ValuesAtPoint - Maple Help
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LREtools

 ValuesAtPoint
 formulas for the values of the solution of difference equation and its derivatives of the given order and at the given point.

 Calling Sequence ValuesAtPoint(L, E, fun, HalfInt_opt, Point_opt, Order_opt)

Parameters

 L - linear difference operator in E with coefficients which are polynomials in x E - name of the shift operator acting on x fun - function f(x) that is a solution of $L\left(f\left(x\right)\right)=0$ HalfInt_opt - (optional) 'HalfInterval'= A, A is a rational number, 0 by default Point_opt - (optional) 'Point'=p, p is a rational number or an algebraic number in the indexed RootOf representation (see,RootOf,indexed), 0 by default Order_opt - (optional) 'OrderDer'=m, m is non-negative integer, 0 by default.

Description

 • The ValuesAtPoint command returns formulas for the values of the function and its derivatives of the given order and at the given point in Point_opt. It also computes conditions for the analyticity of the function at the given point.
 • The input includes a difference operator
 > L := Sum(a[i](x)* E^i,i=0..d);
 ${L}{≔}{\sum }_{{i}{=}{0}}^{{d}}{}{{a}}_{{i}}{}\left({x}\right){}{{E}}^{{i}}$ (1)
 and the point A. Specify the point 'Point'=p to compute the value f(x) and its derivatives at $x=p$, and non-negative integer via the option Order_opt to specify the highest order of required derivatives of f(x) at $x=p.$
 • The procedure returns 2 sets:
 1 The set of conditions. f(x) is assumed to be analytic on some open set which contains a set $A<=\mathrm{Re}\left(x\right). Elements of the set give the conditions of the analyticity of f(x) at $x=p$. They are relations between the values of the function and, possibly several of its derivatives at the points into $A<=\mathrm{Re}\left(x\right).
 2 The set of formulas for computing $f\left(p\right),\frac{ⅆ}{ⅆp}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}f\left(p\right)$,...,$\frac{{ⅆ}^{m}}{ⅆ{p}^{m}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}f\left(p\right)$. (f(x) must satisfy the conditions in the first set.) These formulas give the values of $f\left(p\right),\frac{ⅆ}{ⅆp}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}f\left(p\right)$,...,$\frac{{ⅆ}^{m}}{ⅆ{p}^{m}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}f\left(p\right)$ as linear combinations of f(x) and several of its derivatives in $A<=\mathrm{Re}\left(x\right). For $m=0$, we have one unique formula for $f\left(p\right)$.

Examples

 > $\mathrm{with}\left(\mathrm{LREtools}\right):$
 > $\mathrm{L1}≔x{E}^{2}-\left(3x-3\right)E+\left(2x-3\right)\left(12x+4\right)$
 ${\mathrm{L1}}{≔}{x}{}{{E}}^{{2}}{-}\left({3}{}{x}{-}{3}\right){}{E}{+}\left({2}{}{x}{-}{3}\right){}\left({12}{}{x}{+}{4}\right)$ (2)
 > $\mathrm{ValuesAtPoint}\left(\mathrm{L1},E,f\left(x\right),'\mathrm{HalfInterval}'=2,'\mathrm{Point}'=-\frac{1}{3}\right)$
