{VERSION 3 0 "DEC ALPHA UNIX" "3.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 
0 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 }
{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "2
D Input" 2 19 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "2D Output" 
2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 256 "" 0 1 0 0 
0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 
0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 
-1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 
{CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 
0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 
0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading \+
2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 
8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "
" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }
{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }
1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Title" 0 18 1 {CSTYLE "" -1 
-1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 
}}
{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 33 "Introduction to the Mgfun
 Package" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 4 "The " }{TEXT 256 5 "Mgfun" }{TEXT -1 232 " package \+
consists of procedures to compute with special functions and combinato
rial sequences that are implicitly defined as solutions of systems of \+
linear difference-differential equations.  Its syntax is very close to
 that of the " }{TEXT 257 4 "gfun" }{TEXT -1 67 " package.  In the sam
e spirit as with the latter, a typical use of " }{TEXT 258 5 "Mgfun" }
{TEXT -1 124 " to compute a sum or integral is the following.  First o
ne obtains systems that define the summand or integrand by calls to " 
}{HYPERLNK 17 "dfinite_expr_to_sys" 2 "Mgfun,dfinite_expr_to_sys" "" }
{TEXT -1 40 " or by combining known systems by using " }{HYPERLNK 17 "
`sys+sys`" 2 "Mgfun,`sys+sys`" "" }{TEXT -1 5 " and " }{HYPERLNK 17 "`
sys*sys`" 2 "Mgfun,`sys*sys`" "" }{TEXT -1 53 ".  Then one derives the
 sum or integral by a call to " }{HYPERLNK 17 "sum_of_sys" 2 "Mgfun,su
m_of_sys" "" }{TEXT -1 4 " or " }{HYPERLNK 17 "int_of_sys" 2 "Mgfun,in
t_of_sys" "" }{TEXT -1 79 ".  Finally one has to solve the system to o
btain a closed form (if one exists)." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "with(Mgfun);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7+%7d
finite_expr_to_diffeqG%4dfinite_expr_to_recG%4dfinite_expr_to_sysG%,di
ag_of_sysG%+int_of_sysG%+pol_to_sysG%+sum_of_sysG%(sys*sysG%(sys+sysG
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 111 "In the sequel, we exemplify \+
the use of the package by working out the evaluation of a sum and of t
wo integrals." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 26 "Closed Form Eva
luation of " }{XPPEDIT 18 0 "sum(H[n](x)*H[n](y)*u^n/n!,n = 0 .. infin
ity);" "6#-%$sumG6$**-&%\"HG6#%\"nG6#%\"xG\"\"\"-&F)6#F+6#%\"yGF.)%\"u
GF+F.-%*factorialG6#F+!\"\"/F+;\"\"!%)infinityG" }}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 80 "This is the Mehler formula for Hermite polynomials. \+
 We first input the summand." }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 
1 "expr:=HermiteH(n,x)*HermiteH(n,y)*u^n/n!:" "6#>%%exprG**-%)HermiteH
G6$%\"nG%\"xG\"\"\"-F'6$F)%\"yGF+)%\"uGF)F+-%*factorialG6#F)!\"\"" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Since the sum is over " }{XPPEDIT 
18 0 "n;" "6#%\"nG" }{TEXT -1 52 ", we need the dependency of the summ
and in terms of " }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 61 ".  In th
is session, we focus on the dependency of the sum in " }{XPPEDIT 18 0 
"u;" "6#%\"uG" }{TEXT -1 12 ", and leave " }{XPPEDIT 18 0 "x;" "6#%\"x
G" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT -1 137 " a
s parameters, but other choices could have been made.  The following c
all yields a system that traces the dependency of the summand in " }
{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "u;" "
6#%\"uG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "
sys[1]:=dfinite_expr_to_sys(expr,f(n::shift,u::diff));" }}{PARA 12 "" 
1 "" {XPPMATH 20 "6#>&%$sysG6#\"\"\"<$,,*&,&*$)%\"uG\"\"%\"\"\"\"#;*&F
-F0%\"nGF'F1F'-%\"fG6$F3F.F'F'**)F.\"\"$F0%\"xGF'%\"yGF'-F56$,&F3F'F'F
'F.F'!#;*&,&F3F'F/F'F'-F56$FAF.F'F'*&,**&)F.\"\"#F0F3F0!\")*&FGF0)F:FH
F0\"\")*$FGF0!#?*&FGF0)F;FHF0FLF'-F56$,&F3F'FHF'F.F'F'**F.F'F;F0F:F0-F
56$,&F3F'F9F'F.F'!\"%,&*&F3F0F4F0!\"\"*&F.F0-%%diffG6$F4F.F'F'" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 102 "The summation itself is performed
 by the following call.  Viewing the sum as a formal power series in \+
" }{XPPEDIT 18 0 "u;" "6#%\"uG" }{TEXT -1 40 ", we have that the summa
nd goes to 0 as " }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 62 " tends t
o infinity.  At the other boundary, the recurrence in " }{XPPEDIT 18 
0 "n;" "6#%\"nG" }{TEXT -1 43 " proves that the summand is 0 for negat
ive " }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 25 ".  The use of the op
tion " }{XPPEDIT 18 0 "natural_boundaries;" "6#%3natural_boundariesG" 
}{TEXT -1 30 " is therefore fully justified." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 60 "sys[2]:=sum_of_sys(sys[1],n=0..infinity,natural_
boundaries);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$sysG6#\"\"#<#,&*&,
.*$)%\"uG\"\"$\"\"\"\"#;*&F.\"\"\")%\"yGF'F0\"\")*&F.F0)%\"xGF'F0F6F.!
\"%*&F5F3F9F3F:*()F.F'F0F5F0F9F0!#;F3-%\"fG6#F.F3F3*&,(F3F3*$F=F0!\")*
$)F.\"\"%F0F1F3-%%diffG6$F?F.F3F3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
55 "There only remains to solve this differential equation." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "dsolve(sys[2] union \{f(0)=1
\},f(u));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"uG,$*&*&%\"IG
\"\"\"-%$expG6#,$*&,,*&-%#lnG6#,&!\"\"F,*$)F'\"\"#\"\"\"\"\"%F,F:F<!\"
%*(F'F,%\"yGF,%\"xGF,!\")*$)FAF;F<F;*$)F@F;F<F;F4F,F<*&,&F'F;F,F,\"\"
\",&F'F;F8F,\"\"\"!\"\"#F,F;F,F<-F.6#,&FCF8FEF8FLF8" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 29 "simplify(expand(%),symbolic);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"uG,$*&*&%\"IG\"\"\"-%$expG6#,$*&
*&F'\"\"\",(*&F'F,)%\"xG\"\"#F3F,*&F'F3)%\"yGF8F3F,*&F;F,F7F,!\"\"F,F3
*&,&F'F8F,F,\"\"\",&F'F8F=F,\"\"\"!\"\"\"\"%F,F3*$-%%sqrtG6#,&F=F,*$)F
'F8F3FDF3FCF=" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "In other words, \+
we have obtained: " }{XPPEDIT 18 0 "sum(H[n](x)*H[n](y)*u^n/n!,n = 0 .
. infinity) = exp(4*u*(x*y-u*x^2-u*y^2)/(1-4*u^2))/sqrt(1-4*u^2);" "6#
/-%$sumG6$**-&%\"HG6#%\"nG6#%\"xG\"\"\"-&F*6#F,6#%\"yGF/)%\"uGF,F/-%*f
actorialG6#F,!\"\"/F,;\"\"!%)infinityG*&-%$expG6#**\"\"%F/F6F/,(*&F.F/
F4F/F/*&F6F/*$F.\"\"#F/F:*&F6F/*$F4\"\"#F/F:F/,&\"\"\"F/*&\"\"%F/*$F6
\"\"#F/F:F:F/-%%sqrtG6#,&\"\"\"F/*&\"\"%F/*$F6\"\"#F/F:F:" }{TEXT -1 
1 "." }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 26 "Closed Form Evaluation \+
of " }{XPPEDIT 18 0 "int(u^n*exp(-u*z/2*(1-1/(u^2)))*(BesselJ(n,z)+u*B
esselJ(n-1,z)),z = 0 .. infinity);" "6#-%$intG6$*()%\"uG%\"nG\"\"\"-%$
expG6#,$**F(F*%\"zGF*\"\"#!\"\",&\"\"\"F**&\"\"\"F**$F(\"\"#F2F2F*F2F*
,&-%(BesselJG6$F)F0F**&F(F*-F;6$,&F)F*\"\"\"F2F0F*F*F*/F0;\"\"!%)infin
ityG" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "We study this integral whe
n parameter values make it well-defined, that is to say for " }
{XPPEDIT 18 0 "1 <= n;" "6#1\"\"\"%\"nG" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "abs(u) < 1;" "6#2-%$absG6#%\"uG\"\"\"" }{TEXT -1 1 "." 
