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 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=1..d);
 ${L}{≔}{\sum }_{{i}{=}{1}}^{{d}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}{a}{[}{i}{]}{}\left({x}\right){}{{E}}^{{i}}$ (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 $f\left(p\right),\frac{ⅆ}{ⅆp}f\left(p\right)$,...,$\frac{{ⅆ}^{m}}{ⅆ{p}^{m}}f\left(p\right)$. (f(x) must satisfy the conditions in the first set.) These formulas give the values of $f\left(p\right),\frac{ⅆ}{ⅆp}f\left(p\right)$,...,$\frac{{ⅆ}^{m}}{ⅆ{p}^{m}}f\left(p\right)$ 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)
 > $\mathrm{ValuesAtPoint}\left(\mathrm{L1},E,f\left(x\right),'\mathrm{HalfInterval}'=2,'\mathrm{Point}'=-\frac{1}{3}\right)$
 $\left\{{f}{}\left(\frac{{11}}{{3}}\right){=}{-}\frac{{18}}{{5}}{}{f}{}\left(\frac{{8}}{{3}}\right)\right\}{,}\left\{{f}{}\left({-}\frac{{1}}{{3}}\right){=}\frac{{2}}{{75}}{}{f}{}\left(\frac{{8}}{{3}}\right){+}\frac{{1}}{{440}}{}\genfrac{}{}{0}{}{\frac{{ⅆ}}{{ⅆ}{x}}{}{f}{}\left({x}\right)}{\phantom{{x}{=}\frac{{8}}{{3}}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{|}\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{{ⅆ}}{{ⅆ}{x}}{}{f}{}\left({x}\right)}}{{x}{=}\frac{{8}}{{3}}}{+}\frac{{1}}{{1584}}{}\genfrac{}{}{0}{}{\frac{{ⅆ}}{{ⅆ}{x}}{}{f}{}\left({x}\right)}{\phantom{{x}{=}\frac{{11}}{{3}}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{|}\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{{ⅆ}}{{ⅆ}{x}}{}{f}{}\left({x}\right)}}{{x}{=}\frac{{11}}{{3}}}\right\}$ (3)
 > $\mathrm{ValuesAtPoint}\left(\mathrm{L1},E,f\left(x\right),'\mathrm{HalfInterval}'=2,'\mathrm{Point}'=\mathrm{RootOf}\left({x}^{2}+1,x,\mathrm{index}=1\right),'\mathrm{OrderDer}'=5\right)$
 $\left\{{}\right\}{,}\left\{\genfrac{}{}{0}{}{\frac{{ⅆ}}{{ⅆ}{x}}{}{f}{}\left({x}\right)}{\phantom{{x}{=}{I}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{|}\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{{ⅆ}}{{ⅆ}{x}}{}{f}{}\left({x}\right)}}{{x}{=}{I}}{=}\frac{{1071}}{{422500}}{}{I}{}{f}{}\left({2}{+}{I}\right){+}\frac{{1647}}{{422500}}{}{I}{}{f}{}\left({3}{+}{I}\right){+}\frac{{9}}{{650}}{}{I}{}{\mathrm{D}}{}\left({f}\right){}\left({2}{+}{I}\right){-}\frac{{3}}{{5200}}{}{I}{}{\mathrm{D}}{}\left({f}\right){}\left({3}{+}{I}\right){+}\frac{{1587}}{{422500}}{}{f}{}\left({2}{+}{I}\right){+}\frac{{1731}}{{1690000}}{}{f}{}\left({3}{+}{I}\right){+}\frac{{43}}{{5200}}{}{\mathrm{D}}{}\left({f}\right){}\left({2}{+}{I}\right){+}\frac{{3}}{{1040}}{}{\mathrm{D}}{}\left({f}\right){}\left({3}{+}{I}\right){,}\genfrac{}{}{0}{}{\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{f}{}\left({x}\right)}{\phantom{{x}{=}{I}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{|}\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{f}{}\left({x}\right)}}{{x}{=}{I}}{=}\frac{{9}}{{1300}}{}{I}{}{{\mathrm{D}}}^{\left({2}\right)}{}\left({f}\right){}\left({2}{+}{I}\right){-}\frac{{3}}{{10400}}{}{I}{}{{\mathrm{D}}}^{\left({2}\right)}{}\left({f}\right){}\left({3}{+}{I}\right){+}\frac{{1669431}}{{