 $\left\{{f}{}\left(\frac{{11}}{{3}}\right){=}{-}\frac{{18}{}{f}{}\left(\frac{{8}}{{3}}\right)}{{5}}\right\}{,}\left\{{f}{}\left({-}\frac{{1}}{{3}}\right){=}\frac{{2}{}{f}{}\left(\frac{{8}}{{3}}\right)}{{75}}{+}\frac{\left(\genfrac{}{}{0}{}{\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}{\phantom{\left\{{x}{=}\frac{{8}}{{3}}\right\}}}{|}\genfrac{}{}{0}{}{\phantom{\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}}{\left\{{x}{=}\frac{{8}}{{3}}\right\}}\right)}{{440}}{+}\frac{\left(\genfrac{}{}{0}{}{\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}{\phantom{\left\{{x}{=}\frac{{11}}{{3}}\right\}}}{|}\genfrac{}{}{0}{}{\phantom{\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}}{\left\{{x}{=}\frac{{11}}{{3}}\right\}}\right)}{{1584}}\right\}$ (3)
 > $\mathrm{ValuesAtPoint}\left(\mathrm{L1},E,f\left(x\right),'\mathrm{HalfInterval}'=2,'\mathrm{Point}'=\mathrm{RootOf}\left({x}^{2}+1,x,\mathrm{index}=1\right),'\mathrm{OrderDer}'=5\right)$
 ${\varnothing }{,}\left\{\genfrac{}{}{0}{}{\left(\frac{{{ⅆ}}^{{5}}}{{ⅆ}{{x}}^{{5}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}{\phantom{\left\{{x}{=}{I}\right\}}}{|}\genfrac{}{}{0}{}{\phantom{\left(\frac{{{ⅆ}}^{{5}}}{{ⅆ}{{x}}^{{5}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}}{\left\{{x}{=}{I}\right\}}{=}\frac{{556477}{}{I}{}{{\mathrm{D}}}^{\left({3}\right)}{}\left({f}\right){}\left({2}{+}{I}\right)}{{549250000}}{+}\frac{{219283}{}{I}{}{{\mathrm{D}}}^{\left({3}\right)}{}\left({f}\right){}\left({3}{+}{I}\right)}{{1098500000}}{-}\frac{{I}{}{{\mathrm{D}}}^{\left({5}\right)}{}\left({f}\right){}\left({3}{+}{I}\right)}{{208000}}{+}\frac{{46962840717153}{}{I}{}{f}{}\left({3}{+}{I}\right)}{{18854722656250000}}{+}\frac{{357}{}{I}{}{{\mathrm{D}}}^{\left({4}\right)}{}\left({f}\right){}\left({2}{+}{I}\right)}{{3380000}}{+}\frac{{3}{}{I}{}{{\mathrm{D}}}^{\left({5}\right)}{}\left({f}\right){}\left({2}{+}{I}\right)}{{26000}}{+}\frac{{68810341503}{}{I}{}{\mathrm{D}}{}\left({f}\right){}\left({2}{+}{I}\right)}{{58014531250000}}{-}\frac{{21134484}{}{I}{}{{\mathrm{D}}}^{\left({2}\right)}{}\left({f}\right){}\left({3}{+}{I}\right)}{{11156640625}}{+}\frac{{60416991}{}{I}{}{{\mathrm{D}}}^{\left({2}\right)}{}\left({f}\right){}\left({2}{+}{I}\right)}{{22313281250}}{+}\frac{{549}{}{I}{}{{\mathrm{D}}}^{\left({4}\right)}{}\left({f}\right){}\left({3}{+}{I}\right)}{{3380000}}{-}\frac{{48839499533961}{}{I}{}{f}{}\left({2}{+}{I}\right)}{{18854722656250000}}{-}\frac{{25780729047}{}{I}{}{\mathrm{D}}{}\left({f}\right){}\left({3}{+}{I}\right)}{{29007265625000}}{-}\frac{{250202038329}{}{\mathrm{D}}{}\left({f}\right){}\left({2}{+}{I}\right)}{{58014531250000}}{+}\frac{{190021307517}{}{\mathrm{D}}{}\left({f}\right){}\left({3}{+}{I}\right)}{{58014531250000}}{-}\frac{{204172941}{}{{\mathrm{D}}}^{\left({2}\right)}{}\left({f}\right){}\left({3}{+}{I}\right)}{{357012500000}}{-}\frac{{368697}{}{{\mathrm{D}}}^{\left({3}\right)}{}\left