QECreate - Maple Help

QDifferenceEquations

 QECreate
 create QESol data structure from a q-difference equation

 Calling Sequence QECreate(eq, var)

Parameters

 eq - q-difference equation or a list of such equations (for the system case) var - function variable to solve for, such as $y\left(x\right)$, or a list of such function variables (for the system case)

Description

 • The QECreate(eq,var) command returns QESol data structure, which contains some parameters useful to handle the given q-difference equation.
 • The parameters of QESol are a set of normalized equations, a set of function names, and an information table, INFO. The possible entries in this table are:
 *   linear - true or false
 *   functions - name or a set of names of the recurrence function(s)
 *   vars - name or set of names of the recurrence variable(s)
 *   q_par - name or number used as the parameter q
 *   q_shifts - set of the q-shifts (the orders of q-difference operator in the equation)
 *   order - order of the equation
 *   Qeqn - equation in &Q[vars, q_par](functions) which represents the q-difference operator associated to the linear q-difference equation
 *   coeffs - list of the rhs of the equation and the coefficients of the Qeqn
 *   zero_coeffs - coefficients of the Qeqn
 *   alpha_coeffs - degrees of the coefficients of the Qeqn
 *   systemproperties - result of LinearFunctionalSystems[Properties] (for the system case)
 Note: The first 3 items are always present. The last item is present only in the system case. The 4th item is present only for the scalar equation and the remaining items are there only if the equation is scalar and linear.

