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);

$L≔\sum _{i=0}^d{a}_i\left(x\right)E^i$

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:

The set of conditions. f(x) is assumed to be analytic on some open set which contains a set $A<=\mathrm{Re}\left(x\right)<A+d$. 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 $A<=\mathrm{Re}\left(x\right)<A+d$.

The set of formulas for computing $f\left(p\right),\frac{\&DifferentialD;}{\&DifferentialD;p}f\left(p\right)$,...,$\frac{{\&DifferentialD;}^m}{\&DifferentialD;p^m}f\left(p\right)$. (f(x) must satisfy the conditions in the first set.) These formulas give the values of $f\left(p\right),\frac{\&DifferentialD;}{\&DifferentialD;p}f\left(p\right)$,...,$\frac{{\&DifferentialD;}^m}{\&DifferentialD;p^m}f\left(p\right)$ as linear combinations of f(x) and several of its derivatives in $A<=\mathrm{Re}\left(x\right)<A+d$. For $m=0$, we have one unique formula for $f\left(p\right)$.

$\mathrm{with}\left(\mathrm{LREtools}\right)\&colon;$

$\mathrm{L1}≔x{E}^2-\left(3x-3\right)E+\left(2x-3\right)\left(12x+4\right)$

$\mathrm{L1}≔x{E}^2-\left(3x-3\right)E+\left(2x-3\right)\left(12x+4\right)$

$\mathrm{ValuesAtPoint}\left(\mathrm{L1}\&comma;E\&comma;f\left(x\right)\&comma;'\mathrm{HalfInterval}'=2\&comma;'\mathrm{Point}'=-\frac{1}{3}\right)$

$\left\{f\left(\frac{11}{3}\right)=-\frac{18f\left(\frac{8}{3}\right)}{5}\right\},\left\{f\left(-\frac{1}{3}\right)=\frac{2f\left(\frac{8}{3}\right)}{75}+\frac{\left(\genfrac{}{}{0ex}{}{\left(\frac{\&DifferentialD;}{\&DifferentialD;x}f\left(x\right)\right)}{\phantom{\left\{x=\frac{8}{3}\right\}}}|\genfrac{}{}{0ex}{}{\phantom{\left(\frac{\&DifferentialD;}{\&DifferentialD;x}f\left(x\right)\right)}}{\left\{x=\frac{8}{3}\right\}}\right)}{440}+\frac{\left(\genfrac{}{}{0ex}{}{\left(\frac{\&DifferentialD;}{\&DifferentialD;x}f\left(x\right)\right)}{\phantom{\left\{x=\frac{11}{3}\right\}}}|\genfrac{}{}{0ex}{}{\phantom{\left(\frac{\&DifferentialD;}{\&DifferentialD;x}f\left(x\right)\right)}}{\left\{x=\frac{11}{3}\right\}}\right)}{1584}\right\}$

$\mathrm{ValuesAtPoint}\left(\mathrm{L1}\&comma;E\&comma;f\left(x\right)\&comma;'\mathrm{HalfInterval}'=2\&comma;'\mathrm{Point}'=\mathrm{RootOf}\left(x^2+1\&comma;x\&comma;\mathrm{index}=1\right)\&comma;'\mathrm{OrderDer}'=5\right)$