}}{PARA 0 "" 0 "" {TEXT -1 175 "Here again, we first input the integra
nd and make our choice of variables with respect to which we want to s
tudy the dependency.  This has to contain the integration variable " }
{XPPEDIT 18 0 "z;" "6#%\"zG" }{TEXT -1 50 ".  Here, we made the choice
 to leave no parameter." }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "ex
pr:=u^n*exp(-u*z/2*(1-1/u^2))*(BesselJ(n,z)+u*BesselJ(n-1,z)):" "6#>%%
exprG*()%\"uG%\"nG\"\"\"-%$expG6#,$**F'F)%\"zGF)\"\"#!\"\",&\"\"\"F)*&
\"\"\"F)*$F'\"\"#F1F1F)F1F),&-%(BesselJG6$F(F/F)*&F'F)-F:6$,&F(F)\"\"
\"F1F/F)F)F)" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
62 "sys[1]:=dfinite_expr_to_sys(expr,f(n::shift,z::diff,u::diff));" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#>&%$sysG6#\"\"\"<%,(*&,2*&)%\"zG\"\"#
\"\"\")%\"uGF/F0F/*$F-F0F'*&)F2\"\"%F0F-F0F'*()F2\"\"$F0%\"nGF'F.F'!\"
%*&F1F0F:F0F6*&F1F0)F:F/F0F6*&F2F'F.F0F;*(F:F0F2F0F.F0F;F'-%\"fG6%F:F.
F2F'F'*&,(*&F5F0F.F0F6*&F.F0F1F0F6*&F8F0F:F0!\")F'-%%diffG6$FAF2F'F'*&
F5F0-FK6$FA-%\"$G6$F2F/F'F6,**&,&FGF'*&F2F0F:F0!\"#F'FAF0F'*&F.F0-FB6%
,&F:F'F'F'F.F2F'F'*(F2F0F.F0-FK6$FAF.F'F'*&F1F0FJF0F',**&,2F=!#7F7F6F<
F6F@F6F4F'F,F/F?F;F3F'F'FAF0F'*(F-F0F1F0-FK6$FA-FQ6$F.F/F'F6*&,(*&F-F0
F8F0F6*&F2F0F-F0F;FGF6F'FgnF0F'*&,&FH\"\")*$F8F0F;F'FJF0F'" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 104 "The integration itself takes place in th
e following call.  The justification that we may use the option " }
{XPPEDIT 18 0 "takayama_algo;" "6#%.takayama_algoG" }{TEXT -1 125 ", a
nd thus, that we may perform an integration over natural boundaries is
 the following.  We will consider this integral for " }{XPPEDIT 18 0 "
u;" "6#%\"uG" }{TEXT -1 83 " in a neighbourhood of 0, so that the expo
nential term is exponentially small when " }{XPPEDIT 18 0 "z;" "6#%\"z
G" }{TEXT -1 55 " tends to infinity.  At 0, the integrand has valuatio
n " }{XPPEDIT 18 0 "2*n;" "6#*&\"\"#\"\"\"%\"nGF%" }{TEXT -1 4 " in " 
}{XPPEDIT 18 0 "u;" "6#%\"uG" }{TEXT -1 121 ", so that any linear comb
ination of the integrand and its derivatives with polynomial function \+
coefficients will be 0 at " }{XPPEDIT 18 0 "u = 0;" "6#/%\"uG\"\"!" }
{TEXT -1 11 ", provided " }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 28 "
 becomes sufficiently large." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 55 "sys[2]:=int_of_sys(sys[1],z=0..infinity,takayama_algo);" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#>&%$sysG6#\"\"#<%,,*&,*\"\"(\"\"\"%\"n
G\"#?*$)F.F'\"\"\"\"#7*$)%\"uGF'F2!\"\"F--%\"fG6$F.F6F-F-*&,(!\"%F-F0!