274625000}}{}{I}{}{f}{}\left({2}{+}{I}\right){+}\frac{{657849}}{{549250000}}{}{I}{}{f}{}\left({3}{+}{I}\right){+}\frac{{1071}}{{422500}}{}{I}{}{\mathrm{D}}{}\left({f}\right){}\left({2}{+}{I}\right){+}\frac{{1647}}{{422500}}{}{I}{}{\mathrm{D}}{}\left({f}\right){}\left({3}{+}{I}\right){+}\frac{{43}}{{10400}}{}{{\mathrm{D}}}^{\left({2}\right)}{}\left({f}\right){}\left({2}{+}{I}\right){+}\frac{{3}}{{2080}}{}{{\mathrm{D}}}^{\left({2}\right)}{}\left({f}\right){}\left({3}{+}{I}\right){+}\frac{{987417}}{{274625000}}{}{f}{}\left({2}{+}{I}\right){-}\frac{{1106091}}{{274625000}}{}{f}{}\left({3}{+}{I}\right){+}\frac{{1587}}{{422500}}{}{\mathrm{D}}{}\left({f}\right){}\left({2}{+}{I}\right){+}\frac{{1731}}{{1690000}}{}{\mathrm{D}}{}\left({f}\right){}\left({3}{+}{I}\right){,}\genfrac{}{}{0}{}{\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}{}{f}{}\left({x}\right)}{\phantom{{x}{=}{I}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{|}\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}{}{f}{}\left({x}\right)}}{{x}{=}{I}}{=}\frac{{1647}}{{845000}}{}{I}{}{{\mathrm{D}}}^{\left({2}\right)}{}\left({f}\right){}\left({3}{+}{I}\right){+}\frac{{657849}}{{549250000}}{}{I}{}{\mathrm{D}}{}\left({f}\right){}\left({3}{+}{I}\right){-}\frac{{1}}{{10400}}{}{I}{}{{\mathrm{D}}}^{\left({3}\right)}{}\left({f}\right){}\left({3}{+}{I}\right){-}\frac{{42268968}}{{11156640625}}{}{I}{}{f}{}\left({3}{+}{I}\right){+}\frac{{1071}}{{845000}}{}{I}{}{{\mathrm{D}}}^{\left({2}\right)}{}\left({f}\right){}\left({2}{+}{I}\right){+}\frac{{60416991}}{{11156640625}}{}{I}{}{f}{}\left({2}{+}{I}\right){+}\frac{{3}}{{1300}}{}{I}{}{{\mathrm{D}}}^{\left({3}\right)}{}\left({f}\right){}\left({2}{+}{I}\right){+}\frac{{1669431}}{{274625000}}{}{I}{}{\mathrm{D}}{}\left({f}\right){}\left({2}{+}{I}\right){+}\frac{{1587}}{{845000}}{}{{\mathrm{D}}}^{\left({2}\right)}{}\left({f}\right){}\left({2}{+}{I}\right){+}\frac{{43}}{{31200}}{}{{\mathrm{D}}}^{\left({3}\right)}{}\left({f}\right){}\left({2}{+}{I}\right){+}\frac{{1731}}{{3380000}}{}{{\mathrm{D}}}^{\left({2}\right)}{}\left({f}\right){}\left({3}{+}{I}\right){+}\frac{{1}}{{2080}}{}{{\mathrm{D}}}^{\left({3}\right)}{}\left({f}\right){}\left({3}{+}{I}\right){-}\frac{{178457979}}{{89253125000}}{}{f}{}\left({2}{+}{I}\right){-}\frac{{204172941}}{{178506250000}}{}{f}{}\left({3}{+}{I}\right){+}\frac{{987417}}{{274625000}}{}{\mathrm{D}}{}\left({f}\right){}\left({2}{+}{I}\right){-}\frac{{1106091}}{{274625000}}{}{\mathrm{D}}{}\left({f}\right){}\left({3}{+}{I}\right){,}\genfrac{}{}{0}{}{\frac{{{ⅆ}}^{{4}}}{{ⅆ}{{x}}^{{4}}}{}{f}{}\left({x}\right)}{\phantom{{x}{=}{I}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{|}\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{{{ⅆ}}^{{4}}}{{ⅆ}{{x}}^{{4}}}{}{f}{}\left({x}\right)}}{{x}{=}{I}}{=}\frac{{657849}}{{1098500000}}{}{I}{}{{\mathrm{D}}}^{\left({2}\right)}{}\left({f}\right){}\left({3}{+}{I}\right){-}\frac{{42268968}}{{11156640625}}{}{I}{}{\mathrm{D}}{}\left({f}\right){}\left({3}{+}{I}\right){-}\frac{{25780729047}}{{29007265625000}}{}{I}{}{f}{}\left({3}{+}{I}\