Examples

 > $\mathrm{with}\left(\mathrm{QDifferenceEquations}\right):$
 > $\mathrm{eq1}≔{q}^{3}\left(qx+1\right)y\left({q}^{2}x\right)-2{q}^{2}\left(x+1\right)y\left(qx\right)+y\left(x\right)\left(x+q\right)=\left({q}^{6}-2{q}^{3}+1\right){x}^{2}+x\left({q}^{5}-2{q}^{3}+q\right)$
 ${\mathrm{eq1}}{≔}{{q}}^{{3}}{}\left({q}{}{x}{+}{1}\right){}{y}{}\left({{q}}^{{2}}{}{x}\right){-}{2}{}{{q}}^{{2}}{}\left({x}{+}{1}\right){}{y}{}\left({q}{}{x}\right){+}{y}{}\left({x}\right){}\left({x}{+}{q}\right){=}\left({{q}}^{{6}}{-}{2}{}{{q}}^{{3}}{+}{1}\right){}{{x}}^{{2}}{+}{x}{}\left({{q}}^{{5}}{-}{2}{}{{q}}^{{3}}{+}{q}\right)$ (1)
 > $\mathrm{qe1}≔\mathrm{QECreate}\left(\mathrm{eq1},y\left(x\right)\right)$
 ${\mathrm{qe1}}{≔}{\mathrm{QESol}}{}\left(\left\{{y}{}\left({x}\right){}\left({x}{+}{q}\right){+}\left({-}{2}{}{{q}}^{{2}}{}{x}{-}{2}{}{{q}}^{{2}}\right){}{y}{}\left({q}{}{x}\right){+}\left({{q}}^{{4}}{}{x}{+}{{q}}^{{3}}\right){}{y}{}\left({{q}}^{{2}}{}{x}\right){=}\left({{q}}^{{6}}{-}{2}{}{{q}}^{{3}}{+}{1}\right){}{{x}}^{{2}}{+}{x}{}\left({{q}}^{{5}}{-}{2}{}{{q}}^{{3}}{+}{q}\right)\right\}{,}\left\{{y}{}\left({x}\right)\right\}{,}{\mathrm{INFO}}\right)$ (2)
 > $\mathrm{eval}\left(\mathrm{op}\left(3,\mathrm{qe1}\right)\right)$
 ${table}{}\left(\left[{\mathrm{alpha_coeffs}}{=}\left[{0}{,}{0}{,}{0}\right]{,}{\mathrm{q_par}}{=}{q}{,}{\mathrm{vars}}{=}{x}{,}{\mathrm{linear}}{=}{\mathrm{true}}{,}{\mathrm{order}}{=}{2}{,}{\mathrm{zero_coeffs}}{=}\left[{x}{+}{q}{,}{-}{2}{}{{q}}^{{2}}{}{x}{-}{2}{}{{q}}^{{2}}{,}{{q}}^{{4}}{}{x}{+}{{q}}^{{3}}\right]{,}{\mathrm{coeffs}}{=}\left[{-}\left({{q}}^{{6}}{-}{2}{}{{q}}^{{3}}{+}{1}\right){}{{x}}^{{2}}{-}{x}{}\left({{q}}^{{5}}{-}{2}{}{{q}}^{{3}}{+}{q}\right){,}{x}{+}{q}{,}{-}{2}{}{{q}}^{{2}}{}{x}{-}{2}{}{{q}}^{{2}}{,}{{q}}^{{4}}{}{x}{+}{{q}}^{{3}}\right]{,}{\mathrm{functions}}{=}{y}{,}{\mathrm{qshifts}}{=}\left\{{0}{,}{1}{,}{2}\right\}{,}{\mathrm{Qeqn}}{=}{x}{+}{q}{+}\left({-}{2}{}{{q}}^{{2}}{}{x}{-}{2}{}{{q}}^{{2}}\right){}{{\mathrm{&Q}}}_{{x}{,}{q}}{}\left({y}\right){+}\left({{q}}^{{4}}{}{x}{+}{{q}}^{{3}}\right){}{{{\mathrm{&Q}}}_{{x}{,}{q}}{}\left({y}\right)}^{{2}}\right]\right)$ (3)
 > $\mathrm{sys}≔\left[\mathrm{y2}\left(qx\right)-\mathrm{y1}\left(x\right),100\mathrm{y2}\left(x\right)qx+10000\mathrm{y2}\left(x\right)q-100\mathrm{y2}\left(x\right)x-10000\mathrm{y2}\left(x\right)-100{q}^{3}\mathrm{y1}\left(x\right)x-10000{q}^{2}\mathrm{y1}\left(x\right)+100{q}^{4}\mathrm{y1}\left(qx\right)x+10000\mathrm{y1}\left(qx\right){q}^{2}-100\mathrm{y1}\left(qx\right){q}^{3}x-10000\mathrm{y1}\left(qx\right)q+100\mathrm{y1}\left(x\right)qx+10000\mathrm{y1}\left(x\right)\right]$
 ${\mathrm{sys}}{≔}\left[{\mathrm{y2}}{}\left({q}{}{x}\right){-}{\mathrm{y1}}{}\left({x}\right){,}{100}{}{\mathrm{y2}}{}\left({x}\right){}{q}{}{x}{+}{10000}{}{\mathrm{y2}}{}\left({x}\right){}{q}{-}{100}{}{\mathrm{y2}}{}\left({x}\right){}{x}{-}{10000}{}{\mathrm{y2}}{}\left({x}\right){-}{100}{}{{q}}^{{3}}{}{\mathrm{y1}}{}\left({x}\right){}{x}{-}{10000}{}{{q}}^{{2}}{}{\mathrm{y1}}{}\left({x}\right){+}{100}{}{{q}}^{{4}}{}{\mathrm{y1}}{}\left({q}{}{x}\right){}{x}{+}{10000}{}{\mathrm{y1}}{}\left({q}{}{x}\right){}{{q}}^{{2}}{-}{100}{}{\mathrm{y1}}{}\left({q}{}{x}\right){}{{q}}^{{3}}{}{x}{-}{10000}{}{\mathrm{y1}}{}\left({q}{}{x}\right){}{q}{+}{100}{}{\mathrm{y1}}{}\left({x}\right){}{q}{}{x}{+}{10000}{}{\mathrm{y1}}{}\left({x}\right)\right]$ (4)