#7F.!#?F--F96$,&F.F-F-F-F6F-F-*&,(F6!\"(*&F6F-F.F-!\")*$)F6\"\"$F2F-F-
-%%diffG6$F8F6F-F-*&,&*$)F6\"\"%F2F-F4F-F--FL6$F8-%\"$G6$F6F'F-F-*&,&F
F\"\")F6FRF---&%\"DGF&6#F9FAF-F-,,*&,4\"\"*F-*&FQF2F.F2F7F.\"#:*&F5F2F
.F2!#5F4!\"$*$)F.FJF2F=F0FG*&F5F2F1F2!#C*&FdoF2F5F2F>F-F8F2F-*&,.F`oF3
Feo\"#CF.FfoFgoF3F0F?FcoF=F-F@F2F-*&,,*&)F6\"\"&F2F.F2F-*&FIF2F.F2!\"#
F6!\"*FHFJFFFboF-FKF2F-*&,,FPFJF4FJF`oF-*&F.F2)F6\"\"'F2F-F^oF'F-FSF2F
-*&,(FcoFZF0\"#GF.F3F--F96$,&F.F-F'F-F6F-F-,,*&,2FbpF-F0!#>F.!#HF4F-Fc
oFZ*$)F.FRF2FRF`oFJFeoF_pF-F8F2F-*&,*F.FZFcqF=FcoFGF0FRF-F@F2F-*&,.FHF
7FF\"#@F`pFboF6F]o*&FIF2F1F2!\"&*&F6F2F1F2F_oF-FKF2F-*&,.F^oFJFeoF[rF4
F=F`oFbp*&FQF2F1F2F[rFPF'F-FSF2F-*&,**$F^pF2F-F`pF'FHF-F]pF'F--FL6$F8-
FV6$F6FJF-F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "The above system \+
cancels the integral for sufficiently large " }{XPPEDIT 18 0 "n;" "6#%
\"nG" }{TEXT -1 82 ".  (But unfortunately, we do not know at this poin
t any lower bound for the valid " }{XPPEDIT 18 0 "n;" "6#%\"nG" }
{TEXT -1 2 ".)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "At this point, \+
we face a non-deterministic behaviour of Maple: the above output of " 
}{TEXT 259 10 "int_of_sys" }{TEXT -1 191 " varies, according to the sy
stem on which this session is run.  Avoiding this non-determinism woul
d require that the user inputs some constraints that are not expressab
le at the level of the " }{HYPERLNK 17 "Mgfun" 2 "Mgfun" "" }{TEXT -1 
45 " package, but only at the lower-level of the " }{HYPERLNK 17 "Holo
nomy" 2 "Holonomy" "" }{TEXT -1 76 " package (this relates to choices \+
of term orders in Groebner bases, see the " }{HYPERLNK 17 "Groebner" 
2 "Groebner" "" }{TEXT -1 126 " package).  Due to this, we provide two
 ways to solve the system, depending on the output of the following tw
o Maple commands:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "map(in
dets,sys[2],function);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<%<'-%\"fG6$
%\"nG%\"uG--&%\"DG6#\"\"#6#F&6$,&F(\"\"\"F3F3F)-%%diffG6$F%F)-F&F1-F56
$F%-%\"$G6$F)F/<'F%F4F7-F&6$,&F(F3F/F3F)F8<'-F56$F%-F;6$F)\"\"$F%F4F7F
8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "if nops(remove(has,%,[
D,diff]))=1 then \"Case A\" else \"Case B\" fi;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#Q'Case~B6\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "Ple
ase, directly enter the section with title indicated by the last outpu
t above." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 6 "Case A" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 90 "In this situation, a recurrence is alread
y available in the system.  It is readily solved." }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 85 "eq[0]:=collect(eval(subs(f=unapply(f(n),n,u)
,op(remove(has,sys[2],diff)))),f,factor);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%#eqG6#\"\"!,(*&)%\"uG\"\"#\"\"\"-%\"fG6#%\"nG\"\"\"!
