## maxima-commits

 [Maxima-commits] CVS: maxima/doc/info zeilberger.texi,1.7,1.8 From: Alexey Beshenov - 2008-12-28 18:28:43 Update of /cvsroot/maxima/maxima/doc/info In directory 23jxhf1.ch3.sourceforge.com:/tmp/cvs-serv4584 Modified Files: zeilberger.texi Log Message: TeX formulae Index: zeilberger.texi =================================================================== RCS file: /cvsroot/maxima/maxima/doc/info/zeilberger.texi,v retrieving revision 1.7 retrieving revision 1.8 diff -u -d -r1.7 -r1.8 --- zeilberger.texi 28 Nov 2007 03:36:55 -0000 1.7 +++ zeilberger.texi 28 Dec 2008 18:28:37 -0000 1.8 @@ -23,24 +23,61 @@ @subsubsection The indefinite summation problem -@...{zeilberger} implements Gosper's algorithm -for indefinite hypergeometric summation. +@code{zeilberger} implements Gosper's algorithm for indefinite hypergeometric summation. Given a hypergeometric term @math{F_k} in @math{k} we want to find its hypergeometric -anti-difference, that is, a hypergeometric term @math{f_k} such that @math{F_k = f_(k+1) - f_k}. +anti-difference, that is, a hypergeometric term @math{f_k} such that + +@tex +$$F_k = f_{k+1} - f_k.$$ +@end tex +@ifnottex +@math{F_k = f_(k+1) - f_k}. +@end ifnottex @subsubsection The definite summation problem -@...{zeilberger} implements Zeilberger's algorithm -for definite hypergeometric summation. -Given a proper hypergeometric term (in @math{n} and @math{k}) @math{F_(n,k)} and a -positive integer @math{d} we want to find a @math{d}-th order linear -recurrence with polynomial coefficients (in @math{n}) for @math{F_(n,k)} +@code{zeilberger} implements Zeilberger's algorithm for definite hypergeometric summation. +Given a proper hypergeometric term (in @math{n} and @math{k}) +@tex +$F_{n,k}$ +@end tex +@ifnottex +@math{F_(n,k)} +@end ifnottex +and a positive integer @math{d} we want to find a @math{d}-th order linear +recurrence with polynomial coefficients (in @math{n}) for +@tex +$F_{n,k}$ +@end tex +@ifnottex +@math{F_(n,k)} +@end ifnottex and a rational function @math{R} in @math{n} and @math{k} such that -@...{a_0 F_(n,k) + ... + a_d F_(n+d),k = Delta_K(R(n,k) F_(n,k))} +@tex +$$a_0 \, F_{n,k} + \ldots + a_d \, F_{n+d}, ~ k = \Delta_K \left(R\left(n,k\right) F_{n,k}\right),$$ +@end tex +@ifnottex +@math{a_0 F_(n,k) + ... + a_d F_(n+d),k = Delta_k(R(n,k) F_(n,k))}, +@end ifnottex -where @math{Delta_k} is the @math{k}-forward difference operator, i.e., +@tex +\noindent +@end tex +where +@tex +$\Delta_k$ +@end tex +@ifnottex +@math{Delta_k} +@end ifnottex +is the @math{k}-forward difference operator, i.e., +@tex +$\Delta_k \left(t_k\right) \equiv t_{k+1} - t_k$. +@end tex +@ifnottex @math{Delta_k(t_k) := t_(k+1) - t_k}. +@end ifnottex @subsection Verbosity levels @@ -67,10 +104,9 @@ @node Functions and Variables for zeilberger, , Introduction to zeilberger, zeilberger @section Functions and Variables for zeilberger -@... {Function} AntiDifference (@var{F_k}, @var{k}) +@deffn {Function} AntiDifference (@math{F_k}, @var{k}) -Returns the hypergeometric anti-difference -of @var{F_k}, if it exists. +Returns the hypergeometric anti-difference of @math{F_k}, if it exists. Otherwise @code{AntiDifference} returns @code{no_hyp_antidifference}. @opencatbox @@ -79,12 +115,15 @@ @end deffn -@... {Function} Gosper (@var{F_k}, @var{k}) -Returns the rational certificate @var{R(k)} for @var{F_k}, that is, +@deffn {Function} Gosper (@math{F_k}, @var{k}) +Returns the rational certificate @math{R(k)} for @math{F_k}, that is, a rational function such that - -@...{F_k = R(k+1) F_(k+1) - R(k) F_k} - +@tex +$F_k = R\left(k+1\right) \, F_{k+1} - R\left(k\right) \, F_k$, +@end tex +@ifnottex +@math{F_k = R(k+1) F_(k+1) - R(k) F_k}, +@end ifnottex if it exists. Otherwise, @code{Gosper} returns @code{no_hyp_sol}. @@ -94,10 +133,10 @@ @end deffn -@... {Function} GosperSum (@var{F_k}, @var{k}, @var{a}, @var{b}) +@deffn {Function} GosperSum (@math{F_k}, @var{k}, @var{a}, @var{b}) -Returns the summmation of @var{F_k} from @math{@var{k} = @var{a}} to @math{@var{k} = @var{b}} -if @var{F_k} has a hypergeometric anti-difference. +Returns the summmation of @math{F_k} from @math{@var{k} = @var{a}} to @math{@var{k} = @var{b}} +if @math{F_k} has a hypergeometric anti-difference. Otherwise, @code{GosperSum} returns @code{nongosper_summable}. Examples: @@ -159,8 +198,19 @@ @end deffn -@... {Function} parGosper (@var{F_@{n,k@}}, @var{k}, @var{n}, @var{d}) -Attempts to find a @var{d}-th order recurrence for @var{F_@{n,k@}}. +@iftex +@deffn {Function} parGosper (@math{F_{n,k}}, @var{k}, @var{n}, @var{d}) +@end iftex +@ifnottex +@deffn {Function} parGosper (@math{F_(n,k)}, @var{k}, @var{n}, @var{d}) +@end ifnottex +Attempts to find a @var{d}-th order recurrence for +@tex +$F_{n,k}$. +@end tex +@ifnottex +@math{F_(n,k)}. +@end ifnottex The algorithm yields a sequence @math{[s_1, s_2, ..., s_m]} of solutions. @@ -176,8 +226,19 @@ @end deffn -@... {Function} Zeilberger (@var{F_@{n,k@}}, @var{k}, @var{n}) -Attempts to compute the indefinite hypergeometric summation of @var{F_@{n,k@}}. +@iftex +@deffn {Function} Zeilberger (@math{F_{n,k}}, @var{k}, @var{n}) +@end iftex +@ifnottex +@deffn {Function} Zeilberger (@math{F_(n,k)}, @var{k}, @var{n}) +@end ifnottex +Attempts to compute the indefinite hypergeometric summation of +@tex +$F_{n,k}$. +@end tex +@ifnottex +@math{F_(n,k)}. +@end ifnottex @code{Zeilberger} first invokes @code{Gosper}, and if that fails to find a solution, then invokes @code{parGosper} with order 1, 2, 3, ..., up to @code{MAX_ORD}.