({f}\right){}\left({3}{+}{I}\right)}{{549250000}}{+}\frac{{577}{}{{\mathrm{D}}}^{\left({4}\right)}{}\left({f}\right){}\left({3}{+}{I}\right)}{{13520000}}{+}\frac{{{\mathrm{D}}}^{\left({5}\right)}{}\left({f}\right){}\left({3}{+}{I}\right)}{{41600}}{-}\frac{{3853718024019}{}{f}{}\left({2}{+}{I}\right)}{{2356840332031250}}{+}\frac{{8319818839971}{}{f}{}\left({3}{+}{I}\right)}{{18854722656250000}}{-}\frac{{178457979}{}{{\mathrm{D}}}^{\left({2}\right)}{}\left({f}\right){}\left({2}{+}{I}\right)}{{178506250000}}{+}\frac{{329139}{}{{\mathrm{D}}}^{\left({3}\right)}{}\left({f}\right){}\left({2}{+}{I}\right)}{{549250000}}{+}\frac{{529}{}{{\mathrm{D}}}^{\left({4}\right)}{}\left({f}\right){}\left({2}{+}{I}\right)}{{3380000}}{+}\frac{{43}{}{{\mathrm{D}}}^{\left({5}\right)}{}\left({f}\right){}\left({2}{+}{I}\right)}{{624000}}{,}\genfrac{}{}{0}{}{\left(\frac{{{ⅆ}}^{{4}}}{{ⅆ}{{x}}^{{4}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}{\phantom{\left\{{x}{=}{I}\right\}}}{|}\genfrac{}{}{0}{}{\phantom{\left(\frac{{{ⅆ}}^{{4}}}{{ⅆ}{{x}}^{{4}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}}{\left\{{x}{=}{I}\right\}}{=}\frac{{357}{}{I}{}{{\mathrm{D}}}^{\left({3}\right)}{}\left({f}\right){}\left({2}{+}{I}\right)}{{845000}}{+}\frac{{549}{}{I}{}{{\mathrm{D}}}^{\left({3}\right)}{}\left({f}\right){}\left({3}{+}{I}\right)}{{845000}}{-}\frac{{25780729047}{}{I}{}{f}{}\left({3}{+}{I}\right)}{{29007265625000}}{+}\frac{{3}{}{I}{}{{\mathrm{D}}}^{\left({4}\right)}{}\left({f}\right){}\left({2}{+}{I}\right)}{{5200}}{+}\frac{{60416991}{}{I}{}{\mathrm{D}}{}\left({f}\right){}\left({2}{+}{I}\right)}{{11156640625}}{+}\frac{{657849}{}{I}{}{{\mathrm{D}}}^{\left({2}\right)}{}\left({f}\right){}\left({3}{+}{I}\right)}{{1098500000}}{+}\frac{{1669431}{}{I}{}{{\mathrm{D}}}^{\left({2}\right)}{}\left({f}\right){}\left({2}{+}{I}\right)}{{549250000}}{-}\frac{{I}{}{{\mathrm{D}}}^{\left({4}\right)}{}\left({f}\right){}\left({3}{+}{I}\right)}{{41600}}{+}\frac{{68810341503}{}{I}{}{f}{}\left({2}{+}{I}\right)}{{58014531250000}}{-}\frac{{42268968}{}{I}{}{\mathrm{D}}{}\left({f}\right){}\left({3}{+}{I}\right)}{{11156640625}}{-}\frac{{178457979}{}{\mathrm{D}}{}\left({f}\right){}\left({2}{+}{I}\right)}{{89253125000}}{-}\frac{{204172941}{}{\mathrm{D}}{}\left({f}\right){}\left({3}{+}{I}\right)}{{178506250000}}{-}\frac{{1106091}{}{{\mathrm{D}}}^{\left({2}\right)}{}\left({f}\right){}\left({3}{+}{I}\right)}{{549250000}}{+}\frac{{577}{}{{\mathrm{D}}}^{\left({3}\right)}{}\left({f}\right){}\left({3}{+}{I}\right)}{{3380000}}{+}\frac{{{\mathrm{D}}}^{\left({4}\right)}{}\left({f}\right){}\left({3}{+}{I}\right)}{{8320}}{-}\frac{{250202038329}{}{f}{}\left({2}{+}{I}\right)}{{58014531250000}}{+}\frac{{190021307517}{}{f}{}\left({3}{+}{I}\right)}{{58014531250000}}{+}\frac{{987417}{}{{\mathrm{D}}}^{\left({2}\right)}{}\left({f}\right){}\left({2}{+}{I}\right)}{{549250000}}{+}\frac{{529}{}{{\mathrm{D}}}^{\left({3}\right)}{}\left({f}\right){}\left