$\varnothing ,\left\{\genfrac{}{}{0ex}{}{\left(\frac{{\&DifferentialD;}^5}{\&DifferentialD;x^5}f\left(x\right)\right)}{\phantom{\left\{x=\mathrm{I}\right\}}}|\genfrac{}{}{0ex}{}{\phantom{\left(\frac{{\&DifferentialD;}^5}{\&DifferentialD;x^5}f\left(x\right)\right)}}{\left\{x=\mathrm{I}\right\}}=\frac{60416991\mathrm{I}{\mathrm{D}}^{\left(2\right)}\left(f\right)\left(2+\mathrm{I}\right)}{22313281250}-\frac{21134484\mathrm{I}{\mathrm{D}}^{\left(2\right)}\left(f\right)\left(3+\mathrm{I}\right)}{11156640625}+\frac{549\mathrm{I}{\mathrm{D}}^{\left(4\right)}\left(f\right)\left(3+\mathrm{I}\right)}{3380000}-\frac{\mathrm{I}{\mathrm{D}}^{\left(5\right)}\left(f\right)\left(3+\mathrm{I}\right)}{208000}-\frac{48839499533961\mathrm{I}f\left(2+\mathrm{I}\right)}{18854722656250000}+\frac{46962840717153\mathrm{I}f\left(3+\mathrm{I}\right)}{18854722656250000}+\frac{556477\mathrm{I}{\mathrm{D}}^{\left(3\right)}\left(f\right)\left(2+\mathrm{I}\right)}{549250000}+\frac{219283\mathrm{I}{\mathrm{D}}^{\left(3\right)}\left(f\right)\left(3+\mathrm{I}\right)}{1098500000}-\frac{25780729047\mathrm{I}\mathrm{D}\left(f\right)\left(3+\mathrm{I}\right)}{29007265625000}+\frac{68810341503\mathrm{I}\mathrm{D}\left(f\right)\left(2+\mathrm{I}\right)}{58014531250000}+\frac{357\mathrm{I}{\mathrm{D}}^{\left(4\right)}\left(f\right)\left(2+\mathrm{I}\right)}{3380000}+\frac{3\mathrm{I}{\mathrm{D}}^{\left(5\right)}\left(f\right)\left(2+\mathrm{I}\right)}{26000}-\frac{204172941{\mathrm{D}}^{\left(2\right)}\left(f\right)\left(3+\mathrm{I}\right)}{357012500000}-\frac{368697{\mathrm{D}}^{\left(3\right)}\left(f\right)\left(3+\mathrm{I}\right)}{549250000}+\frac{577{\mathrm{D}}^{\left(4\right)}\left(f\right)\left(3+\mathrm{I}\right)}{13520000}+\frac{{\mathrm{D}}^{\left(5\right)}\left(f\right)\left(3+\mathrm{I}\right)}{41600}-\frac{3853718024019f\left(2+\mathrm{I}\right)}{2356840332031250}+\frac{8319818839971f\left(3+\mathrm{I}\right)}{18854722656250000}-\frac{250202038329\mathrm{D}\left(f\right)\left(2+\mathrm{I}\right)}{58014531250000}+\frac{190021307517\mathrm{D}\left(f\right)\left(3+\mathrm{I}\right)}{58014531250000}-\frac{178457979{\mathrm{D}}^{\left(2\right)}\left(f\right)\left(2+\mathrm{I}\right)}{178506250000}+\frac{329139{\mathrm{D}}^{\left(3\right)}\left(f\right)\left(2+\mathrm{I}\right)}{549250000}+\frac{529{\mathrm{D}}^{\left(4\right)}\left(f\right)\left(2+\mathrm{I}\right)}{3380000}+\frac{43{\mathrm{D}}^{\left(5\right)}\left(f\right)\left(2+\mathrm{I}\right)}{624000}\&comma;\genfrac{}{}{0ex}{}{\left(\frac{{\&DifferentialD;}^4}{\&DifferentialD;x^4}f\left(x\right)\right)}{\phantom{\left\{x=\mathrm{I}\right\}}}|\genfrac{}{}{0ex}{}{\phantom{\left(\frac{{\&DifferentialD;}^4}{\&DifferentialD;x^4}f\left(x\right)\right)}}{\