 > $\mathrm{vars}≔\left[\mathrm{y1}\left(x\right),\mathrm{y2}\left(x\right)\right]:$
 > $\mathrm{qe2}≔\mathrm{QECreate}\left(\mathrm{sys},\mathrm{vars}\right)$
 ${\mathrm{qe2}}{≔}{\mathrm{QESol}}{}\left(\left\{{\mathrm{y2}}{}\left({q}{}{x}\right){-}{\mathrm{y1}}{}\left({x}\right){=}{0}{,}{100}{}{\mathrm{y2}}{}\left({x}\right){}{q}{}{x}{+}{10000}{}{\mathrm{y2}}{}\left({x}\right){}{q}{-}{100}{}{\mathrm{y2}}{}\left({x}\right){}{x}{-}{10000}{}{\mathrm{y2}}{}\left({x}\right){-}{100}{}{{q}}^{{3}}{}{\mathrm{y1}}{}\left({x}\right){}{x}{-}{10000}{}{{q}}^{{2}}{}{\mathrm{y1}}{}\left({x}\right){+}{100}{}{{q}}^{{4}}{}{\mathrm{y1}}{}\left({q}{}{x}\right){}{x}{+}{10000}{}{\mathrm{y1}}{}\left({q}{}{x}\right){}{{q}}^{{2}}{-}{100}{}{\mathrm{y1}}{}\left({q}{}{x}\right){}{{q}}^{{3}}{}{x}{-}{10000}{}{\mathrm{y1}}{}\left({q}{}{x}\right){}{q}{+}{100}{}{\mathrm{y1}}{}\left({x}\right){}{q}{}{x}{+}{10000}{}{\mathrm{y1}}{}\left({x}\right){=}{0}\right\}{,}\left\{{\mathrm{y1}}{}\left({x}\right){,}{\mathrm{y2}}{}\left({x}\right)\right\}{,}{\mathrm{INFO}}\right)$ (5)
 > $\mathrm{eval}\left(\mathrm{op}\left(3,\mathrm{qe2}\right)\right)$
 ${table}{}\left(\left[{\mathrm{q_par}}{=}{q}{,}{\mathrm{systemproperties}}{=}{table}{}\left(\left[{\mathrm{system0}}{=}\left[{\mathrm{y2}}{}\left({q}{}{x}\right){-}{\mathrm{y1}}{}\left({x}\right){,}{100}{}{\mathrm{y2}}{}\left({x}\right){}{q}{}{x}{+}{10000}{}{\mathrm{y2}}{}\left({x}\right){}{q}{-}{100}{}{\mathrm{y2}}{}\left({x}\right){}{x}{-}{10000}{}{\mathrm{y2}}{}\left({x}\right){-}{100}{}{{q}}^{{3}}{}{\mathrm{y1}}{}\left({x}\right){}{x}{-}{10000}{}{{q}}^{{2}}{}{\mathrm{y1}}{}\left({x}\right){+}{100}{}{{q}}^{{4}}{}{\mathrm{y1}}{}\left({q}{}{x}\right){}{x}{+}{10000}{}{\mathrm{y1}}{}\left({q}{}{x}\right){}{{q}}^{{2}}{-}{100}{}{\mathrm{y1}}{}\left({q}{}{x}\right){}{{q}}^{{3}}{}{x}{-}{10000}{}{\mathrm{y1}}{}\left({q}{}{x}\right){}{q}{+}{100}{}{\mathrm{y1}}{}\left({x}\right){}{q}{}{x}{+}{10000}{}{\mathrm{y1}}{}\left({x}\right)\right]{,}{\mathrm{q_par}}{=}{q}{,}{\mathrm{input}}{=}{\mathrm{lists}}{,}{\mathrm{system}}{=}\left[{\mathrm{y2}}{}\left({q}{}{x}\right){-}{\mathrm{y1}}{}\left({x}\right){,}{100}{}{\mathrm{y2}}{}\left({x}\right){}{q}{}{x}{+}{10000}{}{\mathrm{y2}}{}\left({x}\right){}{q}{-}{100}{}{\mathrm{y2}}{}\left({x}\right){}{x}{-}{10000}{}{\mathrm{y2}}{}\left({x}\right){-}{100}{}{{q}}^{{3}}{}{\mathrm{y1}}{}\left({x}\right){}{x}{-}{10000}{}{{q}}^{{2}}{}{\mathrm{y1}}{}\left({x}\right){+}{100}{}{{q}}^{{4}}{}{\mathrm{y1}}{}\left({q}{}{x}\right){}{x}{+}{10000}{}{\mathrm{y1}}{}\left({q}{}{x}\right){}{{q}}^{{2}}{-}{100}{}{\mathrm{y1}}{}\left({q}{}{x}\right){}{{q}}^{{3}}{}{x}{-}{10000}{}{\mathrm{y1}}{}\left({q}{}{x}\right){}{q}{+}{100}{}{\mathrm{y1}}{}\left({x}\right){}{q}{}{x}{+}{10000}{}{\mathrm{y1}}{}\left({x}\right)\right]{,}{\mathrm{order}}{=}{1}{,}{\mathrm{case}}{=}{\mathrm{qdifference}}{,}{\mathrm{method}}{=}{\mathrm{ordinary}}{,}{\mathrm{functions}}{=}\left[{\mathrm{y1}}{}\left({x}\right){,}{\mathrm{y2}}{}\left({x}\right)\right]{,}{\mathrm{var}}{=}{x}{,}{\mathrm{f_names}}{=}\left[{\mathrm{y1}}{,}{\mathrm{y2}}\right]{,}{\mathrm{qshifts}}{=}\left[\left\{{0}{,}{1}\right\}{,}\left\{{0}{,}{1}\right\}\right]\right]\right){,}{\mathrm{vars}}{=}\left\{{x}\right\}{,}{\mathrm{linear}}{=}{\mathrm{true}}{,}{\mathrm{functions}}{=}\left\{{\mathrm{y1}}{,}{\mathrm{y2}}\right\}\right]\right)$ (6)