({2}{+}{I}\right)}{{845000}}{+}\frac{{43}{}{{\mathrm{D}}}^{\left({4}\right)}{}\left({f}\right){}\left({2}{+}{I}\right)}{{124800}}{,}\genfrac{}{}{0}{}{\left(\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}{\phantom{\left\{{x}{=}{I}\right\}}}{|}\genfrac{}{}{0}{}{\phantom{\left(\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}}{\left\{{x}{=}{I}\right\}}{=}\frac{{3}{}{I}{}{{\mathrm{D}}}^{\left({3}\right)}{}\left({f}\right){}\left({2}{+}{I}\right)}{{1300}}{-}\frac{{I}{}{{\mathrm{D}}}^{\left({3}\right)}{}\left({f}\right){}\left({3}{+}{I}\right)}{{10400}}{+}\frac{{1669431}{}{I}{}{\mathrm{D}}{}\left({f}\right){}\left({2}{+}{I}\right)}{{274625000}}{+}\frac{{1647}{}{I}{}{{\mathrm{D}}}^{\left({2}\right)}{}\left({f}\right){}\left({3}{+}{I}\right)}{{845000}}{-}\frac{{42268968}{}{I}{}{f}{}\left({3}{+}{I}\right)}{{11156640625}}{+}\frac{{1071}{}{I}{}{{\mathrm{D}}}^{\left({2}\right)}{}\left({f}\right){}\left({2}{+}{I}\right)}{{845000}}{+}\frac{{60416991}{}{I}{}{f}{}\left({2}{+}{I}\right)}{{11156640625}}{+}\frac{{657849}{}{I}{}{\mathrm{D}}{}\left({f}\right){}\left({3}{+}{I}\right)}{{549250000}}{+}\frac{{987417}{}{\mathrm{D}}{}\left({f}\right){}\left({2}{+}{I}\right)}{{274625000}}{-}\frac{{1106091}{}{\mathrm{D}}{}\left({f}\right){}\left({3}{+}{I}\right)}{{274625000}}{+}\frac{{1731}{}{{\mathrm{D}}}^{\left({2}\right)}{}\left({f}\right){}\left({3}{+}{I}\right)}{{3380000}}{+}\frac{{{\mathrm{D}}}^{\left({3}\right)}{}\left({f}\right){}\left({3}{+}{I}\right)}{{2080}}{-}\frac{{178457979}{}{f}{}\left({2}{+}{I}\right)}{{89253125000}}{-}\frac{{204172941}{}{f}{}\left({3}{+}{I}\right)}{{178506250000}}{+}\frac{{1587}{}{{\mathrm{D}}}^{\left({2}\right)}{}\left({f}\right){}\left({2}{+}{I}\right)}{{845000}}{+}\frac{{43}{}{{\mathrm{D}}}^{\left({3}\right)}{}\left({f}\right){}\left({2}{+}{I}\right)}{{31200}}{,}\genfrac{}{}{0}{}{\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}{\phantom{\left\{{x}{=}{I}\right\}}}{|}\genfrac{}{}{0}{}{\phantom{\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}}{\left\{{x}{=}{I}\right\}}{=}\frac{{1071}{}{I}{}{\mathrm{D}}{}\left({f}\right){}\left({2}{+}{I}\right)}{{422500}}{+}\frac{{1647}{}{I}{}{\mathrm{D}}{}\left({f}\right){}\left({3}{+}{I}\right)}{{422500}}{-}\frac{{3}{}{I}{}{{\mathrm{D}}}^{\left({2}\right)}{}\left({f}\right){}\left({3}{+}{I}\right)}{{10400}}{+}\frac{{1669431}{}{I}{}{f}{}\left({2}{+}{I}\right)}{{274625000}}{+}\frac{{657849}{}{I}{}{f}{}\left({3}{+}{I}\right)}{{549250000}}{+}\frac{{9}{}{I}{}{{\mathrm{D}}}^{\left({2}\right)}{}\left({f}\right){}\left({2}{+}{I}\right)}{{1300}}{+}\frac{{1587}{}{\mathrm{D}}{}\left({f}\right){}\left({2}{+}{I}\right)}{{422500}}{+}\frac{{1731}{}{\mathrm{D}}{}\left({f}\right){}\left({3}{+}{I}\right)}{{1690000}}{+}\frac{{3}{}{{\mathrm{D}}}^{\left({2}\right)}{}\left({f}\right){}\left({3}{+}{I}\right)}{{2080}}{+}\frac{{987417}{}{f}{}\left({2}{+}{I}\right)}{{274625000}}{-}\frac{{1106091}{}{f}{}\left({3}{+}{I}\right)}