left\{x=\mathrm{I}\right\}}=\frac{1669431\mathrm{I}{\mathrm{D}}^{\left(2\right)}\left(f\right)\left(2+\mathrm{I}\right)}{549250000}+\frac{657849\mathrm{I}{\mathrm{D}}^{\left(2\right)}\left(f\right)\left(3+\mathrm{I}\right)}{1098500000}-\frac{\mathrm{I}{\mathrm{D}}^{\left(4\right)}\left(f\right)\left(3+\mathrm{I}\right)}{41600}+\frac{68810341503\mathrm{I}f\left(2+\mathrm{I}\right)}{58014531250000}-\frac{25780729047\mathrm{I}f\left(3+\mathrm{I}\right)}{29007265625000}+\frac{357\mathrm{I}{\mathrm{D}}^{\left(3\right)}\left(f\right)\left(2+\mathrm{I}\right)}{845000}+\frac{549\mathrm{I}{\mathrm{D}}^{\left(3\right)}\left(f\right)\left(3+\mathrm{I}\right)}{845000}-\frac{42268968\mathrm{I}\mathrm{D}\left(f\right)\left(3+\mathrm{I}\right)}{11156640625}+\frac{60416991\mathrm{I}\mathrm{D}\left(f\right)\left(2+\mathrm{I}\right)}{11156640625}+\frac{3\mathrm{I}{\mathrm{D}}^{\left(4\right)}\left(f\right)\left(2+\mathrm{I}\right)}{5200}-\frac{1106091{\mathrm{D}}^{\left(2\right)}\left(f\right)\left(3+\mathrm{I}\right)}{549250000}+\frac{577{\mathrm{D}}^{\left(3\right)}\left(f\right)\left(3+\mathrm{I}\right)}{3380000}+\frac{{\mathrm{D}}^{\left(4\right)}\left(f\right)\left(3+\mathrm{I}\right)}{8320}-\frac{250202038329f\left(2+\mathrm{I}\right)}{58014531250000}+\frac{190021307517f\left(3+\mathrm{I}\right)}{58014531250000}-\frac{178457979\mathrm{D}\left(f\right)\left(2+\mathrm{I}\right)}{89253125000}-\frac{204172941\mathrm{D}\left(f\right)\left(3+\mathrm{I}\right)}{178506250000}+\frac{987417{\mathrm{D}}^{\left(2\right)}\left(f\right)\left(2+\mathrm{I}\right)}{549250000}+\frac{529{\mathrm{D}}^{\left(3\right)}\left(f\right)\left(2+\mathrm{I}\right)}{845000}+\frac{43{\mathrm{D}}^{\left(4\right)}\left(f\right)\left(2+\mathrm{I}\right)}{124800}\&comma;\genfrac{}{}{0ex}{}{\left(\frac{{\&DifferentialD;}^3}{\&DifferentialD;x^3}f\left(x\right)\right)}{\phantom{\left\{x=\mathrm{I}\right\}}}|\genfrac{}{}{0ex}{}{\phantom{\left(\frac{{\&DifferentialD;}^3}{\&DifferentialD;x^3}f\left(x\right)\right)}}{\left\{x=\mathrm{I}\right\}}=\frac{3\mathrm{I}{\mathrm{D}}^{\left(3\right)}\left(f\right)\left(2+\mathrm{I}\right)}{1300}+\frac{1071\mathrm{I}{\mathrm{D}}^{\left(2\right)}\left(f\right)\left(2+\mathrm{I}\right)}{845000}+\frac{1647\mathrm{I}{\mathrm{D}}^{\left(2\right)}\left(f\right)\left(3+\mathrm{I}\right)}{845000}-\frac{\mathrm{I}{\mathrm{D}}^{\left(3\right)}\left(f\right)\left(3+\mathrm{I}\right)}{10400}+\frac{60416991\mathrm{I}f\left(2+\mathrm{I}\right)}{11156640625}+\frac{657849\mathrm{I}\mathrm{D}\left(f\right)\left(3+\mathrm{I}\right)}{549250000}+\frac{1669431\mathrm{I}\mathrm{D}\left(f\right)\left(2+\mathrm{I}\right)}{274625000}-\frac{42268968\mathrm{I}f\left(3+\mathrm{I}\right)}{11156640625}+\frac{1731{\mathrm{D}}^{\left(2\right)}\left(f\right)\left(3+\mathrm{I}\right)}{3380000}+\frac{{\mathrm{D}}^{\left(3\right)}\left(f\right)\left(3+\mathrm{I}\right)}{2080}-\frac{178457979f\left(2+\mathrm{I}\right)}{89253125000}-\frac{204172941f\left(3+\mathrm{I}\right)}{178506250000}+\frac{987417\mathrm{D}\left(f\right)\left(2+\mathrm{I}\right)}{274625000}-\frac{1106091\mathrm{D}\left(f\right)\left(3+\mathrm{I}\right)}{274625000}+\frac{1587{\mathrm{D}}^{\left(2\right)}\left(f\right)\left(2+\mathrm{I}\right)}{845000}+\frac{43{\mathrm{D}}^{\left(3\right)}\left(f\right)\left(2+\mathrm{I}\right)}{31200}\&comma;\genfrac{}{}{0ex}{}{\left(\frac{{\&DifferentialD;}^2}{\&DifferentialD;x^2}f\left(x\right)\right)}{\phantom{\left\{x=\mathrm{I}\right\}}}|\genfrac{}{}{0ex}{}{\phantom{\left(\frac{{\&DifferentialD;}^2}{\&DifferentialD;x^2}f\left(x\right)\right)}}{\left\{x=\mathrm{I}\right\}}=-\frac{3\mathrm{I}{\mathrm{D}}^{\left(2\right)}\left(f\right)\left(3+\mathrm{I}\right)}{10400}+\frac{1669431\mathrm{I}f\left(2+\mathrm{I}\right)}{274625000}+\frac{657849\mathrm{I}f\left(3+\mathrm{I}\right)}{549250000}+\frac{1071\mathrm{I}\mathrm{D}\left(f\right)\left(2+\mathrm{I}\right)}{422500}+\frac{1647\mathrm{I}\mathrm{D}\left(f\right)\left(3+\mathrm{I}\right)}{422500}+\frac{9\mathrm{I}{\mathrm{D}}^{\left(2\right)}\left(f\right)\left(2+\mathrm{I}\right)}{1300}+\frac{3{\mathrm{D}}^{\left(2\right)}\left(f\right)\left(3+\mathrm{I}\right)}{2080}+\frac{987417f\left(2+\mathrm{I}\right)}{274625000}-\frac{1106091f\left(3+\mathrm{I}\right)}{274625000}+\frac{1587\mathrm{D}\left(f\right)\left(2+\mathrm{I}\right)}{422500}+\frac{1731\mathrm{D}\left(f\right)\left(3+\mathrm{I}\right)}{1690000}+\frac{43{\mathrm{D}}^{\left(2\right)}\left(f\right)\left(2+\mathrm{I}\right)}{10400}\&comma;\genfrac{}{}{0ex}{}{\left(\frac{\&DifferentialD;}{\&DifferentialD;x}f\left(x\right)\right)}{\phantom{\left\{x=\mathrm{I}\right\}}}|\genfrac{}{}{0ex}{}{\phantom{\left(\frac{\&DifferentialD;}{\&DifferentialD;x}f\left(x\right)\right)}}{\left\{x=\mathrm{I}\right\}}=\frac{1071\mathrm{I}f\left(2+\mathrm{I}\right)}{422500}+\frac{1647\mathrm{I}f\left(3+\mathrm{I}\right)}{422500}+\frac{9\mathrm{I}\mathrm{D}\left(f\right)\left(2+\mathrm{I}\right)}{650}-\frac{3\mathrm{I}\mathrm{D}\left(f\right)\left(3+\mathrm{I}\right)}{5200}+\frac{1587f\left(2+\mathrm{I}\right)}{422500}+\frac{1731f\left(3+\mathrm{I}\right)}{1690000}+\frac{43\mathrm{D}\left(f\right)\left(2+\mathrm{I}\right)}{5200}+\frac{3\mathrm{D}\left(f\right)\left(3+\mathrm{I}\right)}{1040}\&comma;f\left(\mathrm{I}\right)=\frac{9\mathrm{I}f\left(2+\mathrm{I}\right)}{650}-\frac{3\mathrm{I}f\left(3+\mathrm{I}\right)}{5200}+\frac{43f\left(2+\mathrm{I}\right)}{5200}+\frac{3f\left(3+\mathrm{I}\right)}{1040}\right\}$