\"\"*(,&F+F2F3F2F2,&F+F2F2F2F2-F/6#,&F1F2F2F2F2F2-F/6#,&F1F2F,F2F2" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "normal(LREtools[hypergeomso
ls](eq[0],f(n),\{\},output=basis),expanded);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,&&%#_CG6#\"\"\"F'*&&F%6#\"\"#F'),$*$)%\"uGF+\"\"\"!\"
\"%\"nGF'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "solve(expand
(\{subs(n=1,%)=2*u,subs(n=2,%)=2*u,subs(n=3,%)=2*u\}),\{_C[1],_C[2]\})
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/&%#_CG6#\"\"\",$%\"uG\"\"#/&F&
6#F+\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "rsol:=normal(s
ubs(%,%%),expanded);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%rsolG,$%\"u
G\"\"#" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 6 "Case B" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 57 "In this situation, we proceed to eliminat
e all shifts in " }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 48 " so as t
o obtain a purely differential relation." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 47 "eq[1]:=op(select(has,sys[2],diff(f(n,u),u$3)));" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#>&%#eqG6#\"\"\",,*&,2!\"*F'*$)%\"nG\"
\"#\"\"\"!#>F.!#H*$)%\"uGF/F0F'*$)F.\"\"$F0\"\")*$)F.\"\"%F0F<*&F4F0F.
F'F8*&F4F0F-F0\"\"&F'-%\"fG6$F.F5F'F'*&,*F.F9F:!\"%F6!\")F,F<F'-FA6$,&
F.F'F'F'F5F'F'*&,.*$)F5F8F0!\"\"*&F5F'F.F0\"#@*&FMF0F.F0!\"$F5\"\"**&F
MF0F-F0!\"&*&F5F0F-F0\"#:F'-%%diffG6$F@F5F'F'*&,.*&)F5F<F0F.F0F8F>FUF3
FEF=F+*&FhnF0F-F0FU*$FhnF0F/F'-FY6$F@-%\"$G6$F5F/F'F'*&,**$)F5F?F0F'FQ
F/FLF'*&FcoF0F.F0F/F'-FY6$F@-F^o6$F5F8F'F'" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 45 "eq[2]:=op(select(has,sys[2],D[2](f)(n+1,u)));" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#>&%#eqG6#\"\"#,,*&,*\"\"(\"\"\"%\"nG\"
#?*$)F-F'\"\"\"\"#7*$)%\"uGF'F1!\"\"F,-%\"fG6$F-F5F,F,*&,(!\"%F,F/!#7F
-!#?F,-F86$,&F-F,F,F,F5F,F,*&,(F5!\"(*&F5F,F-F,!\")*$)F5\"\"$F1F,F,-%%
diffG6$F7F5F,F,*&,&*$)F5\"\"%F1F,F3F,F,-FK6$F7-%\"$G6$F5F'F,F,*&,&FE\"
\")F5FQF,--&%\"DGF&6#F8F@F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 39 "eq[3]:=op(select(has,sys[2],f(n+2,u)));" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>&%#eqG6#\"\"$,,*&,4\"\"*\"\"\"*&)%\"uG\"\"%\"\"\"%\"nG
F,!\"\"F2\"#:*&)F/\"\"#F1F2F1!#5*$F6F1!\"$*$)F2F'F1!\"%*$)F2F7F1!\")*&
F6F1F?F1!#C*&F<F1F6F1!#7F,-%\"fG6$F2F/F,F,*&,.F5\"#7FA\"#CF2FBFCFJF>!#
?F;F=F,-FF6$,&F2F,F,F,F/F,F,*&,,*&)F/\"\"&F1F2F1F,*&)F/F'F1F2F1!\"#F/!