right){-}\frac{{1}}{{41600}}{}{I}{}{{\mathrm{D}}}^{\left({4}\right)}{}\left({f}\right){}\left({3}{+}{I}\right){+}\frac{{1669431}}{{549250000}}{}{I}{}{{\mathrm{D}}}^{\left({2}\right)}{}\left({f}\right){}\left({2}{+}{I}\right){+}\frac{{68810341503}}{{58014531250000}}{}{I}{}{f}{}\left({2}{+}{I}\right){+}\frac{{357}}{{845000}}{}{I}{}{{\mathrm{D}}}^{\left({3}\right)}{}\left({f}\right){}\left({2}{+}{I}\right){+}\frac{{60416991}}{{11156640625}}{}{I}{}{\mathrm{D}}{}\left({f}\right){}\left({2}{+}{I}\right){+}\frac{{3}}{{5200}}{}{I}{}{{\mathrm{D}}}^{\left({4}\right)}{}\left({f}\right){}\left({2}{+}{I}\right){+}\frac{{549}}{{845000}}{}{I}{}{{\mathrm{D}}}^{\left({3}\right)}{}\left({f}\right){}\left({3}{+}{I}\right){+}\frac{{987417}}{{549250000}}{}{{\mathrm{D}}}^{\left({2}\right)}{}\left({f}\right){}\left({2}{+}{I}\right){+}\frac{{529}}{{845000}}{}{{\mathrm{D}}}^{\left({3}\right)}{}\left({f}\right){}\left({2}{+}{I}\right){+}\frac{{43}}{{124800}}{}{{\mathrm{D}}}^{\left({4}\right)}{}\left({f}\right){}\left({2}{+}{I}\right){-}\frac{{1106091}}{{549250000}}{}{{\mathrm{D}}}^{\left({2}\right)}{}\left({f}\right){}\left({3}{+}{I}\right){+}\frac{{577}}{{3380000}}{}{{\mathrm{D}}}^{\left({3}\right)}{}\left({f}\right){}\left({3}{+}{I}\right){+}\frac{{1}}{{8320}}{}{{\mathrm{D}}}^{\left({4}\right)}{}\left({f}\right){}\left({3}{+}{I}\right){-}\frac{{250202038329}}{{58014531250000}}{}{f}{}\left({2}{+}{I}\right){+}\frac{{190021307517}}{{58014531250000}}{}{f}{}\left({3}{+}{I}\right){-}\frac{{178457979}}{{89253125000}}{}{\mathrm{D}}{}\left({f}\right){}\left({2}{+}{I}\right){-}\frac{{204172941}}{{178506250000}}{}{\mathrm{D}}{}\left({f}\right){}\left({3}{+}{I}\right){,}\genfrac{}{}{0}{}{\frac{{{ⅆ}}^{{5}}}{{ⅆ}{{x}}^{{5}}}{}{f}{}\left({x}\right)}{\phantom{{x}{=}{I}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{|}\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{{{ⅆ}}^{{5}}}{{ⅆ}{{x}}^{{5}}}{}{f}{}\left({x}\right)}}{{x}{=}{I}}{=}{-}\frac{{21134484}}{{11156640625}}{}{I}{}{{\mathrm{D}}}^{\left({2}\right)}{}\left({f}\right){}\left({3}{+}{I}\right){-}\frac{{25780729047}}{{29007265625000}}{}{I}{}{\mathrm{D}}{}\left({f}\right){}\left({3}{+}{I}\right){+}\frac{{3}}{{26000}}{}{I}{}{{\mathrm{D}}}^{\left({5}\right)}{}\left({f}\right){}\left({2}{+}{I}\right){+}\frac{{46962840717153}}{{18854722656250000}}{}{I}{}{f}{}\left({3}{+}{I}\right){+}\frac{{549}}{{3380000}}{}{I}{}{{\mathrm{D}}}^{\left({4}\right)}{}\left({f}\right){}\left({3}{+}{I}\right){+}\frac{{60416991}}{{22313281250}}{}{I}{}{{\mathrm{D}}}^{\left({2}\right)}{}\left({f}\right){}\left({2}{+}{I}\right){-}\frac{{48839499533961}}{{18854722656250000}}{}{I}{}{f}{}\left({2}{+}{I}\right){+}\frac{{556477}}{{549250000}}{}{I}{}{{\mathrm{D}}}^{\left({3}\right)}{}\left({f}\right){}\left({2}{+}{I}\right){+}\frac{{68810341503}}{{58014531250000}}{}{I}{}{\mathrm{D}}{}\left({f}\right){}\left({2}{+}{I}\right){+}\frac{{357}}{{3380000}}{}{I}{}{{\mathrm{D}}}^{\left({4}\right)}{}\left({f}\right){}\left({2}{+}{I}\right){+}\frac{{219283}}{{1098500000}}{}{I}{}{{\mathrm{D}}}^{\left({3}\right)}{