$\mathrm{ValuesAtPoint}\left(\mathrm{L1}\&comma;E\&comma;f\left(x\right)\&comma;'\mathrm{HalfInterval}'=0\&comma;'\mathrm{Point}'=2\right)$

$\left\{f\left(1\right)=4f\left(0\right)\right\},\left\{f\left(2\right)=40f\left(0\right)+12\left(\genfrac{}{}{0ex}{}{\left(\frac{\&DifferentialD;}{\&DifferentialD;x}f\left(x\right)\right)}{\phantom{\left\{x=0\right\}}}|\genfrac{}{}{0ex}{}{\phantom{\left(\frac{\&DifferentialD;}{\&DifferentialD;x}f\left(x\right)\right)}}{\left\{x=0\right\}}\right)-3\left(\genfrac{}{}{0ex}{}{\left(\frac{\&DifferentialD;}{\&DifferentialD;x}f\left(x\right)\right)}{\phantom{\left\{x=1\right\}}}|\genfrac{}{}{0ex}{}{\phantom{\left(\frac{\&DifferentialD;}{\&DifferentialD;x}f\left(x\right)\right)}}{\left\{x=1\right\}}\right)\right\}$

$\mathrm{ValuesAtPoint}\left(\mathrm{L1}\&comma;E\&comma;f\left(x\right)\&comma;'\mathrm{HalfInterval}'=0\&comma;'\mathrm{Point}'=10\&comma;'\mathrm{OrderDer}'=3\right)$

