href="#solve_005frec_005frat">solve_rec_rat</a></code>, <code><a href="#simplify_005fproducts">simplify_products</a></code> and <code><a href="#product_005fuse_005fgamma">product_use_gamma</a></code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-solve_005frec">Package solve_rec</a>
·</div>
</dd></dl>
<a name="solve_005frec_005frat"></a><a name="Item_003a-solve_005frec_002fdeffn_002fsolve_005frec_005frat"></a><dl>
<dt><a name="index-solve_005frec_005frat"></a>Function: <strong>solve_rec_rat</strong> <em>(<var>eqn</var>, <var>var</var>, [<var>init</var>])</em></dt>
<dd>
<p>Solves for rational solutions to linear recurrences. See solve_rec for
description of arguments.
</p>
<p>To use this function first load the <code>solve_rec</code> package with
<code>load("solve_rec");</code>.
</p>
<p>Example:
</p>
<div class="example">
<pre class="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
</pre><pre class="example">(%i2) solve_rec_rat(% = (x+2)/(x+1), a[x]);
1
(%o2) a = ---------------
x (x - 1) (x + 1)
</pre></div>
<p>See also: <code><a href="#solve_005frec">solve_rec</a></code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-solve_005frec">Package solve_rec</a>
·</div>
</dd></dl>
<a name="product_005fuse_005fgamma"></a><a name="Item_003a-solve_005frec_002fdefvr_002fproduct_005fuse_005fgamma"></a><dl>
<dt><a name="index-product_005fuse_005fgamma"></a>Option variable: <strong>product_use_gamma</strong></dt>
<dd><p>Default value: <code>true</code>
</p>
<p>When simplifying products, <code>solve_rec</code> introduces gamma function
into the expression if <code>product_use_gamma</code> is <code>true</code>.
</p>
<p>See also: <code><a href="#simplify_005fproducts">simplify_products</a></code>, <code><a href="#solve_005frec">solve_rec</a></code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-solve_005frec">Package solve_rec</a>
·</div>
</dd></dl>
<a name="summand_005fto_005frec"></a><a name="Item_003a-solve_005frec_002fdeffn_002fsummand_005fto_005frec"></a><dl>
<dt><a name="index-summand_005fto_005frec"></a>Function: <strong>summand_to_rec</strong> <em><br> <tt>summand_to_rec</tt> (<var>summand</var>, <var>k</var>, <var>n</var>) <br> <tt>summand_to_rec</tt> (<var>summand</var>, [<var>k</var>, <var>lo</var>, <var>hi</var>], <var>n</var>)</em></dt>
<dd>
<p>Returns the recurrence satisfied by the sum
</p>
<div class="example">
<pre class="example"> hi
====
\
> summand
/
====
k = lo
</pre></div>
<p>where summand is hypergeometrical in <var>k</var> and <var>n</var>. If <var>lo</var> and <var>hi</var>
are omitted, they are assumed to be <code>lo = -inf</code> and <code>hi = inf</code>.
</p>
<p>To use this function first load the <code>simplify_sum</code> package with
<code>load("simplify_sum")</code>.
</p>
<p>Example:
</p>
<div class="example">
<pre class="example">(%i1) load("simplify_sum")$
</pre><pre class="example">(%i2) summand: binom(n,k);
(%o2) binomial(n, k)
</pre><pre class="example">(%i3) summand_to_rec(summand,k,n);
(%o3) 2 sm - sm = 0
n n + 1
</pre><pre class="example">(%i7) summand: binom(n, k)/(k+1);
binomial(n, k)
(%o7) --------------
k + 1
</pre><pre class="example">(%i8) summand_to_rec(summand, [k, 0, n], n);
(%o8) 2 (n + 1) sm - (n + 2) sm = - 1
n n + 1
</pre></div>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-solve_005frec">Package solve_rec</a>
·</div>
</dd></dl>
<hr>
<div class="header">
<p>
Previous: <a href="maxima_365.html#Introduction-to-solve_005frec" accesskey="p" rel="previous">Introduction to solve_rec</a>, Up: <a href="maxima_364.html#solve_005frec_002dpkg" accesskey="u" rel="up">solve_rec-pkg</a> [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_423.html#Function-and-Variable-Index" title="Index" rel="index">Index</a>]</p>
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