{{274625000}}{+}\frac{{43}{}{{\mathrm{D}}}^{\left({2}\right)}{}\left({f}\right){}\left({2}{+}{I}\right)}{{10400}}{,}\genfrac{}{}{0}{}{\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}{\phantom{\left\{{x}{=}{I}\right\}}}{|}\genfrac{}{}{0}{}{\phantom{\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}}{\left\{{x}{=}{I}\right\}}{=}\frac{{9}{}{I}{}{\mathrm{D}}{}\left({f}\right){}\left({2}{+}{I}\right)}{{650}}{-}\frac{{3}{}{I}{}{\mathrm{D}}{}\left({f}\right){}\left({3}{+}{I}\right)}{{5200}}{+}\frac{{1071}{}{I}{}{f}{}\left({2}{+}{I}\right)}{{422500}}{+}\frac{{1647}{}{I}{}{f}{}\left({3}{+}{I}\right)}{{422500}}{+}\frac{{43}{}{\mathrm{D}}{}\left({f}\right){}\left({2}{+}{I}\right)}{{5200}}{+}\frac{{3}{}{\mathrm{D}}{}\left({f}\right){}\left({3}{+}{I}\right)}{{1040}}{+}\frac{{1587}{}{f}{}\left({2}{+}{I}\right)}{{422500}}{+}\frac{{1731}{}{f}{}\left({3}{+}{I}\right)}{{1690000}}{,}{f}{}\left({I}\right){=}\frac{{9}{}{I}{}{f}{}\left({2}{+}{I}\right)}{{650}}{-}\frac{{3}{}{I}{}{f}{}\left({3}{+}{I}\right)}{{5200}}{+}\frac{{43}{}{f}{}\left({2}{+}{I}\right)}{{5200}}{+}\frac{{3}{}{f}{}\left({3}{+}{I}\right)}{{1040}}\right\}$ (4)
 > $\mathrm{ValuesAtPoint}\left(\mathrm{L1},E,f\left(x\right),'\mathrm{HalfInterval}'=0,'\mathrm{Point}'=2\right)$
 $\left\{{f}{}\left({1}\right){=}{4}{}{f}{}\left({0}\right)\right\}{,}\left\{{f}{}\left({2}\right){=}{40}{}{f}{}\left({0}\right){+}{12}{}\left(\genfrac{}{}{0}{}{\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}{\phantom{\left\{{x}{=}{0}\right\}}}{|}\genfrac{}{}{0}{}{\phantom{\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}}{\left\{{x}{=}{0}\right\}}\right){-}{3}{}\left(\genfrac{}{}{0}{}{\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}{\phantom{\left\{{x}{=}{1}\right\}}}{|}\genfrac{}{}{0}{}{\phantom{\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}}{\left\{{x}{=}{1}\right\}}\right)\right\}$ (5)
 > $\mathrm{ValuesAtPoint}\left(\mathrm{L1},E,f\left(x\right),'\mathrm{HalfInterval}'=0,'\mathrm{Point}'=10,'\mathrm{OrderDer}'=3\right)$
 $\left\{{f}{}\left({1}\right){=}{4}{}{f}{}\left({0}\right)\right\}{,}\left\{\genfrac{}{}{0}{}{\left(\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}{\phantom{\left\{{x}{=}{10}\right\}}}{|}\genfrac{}{}{0}{}{\phantom{\left(\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}}{\left\{{x}{=}{10}\right\}}{=}\frac{{58109611}{}\left(\genfrac{}{}{0}{}{\left(\frac{{{ⅆ}}^{{4}}}{{ⅆ}{{x}}^{{4}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}{\phantom{\left\{{x}{=}{0}\right\}}}{|}\genfrac{}{}{0}{}{\phantom{\left(\frac{{{ⅆ}}^{{4}}}{{ⅆ}{{x}}^{{4}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}}{\left\{{x}{=}{0}\right\}}\right)}{{10}}{-}\frac{{58109611}{}\left(\genfrac{}{}{0}{}{\left(\frac{{{ⅆ}}^{{4}}}{{ⅆ}{{x}}^{{4}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}{\phantom{\left\{{x}{=}{1}\right\}}}{|}\genfrac{}{}{0}{}{\phantom{\left(\frac{{{ⅆ}}^{{4}}}{{ⅆ}{{x}}^{{4}