$\left\{f\left(1\right)=4f\left(0\right)\right\},\left\{\genfrac{}{}{0ex}{}{\left(\frac{{\&DifferentialD;}^3}{\&DifferentialD;x^3}f\left(x\right)\right)}{\phantom{\left\{x=10\right\}}}|\genfrac{}{}{0ex}{}{\phantom{\left(\frac{{\&DifferentialD;}^3}{\&DifferentialD;x^3}f\left(x\right)\right)}}{\left\{x=10\right\}}=\frac{355444180401\left(\genfrac{}{}{0ex}{}{\left(\frac{{\&DifferentialD;}^2}{\&DifferentialD;x^2}f\left(x\right)\right)}{\phantom{\left\{x=0\right\}}}|\genfrac{}{}{0ex}{}{\phantom{\left(\frac{{\&DifferentialD;}^2}{\&DifferentialD;x^2}f\left(x\right)\right)}}{\left\{x=0\right\}}\right)}{2000}+\frac{12791427403\left(\genfrac{}{}{0ex}{}{\left(\frac{{\&DifferentialD;}^3}{\&DifferentialD;x^3}f\left(x\right)\right)}{\phantom{\left\{x=0\right\}}}|\genfrac{}{}{0ex}{}{\phantom{\left(\frac{{\&DifferentialD;}^3}{\&DifferentialD;x^3}f\left(x\right)\right)}}{\left\{x=0\right\}}\right)}{150}-\frac{3257675041\left(\genfrac{}{}{0ex}{}{\left(\frac{{\&DifferentialD;}^3}{\&DifferentialD;x^3}f\left(x\right)\right)}{\phantom{\left\{x=1\right\}}}|\genfrac{}{}{0ex}{}{\phantom{\left(\frac{{\&DifferentialD;}^3}{\&DifferentialD;x^3}f\left(x\right)\right)}}{\left\{x=1\right\}}\right)}{200}+\frac{367470002559\left(\genfrac{}{}{0ex}{}{\left(\frac{{\&DifferentialD;}^2}{\&DifferentialD;x^2}f\left(x\right)\right)}{\phantom{\left\{x=1\right\}}}|\genfrac{}{}{0ex}{}{\phantom{\left(\frac{{\&DifferentialD;}^2}{\&DifferentialD;x^2}f\left(x\right)\right)}}{\left\{x=1\right\}}\right)}{8000}-\frac{2713158528557f\left(0\right)}{20000}+\frac{13102438497001\left(\genfrac{}{}{0ex}{}{\left(\frac{\&DifferentialD;}{\&DifferentialD;x}f\left(x\right)\right)}{\phantom{\left\{x=0\right\}}}|\genfrac{}{}{0ex}{}{\phantom{\left(\frac{\&DifferentialD;}{\&DifferentialD;x}f\left(x\right)\right)}}{\left\{x=0\right\}}\right)}{120000}+\frac{83425799085959\left(\genfrac{}{}{0ex}{}{\left(\frac{\&DifferentialD;}{\&DifferentialD;x}f\left(x\right)\right)}{\phantom{\left\{x=1\right\}}}|\genfrac{}{}{0ex}{}{\phantom{\left(\frac{\&DifferentialD;}{\&DifferentialD;x}f\left(x\right)\right)}}{\left\{x=1\right\}}\right)}{480000}+\frac{58109611\left(\genfrac{}{}{0ex}{}{\left(\frac{{\&DifferentialD;}^4}{\&DifferentialD;x^4}f\left(x\right)\right)}{\phantom{\left\{x=0\right\}}}|\genfrac{}{}{0ex}{}{\phantom{\left(\frac{{\&DifferentialD;}^4}{\&DifferentialD;x^4}f\left(x\right)\right)}}{\left\{x=0\right\}}\right)}{10}-\frac{58109611\left(\genfrac{}{}{0ex}{}{\left(\frac{{\&DifferentialD;}^4}{\&DifferentialD;x^4}f\left(x\right)\right)}{\phantom{\left\{x=1\right\}}}|\genfrac{}{}{0ex}{}{\phantom{\left(\frac{{\&DifferentialD;}^4}{\&DifferentialD;x^4}f\left(x\right)\right)}}{\left\{x=1\right\}}\right)}{40}\&comma;\genfrac{}{}{0ex}{}{\left(\frac{{\&DifferentialD;}^2}{\&DifferentialD;x^2}f\left(x\right)\right)}{\phantom{\left\{x=10\right\}}}|\genfrac{}{}{0ex}{}{\phantom{\left(\frac{{\&DifferentialD;}^2}{\&DifferentialD;x^2}f\left(x\right)\right)}}{\left\{x=10\right\}}=\frac{355444180401\left(\genfrac{}{}{0ex}{}{\left(\frac{\&DifferentialD;}{\&DifferentialD;x}f\left(x\right)\right)}{\phantom{\left\{x=0\right\}}}|\genfrac{}{}{0ex}{}{\phantom{\left(\frac{\&DifferentialD;}{\&DifferentialD;x}f\left(x\right)\right)}}{\left\{x=0\right\}}\right)}{1000}+\frac{12791427403\left(\genfrac{}{}{0ex}{}{\left(\frac{{\&DifferentialD;}^2}{\&DifferentialD;x^2}f\left(x\right)\right)}{\phantom{\left\{x=0\right\}}}|\genfrac{}{}{0ex}{}{\phantom