}\left({f}\right){}\left({3}{+}{I}\right){-}\frac{{1}}{{208000}}{}{I}{}{{\mathrm{D}}}^{\left({5}\right)}{}\left({f}\right){}\left({3}{+}{I}\right){-}\frac{{178457979}}{{178506250000}}{}{{\mathrm{D}}}^{\left({2}\right)}{}\left({f}\right){}\left({2}{+}{I}\right){+}\frac{{329139}}{{549250000}}{}{{\mathrm{D}}}^{\left({3}\right)}{}\left({f}\right){}\left({2}{+}{I}\right){+}\frac{{529}}{{3380000}}{}{{\mathrm{D}}}^{\left({4}\right)}{}\left({f}\right){}\left({2}{+}{I}\right){+}\frac{{43}}{{624000}}{}{{\mathrm{D}}}^{\left({5}\right)}{}\left({f}\right){}\left({2}{+}{I}\right){-}\frac{{204172941}}{{357012500000}}{}{{\mathrm{D}}}^{\left({2}\right)}{}\left({f}\right){}\left({3}{+}{I}\right){-}\frac{{368697}}{{549250000}}{}{{\mathrm{D}}}^{\left({3}\right)}{}\left({f}\right){}\left({3}{+}{I}\right){+}\frac{{577}}{{13520000}}{}{{\mathrm{D}}}^{\left({4}\right)}{}\left({f}\right){}\left({3}{+}{I}\right){+}\frac{{1}}{{41600}}{}{{\mathrm{D}}}^{\left({5}\right)}{}\left({f}\right){}\left({3}{+}{I}\right){-}\frac{{3853718024019}}{{2356840332031250}}{}{f}{}\left({2}{+}{I}\right){+}\frac{{8319818839971}}{{18854722656250000}}{}{f}{}\left({3}{+}{I}\right){-}\frac{{250202038329}}{{58014531250000}}{}{\mathrm{D}}{}\left({f}\right){}\left({2}{+}{I}\right){+}\frac{{190021307517}}{{58014531250000}}{}{\mathrm{D}}{}\left({f}\right){}\left({3}{+}{I}\right){,}{f}{}\left({I}\right){=}\frac{{9}}{{650}}{}{I}{}{f}{}\left({2}{+}{I}\right){-}\frac{{3}}{{5200}}{}{I}{}{f}{}\left({3}{+}{I}\right){+}\frac{{43}}{{5200}}{}{f}{}\left({2}{+}{I}\right){+}\frac{{3}}{{1040}}{}{f}{}\left({3}{+}{I}\right)\right\}$ (4)
 > $\mathrm{ValuesAtPoint}\left(\mathrm{L1},E,f\left(x\right),'\mathrm{HalfInterval}'=0,'\mathrm{Point}'=2\right)$
 $\left\{{f}{}\left({1}\right){=}{4}{}{f}{}\left({0}\right)\right\}{,}\left\{{f}{}\left({2}\right){=}{40}{}{f}{}\left({0}\right){+}{12}{}\left(\genfrac{}{}{0}{}{\frac{{ⅆ}}{{ⅆ}{x}}{}{f}{}\left({x}\right)}{\phantom{{x}{=}{0}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{|}\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{{ⅆ}}{{ⅆ}{x}}{}{f}{}\left({x}\right)}}{{x}{=}{0}}\right){-}{3}{}\left(\genfrac{}{}{0}{}{\frac{{ⅆ}}{{ⅆ}{x}}{}{f}{}\left({x}\right)}{\phantom{{x}{=}{1}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{|}\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{{ⅆ}}{{ⅆ}{x}}{}{f}{}\left({x}\right)}}{{x}{=}{1}}\right)\right\}$ (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}{}{\frac{{ⅆ}}{{ⅆ}{x}}{}{f}{}\left({x}\right)}{\phantom{{x}{=}{10}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{|}\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{{ⅆ}}{{ⅆ}{x}}{}{f}{}\left({x}\right)}}{{x}{=}{10}}{=}\frac{{18072854574}}{{25}}{}{f}{}\left({0}\right){+}\frac{{12791427403}}{{25}}{}\genfrac{}{}{0}{}{\frac{{ⅆ}}{{ⅆ}{x}}{}{f}{}\left({x}\right)}{\phantom{{x}{=}{0}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{|}\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{{ⅆ}}{{ⅆ}{x}}{}{f}{}\left({x}\right)}}{{x}{=}{0}}{-}\frac{{9773025123}}{{100}}{}\genfrac{}{}{0}{}{\frac{{ⅆ}}{{ⅆ}{x}}{}{f}{}\left({