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}}{\left\{{x}{=}{1}\right\}}\right)}{{40}}{+}\frac{{355444180401}{}\left(\genfrac{}{}{0}{}{\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}{\phantom{\left\{{x}{=}{0}\right\}}}{|}\genfrac{}{}{0}{}{\phantom{\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}}{\left\{{x}{=}{0}\right\}}\right)}{{2000}}{+}\frac{{12791427403}{}\left(\genfrac{}{}{0}{}{\left(\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}{\phantom{\left\{{x}{=}{0}\right\}}}{|}\genfrac{}{}{0}{}{\phantom{\left(\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}}{\left\{{x}{=}{0}\right\}}\right)}{{150}}{-}\frac{{3257675041}{}\left(\genfrac{}{}{0}{}{\left(\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}{\phantom{\left\{{x}{=}{1}\right\}}}{|}\genfrac{}{}{0}{}{\phantom{\left(\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}}{\left\{{x}{=}{1}\right\}}\right)}{{200}}{+}\frac{{367470002559}{}\left(\genfrac{}{}{0}{}{\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}{\phantom{\left\{{x}{=}{1}\right\}}}{|}\genfrac{}{}{0}{}{\phantom{\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}}{\left\{{x}{=}{1}\right\}}\right)}{{8000}}{-}\frac{{2713158528557}{}{f}{}\left({0}\right)}{{20000}}{+}\frac{{13102438497001}{}\left(\genfrac{}{}{0}{}{\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}{\phantom{\left\{{x}{=}{0}\right\}}}{|}\genfrac{}{}{0}{}{\phantom{\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}}{\left\{{x}{=}{0}\right\}}\right)}{{120000}}{+}\frac{{83425799085959}{}\left(\genfrac{}{}{0}{}{\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}{\phantom{\left\{{x}{=}{1}\right\}}}{|}\genfrac{}{}{0}{}{\phantom{\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}}{\left\{{x}{=}{1}\right\}}\right)}{{480000}}{,}\genfrac{}{}{0}{}{\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}{\phantom{\left\{{x}{=}{10}\right\}}}{|}\genfrac{}{}{0}{}{\phantom{\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}}{\left\{{x}{=}{10}\right\}}{=}\frac{{116219222}{}\left(\genfrac{}{}{0}{}{\left(\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}{\phantom{\left\{{x}{=}{0}\right\}}}{|}\genfrac{}{}{0}{}{\phantom{\left(\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}}{\left\{{x}{=}{0}\right\}}\right)}{{5}}{-}\frac{{58109611}{}\left(\genfrac{}{}{0}{}{\left(\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}{\phantom{\left\{{x}{=}{1}\right\}}}{|}\genfrac{}{}{0}{}{\phantom{\left(\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}}{\left\{{x}{=}{1}\right\}}\right)}{{10}}{+}\frac{{355444180401}{}\left(\genfrac{}{}{0}{}{\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}{\phantom{\left\{{x}{=}{0}\right\}}}{|}\genfrac{}{}{0}{}{\phantom{\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}}{\left\{{x}{=}{0}\right\}}\right)}{{1000}}{+}\frac{{12791427403}{}\left(\genfrac{}{}{0}{}{\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}{\phantom{\left\{{x}{=}{0}\right\}}}{|}\genfrac{}{}{0}{}{\phantom{\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}}{\left\{{x}{=}{0}\right\}}\right)}{{50}}{-}\frac{{9773025123}{}\left(\genfrac{}{}{0}{}{\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}{\phantom{\left\{{x}{=}{1}\right\}}}{|}\genfrac{}{}{0}{}{\phantom{\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}}{\left\{{x}{=}{1}\right\}}\right)}{{200}}{+}\frac{{367470002559}{}\left(\genfrac{}{}{0}{}{\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}{\phantom{\left\{{x}{=}{1}\right\}}}{|}\genfrac{}{}{0}{}{\phantom{\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}}{\left\{{x}{=}{1}\right\}}\right)}{{4000}}{+}\frac{{402200989929}{}{f}{}\left({0}\right)}{{500}}{,}\genfrac{}{}{0}{}{\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}{\phantom{\left\{{x}{=}{10}\right\}}}{|}\genfrac{}{}{0}{}{\phantom{\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}}{\left\{{x}{=}{10}\right\}}{=}\frac{{348657666}{}\left(\genfrac{}{}{0}{}{\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}{\phantom{\left\{{x}{=}{0}\right\}}}{|}\genfrac{}{}{0}{}{\phantom{\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}}{\left\{{x}{=}{0}\right\}}\right)}{{5}}{-}\frac{{174328833}{}\left(\genfrac{}{}{0}{}{\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}{\phantom{\left\{{x}{=}{1}\right\}}}{|}\genfrac{}{}{0}{}{\phantom{\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}}{\left\{{x}{=}{1}\right\}}\right)}{{10}}{+}\frac{{18072854574}{}{f}{}\left({0}\right)}{{25}}{+}\frac{{12791427403}{}\left(\genfrac{}{}{0}{}{\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}{\phantom{\left\{{x}{=}{0}\right\}}}{|}\genfrac{}{}{0}{}{\phantom{\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}}{\left\{{x}{=}{0}\right\}}\right)}{{25}}{-}\frac{{9773025123}{}\left(\genfrac{}{}{0}{}{\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}{\phantom{\left\{{x}{=}{1}\right\}}}{|}\genfrac{}{}{0}{}{\phantom{\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}}{\left\{{x}{=}{1}\right\}}\right)}{{100}}{,}{f}{}\left({10}\right){=}\frac{{697315332}{}\left(\genfrac{}{}{0}{}{\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}{\phantom{\left\{{x}{=}{0}\right\}}}{|}\genfrac{}{}{0}{}{\phantom{\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}}{\left\{{x}{=}{0}\right\}}\right)}{{5}}{-}\frac{{174328833}{}\left(\genfrac{}{}{0}{}{\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}{\phantom{\left\{{x}{=}{1}\right\}}}{|}\genfrac{}{}{0}{}{\phantom{\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}\right)\right)}}{\left\{{x}{=}{1}\right\}}\right)}{{5}}{+}\frac{{603680456}{}{f}{}\left({0}\right)}{{5}}\right\}$ (6)

References

 Abramov, S.A., and van Hoeij, M. "Set of Poles of Solutions of Linear Difference Equations with Polynomial Coefficients." Computation Mathematics and Mathematical Physics. Vol. 43 No. 1. (2003): 57-62.