\"**$FVF1F'*&F/F,F2F1F:F,-%%diffG6$FEF/F,F,*&,,*$F.F1F'F9F'F5F,*&F2F1)
F/\"\"'F1F,F-F7F,-Ffn6$FE-%\"$G6$F/F7F,F,*&,(F;\"\")F>\"#GF2FJF,-FF6$,
&F2F,F7F,F/F,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "[Procedure to \+
replace var in expr with its value solved from relation.]" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "plug:=proc(relation,expr,var)\n    \+
eval(subs(var=solve(relation,var),expr))\nend:" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 48 "[Procedure that collects in each function call.]" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "normalize:=proc(expr)\n    \+
collect(expr,indets(expr,function),distributed,factor)\nend:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "eq[4]:=normalize(plug(diff(e
q[1],u),eq[2],D[2](f)(n+1,u)));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>&%
#eqG6#\"\"%,.*&**)%\"uGF'\"\"\"),&%\"nG\"\"#\"\"\"F2F1F-,&*$)F,F1F-F2F
2F2F2-%%diffG6$-%\"fG6$F0F,-%\"$G6$F,F'F2F-**F0\"\"\",&F0F2!\"\"F2\"\"
\",&F0F2F1F2\"\"\",&F0F2F2F2\"\"\"!\"\"F2*&*&,8F0!#9*$)F0F'F-\"#N*$)F0
\"\"$F-!#I*$)F0F1F-!#Z*$)F0\"\"&F-\"#W*$)F0\"\"'F-\"#7*&F5F-F0F2Fhn*&F
5F-FNF-FB*&FQF-F5F-\"#=*&F5F-FUF-\"#BF4F1F2F9F2F-**F0\"\"\"FA\"\"\"FD
\"\"\"FF\"\"\"FHF2*&*(F5F-,6F0FRFin\"#@F[o!#[F]o!\")FT\"#5FPFhnF2F2F4
\"\"(FjnF2FMF2F2-F76$F9-F=6$F,F1F2F-**F0\"\"\"FA\"\"\"FD\"\"\"FF\"\"\"
FHF2*&*(F,F2,4FTFRFin!#7FM!\"$F[o!#=F]o!#BF0FgnF4!\"#FP!\"'FjnF2F2-F76
$F9F,F2F-**F0\"\"\"FA\"\"\"FD\"\"\"FF\"\"\"FHF2*&**)F,FRF-F/F2,.F2F2F4
!\"(Fin!#8F0FRFTFYF]oFYF2-F76$F9-F=6$F,FRF2F-**F0\"\"\"FA\"\"\"FD\"\"
\"FF\"\"\"FHFB*&,(!\"%F2FTFhpF0!#?F2-F:6$FFF,F2F2" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 45 "eq[5]:=normalize(plug(eq[1],eq[4],f(n+1,u)))
;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>&%#eqG6#\"\"&,,*&**)%\"uG\"\"%\"
\"\"),&%\"nG\"\"#\"\"\"F3F2F.,&*$)F,F2F.F3F3F3F3-%%diffG6$-%\"fG6$F1F,
-%\"$G6$F,F-F3F.**F1\"\"\",&F1F3!\"\"F3\"\"\",&F1F3F2F3\"\"\",&F1F3F3F
3\"\"\"!\"\"F3*&*()F0\"\"$F.,*F1!\"'*&F6F.F1F3F2!\"*F3F5FCF3F:F3F.**F1
\"\"\"FB\"\"\"FE\"\"\"FG\"\"\"FIFC*&**F/F.,.*$)F1F2F.F-*&F6F.FenF.F-F1
\"#7FP!#7F'F3F5F'F3F6F.-F86$F:-F>6$F,F2F3F.**F1\"\"\"FB\"\"\"FE\"\"\"F
G\"\"\"FIF3*&**FLF.FNF.F,F3-F86$F:F,F3F.**F1\"\"\"FB\"\"\"FE\"\"\"FG\"
\"\"FIF3*&**F/F.,*F1F2FPF2F3F3F5!\"$F3)F,FMF.-F86$F:-F>6$F,FMF3F.**F1
\"\"\"FB\"\"\"FE\"\"\"FG\"\"\"FI!\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 74 "The following is a purely differential equation satisfied by th
e integral." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "eq[5]:=norma
lize(primpart(subs(f(n,u)=f(u),eq[5]),diff(f(u),u$4)));" }}{PARA 12 "
" 1 "" {XPPMATH 20 "6#>&%#eqG6#\"\"&,,**%\"uG\"\"\",&%\"nG\"\"#F+F+F+,
*F-!\"'*&)F*F.\"\"\"F-F+F.!\"*F+*$F2F3!\"\"F+-%%diffG6$-%\"fG6#F*F*F+F
+*(F2F3,.*$)F-F.F3\"\"%*&F2F3F@F3FAF-\"#7F1!#7F'F+F5F'F+-F86$F:-%\"$G6
$F*F.F+F+*(F,F3F/F3F:F+F6*()F*FAF3,&F5F+F+F+F+-F86$F:-FH6$F*FAF+F+*()F
*\"\"$F3,*F-F.F1F.F+F+F5!\"$F+-F86$F:-FH6$F*FTF+!\"#" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 107 "There only remains to solve it.  First, a gene
ral solution is obtained by a call to Maple's general dsolve." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "dsol:=op(2,dsolve(eq[5],f(u)
));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%dsolG,**&*&%$_C1G\"\"\"%\"uG
F)\"\"\",&*$)F*\"\"#F+F)F)F)!\"\"F)*&*&%$_C2GF))F*\"\"$F+F+F,F0F)*&*&%