x}\right)}{\phantom{{x}{=}{1}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{|}\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{{ⅆ}}{{ⅆ}{x}}{}{f}{}\left({x}\right)}}{{x}{=}{1}}{+}\frac{{348657666}}{{5}}{}\genfrac{}{}{0}{}{\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{f}{}\left({x}\right)}{\phantom{{x}{=}{0}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{|}\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{f}{}\left({x}\right)}}{{x}{=}{0}}{-}\frac{{174328833}}{{10}}{}\genfrac{}{}{0}{}{\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{f}{}\left({x}\right)}{\phantom{{x}{=}{1}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{|}\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{f}{}\left({x}\right)}}{{x}{=}{1}}{,}\genfrac{}{}{0}{}{\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{f}{}\left({x}\right)}{\phantom{{x}{=}{10}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{|}\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{f}{}\left({x}\right)}}{{x}{=}{10}}{=}\frac{{402200989929}}{{500}}{}{f}{}\left({0}\right){+}\frac{{367470002559}}{{4000}}{}\genfrac{}{}{0}{}{\frac{{ⅆ}}{{ⅆ}{x}}{}{f}{}\left({x}\right)}{\phantom{{x}{=}{1}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{|}\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{{ⅆ}}{{ⅆ}{x}}{}{f}{}\left({x}\right)}}{{x}{=}{1}}{+}\frac{{355444180401}}{{1000}}{}\genfrac{}{}{0}{}{\frac{{ⅆ}}{{ⅆ}{x}}{}{f}{}\left({x}\right)}{\phantom{{x}{=}{0}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{|}\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{{ⅆ}}{{ⅆ}{x}}{}{f}{}\left({x}\right)}}{{x}{=}{0}}{+}\frac{{12791427403}}{{50}}{}\genfrac{}{}{0}{}{\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{f}{}\left({x}\right)}{\phantom{{x}{=}{0}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{|}\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{f}{}\left({x}\right)}}{{x}{=}{0}}{-}\frac{{9773025123}}{{200}}{}\genfrac{}{}{0}{}{\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{f}{}\left({x}\right)}{\phantom{{x}{=}{1}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{|}\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{f}{}\left({x}\right)}}{{x}{=}{1}}{+}\frac{{116219222}}{{5}}{}\genfrac{}{}{0}{}{\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}{}{f}{}\left({x}\right)}{\phantom{{x}{=}{0}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{|}\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}{}{f}{}\left({x}\right)}}{{x}{=}{0}}{-}\frac{{58109611}}{{10}}{}\genfrac{}{}{0}{}{\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}{}{f}{}\left({x}\right)}{\phantom{{x}{=}{1}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{|}\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}{}{f}{}\left({x}\right)}}{{x}{=}{1}}{,}\genfrac{}{}{0}{}{\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}{}{f}{}\left({x}\right)}{\phantom{{x}{=}{10}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{|}\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}{}{f}{}\left({x}\right)}}{{x}{=}{10