{\left(\frac{{\&DifferentialD;}^2}{\&DifferentialD;x^2}f\left(x\right)\right)}}{\left\{x=0\right\}}\right)}{50}-\frac{9773025123\left(\genfrac{}{}{0ex}{}{\left(\frac{{\&DifferentialD;}^2}{\&DifferentialD;x^2}f\left(x\right)\right)}{\phantom{\left\{x=1\right\}}}|\genfrac{}{}{0ex}{}{\phantom{\left(\frac{{\&DifferentialD;}^2}{\&DifferentialD;x^2}f\left(x\right)\right)}}{\left\{x=1\right\}}\right)}{200}+\frac{367470002559\left(\genfrac{}{}{0ex}{}{\left(\frac{\&DifferentialD;}{\&DifferentialD;x}f\left(x\right)\right)}{\phantom{\left\{x=1\right\}}}|\genfrac{}{}{0ex}{}{\phantom{\left(\frac{\&DifferentialD;}{\&DifferentialD;x}f\left(x\right)\right)}}{\left\{x=1\right\}}\right)}{4000}+\frac{402200989929f\left(0\right)}{500}+\frac{116219222\left(\genfrac{}{}{0ex}{}{\left(\frac{{\&DifferentialD;}^3}{\&DifferentialD;x^3}f\left(x\right)\right)}{\phantom{\left\{x=0\right\}}}|\genfrac{}{}{0ex}{}{\phantom{\left(\frac{{\&DifferentialD;}^3}{\&DifferentialD;x^3}f\left(x\right)\right)}}{\left\{x=0\right\}}\right)}{5}-\frac{58109611\left(\genfrac{}{}{0ex}{}{\left(\frac{{\&DifferentialD;}^3}{\&DifferentialD;x^3}f\left(x\right)\right)}{\phantom{\left\{x=1\right\}}}|\genfrac{}{}{0ex}{}{\phantom{\left(\frac{{\&DifferentialD;}^3}{\&DifferentialD;x^3}f\left(x\right)\right)}}{\left\{x=1\right\}}\right)}{10}\&comma;\genfrac{}{}{0ex}{}{\left(\frac{\&DifferentialD;}{\&DifferentialD;x}f\left(x\right)\right)}{\phantom{\left\{x=10\right\}}}|\genfrac{}{}{0ex}{}{\phantom{\left(\frac{\&DifferentialD;}{\&DifferentialD;x}f\left(x\right)\right)}}{\left\{x=10\right\}}=\frac{18072854574f\left(0\right)}{25}+\frac{12791427403\left(\genfrac{}{}{0ex}{}{\left(\frac{\&DifferentialD;}{\&DifferentialD;x}f\left(x\right)\right)}{\phantom{\left\{x=0\right\}}}|\genfrac{}{}{0ex}{}{\phantom{\left(\frac{\&DifferentialD;}{\&DifferentialD;x}f\left(x\right)\right)}}{\left\{x=0\right\}}\right)}{25}-\frac{9773025123\left(\genfrac{}{}{0ex}{}{\left(\frac{\&DifferentialD;}{\&DifferentialD;x}f\left(x\right)\right)}{\phantom{\left\{x=1\right\}}}|\genfrac{}{}{0ex}{}{\phantom{\left(\frac{\&DifferentialD;}{\&DifferentialD;x}f\left(x\right)\right)}}{\left\{x=1\right\}}\right)}{100}+\frac{348657666\left(\genfrac{}{}{0ex}{}{\left(\frac{{\&DifferentialD;}^2}{\&DifferentialD;x^2}f\left(x\right)\right)}{\phantom{\left\{x=0\right\}}}|\genfrac{}{}{0ex}{}{\phantom{\left(\frac{{\&DifferentialD;}^2}{\&DifferentialD;x^2}f\left(x\right)\right)}}{\left\{x=0\right\}}\right)}{5}-\frac{174328833\left(\genfrac{}{}{0ex}{}{\left(\frac{{\&DifferentialD;}^2}{\&DifferentialD;x^2}f\left(x\right)\right)}{\phantom{\left\{x=1\right\}}}|\genfrac{}{}{0ex}{}{\phantom{\left(\frac{{\&DifferentialD;}^2}{\&DifferentialD;x^2}f\left(x\right)\right)}}{\left\{x=1\right\}}\right)}{10}\&comma;f\left(10\right)=\frac{603680456f\left(0\right)}{5}+\frac{697315332\left(\genfrac{}{}{0ex}{}{\left(\frac{\&DifferentialD;}{\&DifferentialD;x}f\left(x\right)\right)}{\phantom{\left\{x=0\right\}}}|\genfrac{}{}{0ex}{}{\phantom{\left(\frac{\&DifferentialD;}{\&DifferentialD;x}f\left(x\right)\right)}}{\left\{x=0\right\}}\right)}{5}-\frac{174328833\left(\genfrac{}{}{0ex}{}{\left(\frac{\&DifferentialD;}{\&DifferentialD;x}f\left(x\right)\right)}{\phantom{\left\{x=1\right\}}}|\genfrac{}{}{0ex}{}{\phantom{\left(\frac{\&DifferentialD;}{\&DifferentialD;x}f\left(x\right)\right)}}{\left\{x=1\right\}}\right)}{5}\right\}$

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.