$_C3GF))F*,&%\"nGF/F5F)F)F+F,F0F)*&*&%$_C4GF))F*,&F;F/F)F)F)F+F,F0F)" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Next, we use initial conditions
." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "solve(\{seq(subs(n=i,
dsol)=simplify(int(subs(n=i,expr),z=0..infinity),symbolic),i=1..4)\},
\{_C1,_C2,_C3,_C4\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<&/%$_C2GF%/%
$_C3GF'/%$_C4G,$*&F'\"\"\")%\"uG\"\"#\"\"\"!\"\"/%$_C1G,(*&F%F,F-F0F1*
$F-F0F/F/F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "normal(subs(
%,dsol),expanded);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%\"uG\"\"#" }}
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "We have just obtained: " }
{XPPEDIT 18 0 "int(u^n*exp(-u*z/2*(1-1/(u^2)))*(BesselJ(n,z)+u*BesselJ
(n-1,z)),z = 0 .. infinity) = 2*u;" "6#/-%$intG6$*()%\"uG%\"nG\"\"\"-%
$expG6#,$**F)F+%\"zGF+\"\"#!\"\",&\"\"\"F+*&\"\"\"F+*$F)\"\"#F3F3F+F3F
+,&-%(BesselJG6$F*F1F+*&F)F+-F<6$,&F*F+\"\"\"F3F1F+F+F+/F1;\"\"!%)infi
nityG*&\"\"#F+F)F+" }{TEXT -1 10 " whenever " }{XPPEDIT 18 0 "1 <= n;
" "6#1\"\"\"%\"nG" }{TEXT -1 1 "." }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 
-1 26 "Chebyshev Coefficients of " }{XPPEDIT 18 0 "x*exp(x);" "6#*&%\"
xG\"\"\"-%$expG6#F$F%" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 30 "Again, we input the integrand." }}}{EXCHG {PARA 0 "> " 0 "" 
{XPPEDIT 19 1 "expr:=x*exp(x)*ChebyshevT(n,x)/sqrt(1-x^2):" "6#>%%expr
G**%\"xG\"\"\"-%$expG6#F&F'-%+ChebyshevTG6$%\"nGF&F'-%%sqrtG6#,&\"\"\"
F'*$F&\"\"#!\"\"F6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "The followi
ng differential difference system keeps track of all the dependencies.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "sys[1]:=dfinite_expr_to
_sys(expr,f(n::shift,x::diff));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>&%
$sysG6#\"\"\"<$,(*&,**&)%\"xG\"\"#\"\"\"%\"nGF'!\"\"F2F'F.F2*$)F.\"\"$
F0F'F'-%\"fG6$F1F.F'F'*&,&F.F'F3F2F'-%%diffG6$F6F.F'F'*(F.F'F1F0-F76$,
&F1F'F'F'F.F'F',(*&,.*&F-F0)F1F/F0F2*$)F.\"\"%F0F'F3F2*$F-F0F2F.!\"#FK
F'F'F6F0F'*&,*FGFKF3F'FJF/F.F/F'F;F0F'*&,&FGF'FJF2F'-F<6$F6-%\"$G6$F.F
/F'F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 121 "The following call comp
utes a recurrence satisfied by the residue at any singularity of any s
olution of the above system." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 38 "sys[2]:=int_of_sys(sys[1],x=residues);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%$sysG6#\"\"#<#,*-%\"fG6#%\"nG\"\"\"-F+6#,&F-F.\"\"%F
.!\"\"*&F-F.-F+6#,&F-F.F.F.F.!\"#*&,&F-F8!\")F.F.-F+6#,&F-F.\"\"$F.F.F
." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "This fourth order recurrence
 has no simple solution." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 
"rec[4]:=op(sys[2]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "LRE
tools[hypergeomsols](rec[4],f(n),\{\},output=basis);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 153 "This \+
is no completely clear at the time why the following second order recu
rrence, also satisfied by the integrals, cannot be found by holonomic \+
methods." }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "rec[2]:=(n^2+n+1)
*f(n+2)+2*(n+1)^3*f(n+1)-(n^2+3*n+3)*f(n):" "6#>&%$recG6#\"\"#,(*&,(*$
%\"nG\"\"#\"\"\"F,F.\"\"\"F.F.-%\"fG6#,&F,F.\"\"#F.F.F.*(\"\"#F.*$,&F,
F.\"\"\"F.\"\"$F.-F16#,&F,F.\"\"\"F.F.F.*&,(*$F,\"\"#F.*&\"\"$F.F,F.F.