}}{=}{-}\frac{{2713158528557}}{{20000}}{}{f}{}\left({0}\right){+}\frac{{13102438497001}}{{120000}}{}\genfrac{}{}{0}{}{\frac{{ⅆ}}{{ⅆ}{x}}{}{f}{}\left({x}\right)}{\phantom{{x}{=}{0}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{|}\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{{ⅆ}}{{ⅆ}{x}}{}{f}{}\left({x}\right)}}{{x}{=}{0}}{+}\frac{{83425799085959}}{{480000}}{}\genfrac{}{}{0}{}{\frac{{ⅆ}}{{ⅆ}{x}}{}{f}{}\left({x}\right)}{\phantom{{x}{=}{1}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{|}\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{{ⅆ}}{{ⅆ}{x}}{}{f}{}\left({x}\right)}}{{x}{=}{1}}{+}\frac{{367470002559}}{{8000}}{}\genfrac{}{}{0}{}{\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{f}{}\left({x}\right)}{\phantom{{x}{=}{1}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{|}\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{f}{}\left({x}\right)}}{{x}{=}{1}}{+}\frac{{355444180401}}{{2000}}{}\genfrac{}{}{0}{}{\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{f}{}\left({x}\right)}{\phantom{{x}{=}{0}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{|}\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{f}{}\left({x}\right)}}{{x}{=}{0}}{+}\frac{{12791427403}}{{150}}{}\genfrac{}{}{0}{}{\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}{}{f}{}\left({x}\right)}{\phantom{{x}{=}{0}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{|}\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}{}{f}{}\left({x}\right)}}{{x}{=}{0}}{-}\frac{{3257675041}}{{200}}{}\genfrac{}{}{0}{}{\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}{}{f}{}\left({x}\right)}{\phantom{{x}{=}{1}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{|}\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}{}{f}{}\left({x}\right)}}{{x}{=}{1}}{+}\frac{{58109611}}{{10}}{}\genfrac{}{}{0}{}{\frac{{{ⅆ}}^{{4}}}{{ⅆ}{{x}}^{{4}}}{}{f}{}\left({x}\right)}{\phantom{{x}{=}{0}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{|}\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{{{ⅆ}}^{{4}}}{{ⅆ}{{x}}^{{4}}}{}{f}{}\left({x}\right)}}{{x}{=}{0}}{-}\frac{{58109611}}{{40}}{}\genfrac{}{}{0}{}{\frac{{{ⅆ}}^{{4}}}{{ⅆ}{{x}}^{{4}}}{}{f}{}\left({x}\right)}{\phantom{{x}{=}{1}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{|}\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{{{ⅆ}}^{{4}}}{{ⅆ}{{x}}^{{4}}}{}{f}{}\left({x}\right)}}{{x}{=}{1}}{,}{f}{}\left({10}\right){=}\frac{{603680456}}{{5}}{}{f}{}\left({0}\right){+}\frac{{697315332}}{{5}}{}\genfrac{}{}{0}{}{\frac{{ⅆ}}{{ⅆ}{x}}{}{f}{}\left({x}\right)}{\phantom{{x}{=}{0}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{|}\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{{ⅆ}}{{ⅆ}{x}}{}{f}{}\left({x}\right)}}{{x}{=}{0}}{-}\frac{{174328833}}{{5}}{}\genfrac{}{}{0}{}{\frac{{ⅆ}}{{ⅆ}{x}}{}{f}{}\left({x}\right)}{\phantom{{x}{=}{1}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{|}\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{{ⅆ}}{{ⅆ}{x}}{}{f}{}\left({x}\right)}}{{x}{=}{1}}\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.