\"\"$F.F.-F16#F,F.!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "In oth
er words, the integral of the following " }{XPPEDIT 18 0 "Z;" "6#%\"ZG
" }{TEXT -1 9 " is zero." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 
"Z:=eval(subs(f=unapply(expr,n),rec[2]));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"ZG,(*&**,(*$)%\"nG\"\"#\"\"\"\"\"\"F+F.F.F.F.%\"xGF
.-%$expG6#F/F.-%+ChebyshevTG6$,&F+F.F,F.F/F.F-*$-%%sqrtG6#,&F.F.*$)F/F
,F-!\"\"F-!\"\"F.*&**),&F+F.F.F.\"\"$F-F/F-F0F--F46$FCF/F.F-*$-F96#F;F
-F?F,*&**,(F)F.F+FDFDF.F.F/F-F0F--F46$F+F/F.F-*$-F96#F;F-F?F>" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "This is easily checked numerically
." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 158 "for i from 0 to 3 do
\n    'Z'(n=i)=Int(normal(eval(subs(n=i,subs(ChebyshevT=orthopoly[T],Z
)))),x=-1..1);\n    \"is approximately equal to:\",evalf(op(2,%),30)\n
od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"ZG6#/%\"nG\"\"!-%$IntG6$,$
*&*(%\"xG\"\"\"-%$expG6#F0F1,(*$)F0\"\"#\"\"\"F1!\"#F1F0F1F1F9*$-%%sqr
tG6#,&F1F1F6!\"\"F9!\"\"F8/F0;F@F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$
Q;is~approximately~equal~to:6\"$\"#Y!#K" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/-%\"ZG6#/%\"nG\"\"\"-%$IntG6$,$*&*(%\"xGF)-%$expG6#F0F),**$)F0
\"\"$\"\"\"F7F0!\"%*$)F0\"\"#F8\"\")F9F)F)F8*$-%%sqrtG6#,&F)F)F:!\"\"F
8!\"\"\"\"%/F0;FCF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$Q;is~approximat
ely~equal~to:6\"$\"$%y!#K" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"ZG6#
/%\"nG\"\"#-%$IntG6$,$*&*(%\"xG\"\"\"-%$expG6#F0F1,,*$)F0\"\"%\"\"\"\"
#G*$)F0F)F9!#T\"#5F1*$)F0\"\"$F9\"$3\"F0!#\")F1F9*$-%%sqrtG6#,&F1F1F;!
\"\"F9!\"\"F)/F0;FIF1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$Q;is~approxim
ately~equal~to:6\"$\"%xZ!#K" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"ZG
6#/%\"nG\"\"$-%$IntG6$,$*&*(%\"xG\"\"\"-%$expG6#F0F1,.*$)F0\"\"&\"\"\"
\"#E*$)F0F)F9!#VF0\"#;*$)F0\"\"%F9\"$G\"*$)F0\"\"#F9!$G\"F>F1F1F9*$-%%
sqrtG6#,&F1F1FC!\"\"F9!\"\"\"\")/F0;FLF1" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6$Q;is~approximately~equal~to:6\"$\"%#z)!#K" }}}}}{MARK "0 0 0" 0 
}{VIEWOPTS 1 1 0 3 2 1804 }