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 IsDesingularizable
 test for desingularizable linear recurrence equations

 Calling Sequence IsDesingularizable(eqn,fcn,opts) IsDesingularizable['both'](eqn,fcn,opts) IsDesingularizable['trailing'](eqn,fcn,opts) IsDesingularizable['leading'](eqn,fcn,opts)

Parameters

 eqn - linear recurrence equation with coefficients which are polynomials in n fcn - function name, for example, v(n) opts - sequence of optional arguments

Description

 • The IsDesingularizable['trailing'],  IsDesingularizable['leading'], IsDesingularizable['both'] commands determine if eqn is desingularizable related to trailing, leading or both coefficients respectively. If the IsDesingularizable command is called without an index, the IsDesingularizable['both'] is meant.
 • For the given eqn=P v(n) (where P is a difference operator with polynomial coefficients) integer roots $\mathrm{x1}<\mathrm{x2}<\mathrm{...}<\mathrm{xk}$ of its trailing coefficient are called t-singularities. For all integer roots z1
 • A recurrence T v(n) with polynomial coefficients, such that the operator T is right divisible by P and either T v(n) has no t-singularities or its t-singularities are $\mathrm{x1}<\mathrm{x2}<\mathrm{...}<\mathrm{xm}$ where $m and m is minimal, is called t-desingularization of P v(n). Similarly, T v(n), such that T is right divisible by P and either T v(n) has no l-singularities or its l-singularities are $\mathrm{zr}+d<\mathrm{...}<\mathrm{zp}+d$, where $1 and r is maximal, is called l-desingularization of P v(n).
 • If there is t-(resp. l-)-desingularization then P v(n) is called t-(resp. l)-desingularizable.
 • If P v(n) is desingularizable related to both trailing and leading coefficients (i.e. lt-desingularizable) there is its lt-desingularization which has only $\mathrm{x1}<=\mathrm{x2}<=\mathrm{...}<=\mathrm{xm}$ t-singularities and $\mathrm{zr}+d<=\mathrm{...}<=\mathrm{zp}+d$ l-singularities.
 • For example, it is useful to have a desingularization for solving the continuation problem. The singularities of the recurrence may present obstacles to continuing sequences which satisfy it. The desingularization can overcome those obstacles by removing these singularities.

Options

 • Each optional argument is of the type option = value. The following options are supported.
 'desingularization'
 Specifies the name T that is assigned to the t-(resp. l-,lt-)-desingularization if P v(n) is t-(resp. l-,lt-)-desingularizable, or is assigned to NULL otherwise.
 'remaining_singularities'
 Specifies the name S that is assigned to a set of t-(resp. l-)-singularities of T v(n) if P v(n) is t-(resp. l-)-desingularizable, or is assigned to t-(resp. l-)-singularities of P v(n) otherwise. In the case of lt-desingularization this option is ignored.
 •

Examples

 > $\mathrm{with}\left(\mathrm{LREtools}\right):$
 > $p≔n{\left(n-1\right)}^{2}v\left(n\right)+\left(n-5\right)\left(n-2\right)v\left(n+1\right)$
 ${p}{≔}{n}{}{\left({n}{-}{1}\right)}^{{2}}{}{v}{}\left({n}\right){+}\left({n}{-}{5}\right){}\left({n}{-}{2}\right){}{v}{}\left({n}{+}{1}\right)$ (1)

 > $\mathrm{IsDesingularizable}['\mathrm{trailing}']\left(p,v\left(n\right),'\mathrm{desingularization}'='\mathrm{T1}','\mathrm{remaining_singularities}'='\mathrm{S1}'\right)$
 ${\mathrm{true}}$ (2)

 > $\mathrm{T1},\mathrm{S1}$
 ${n}{}{v}{}\left({n}\right){+}\left(\frac{{15}}{{2}}{}{{n}}^{{2}}{+}{13}{}{n}{+}{10}\right){}{v}{}\left({n}{+}{1}\right){+}\left({-}\frac{{1}}{{2}}{}{{n}}^{{2}}{+}{5}{}{n}{-}{37}\right){}{v}{}\left({n}{+}{2}\right){+}\left({-}\frac{{1}}{{36}}{}{{n}}^{{2}}{-}\frac{{13}}{{18}}{}{n}{+}\frac{{1}}{{3}}\right){}{v}{}\left({n}{+}{3}\right){+}\left({-}\frac{{1}}{{1440}}{}{{n}}^{{2}}{-}\frac{{17}}{{480}}{}{n}{-}\frac{{11}}{{360}}\right){}{v}{}\left({n}{+}{4}\right){+}\left({-}\frac{{1}}{{1440}}{}{n}{-}\frac{{1}}{{720}}\right){}{v}{}\left({n}{+}{5}\right){,}\left\{{0}\right\}$ (3)

 > $\mathrm{IsDesingularizable}['\mathrm{leading}']\left(p,v\left(n\right),'\mathrm{desingularization}'='\mathrm{T2}','\mathrm{remaining_singularities}'='\mathrm{S2}'\right)$
 ${\mathrm{true}}$ (4)

 > $\mathrm{T2},\mathrm{S2}$
 $\left({-}{3600}{}{{n}}^{{2}}{+}{2640}{}{n}\right){}{v}{}\left({n}\right){+}\left({-}{1680}{}{n}{+}{26400}\right){}{v}{}\left({n}{+}{1}\right){+}\left({480}{}{{n}}^{{2}}{+}{2400}{}{n}{+}{6720}\right){}{v}{}\left({n}{+}{2}\right){+}\left({27}{}{{n}}^{{2}}{+}{695}{}{n}{-}{318}\right){}{v}{}\left({n}{+}{3}\right){+}\left({29}{}{n}{+}{17}\right){}{v}{}\left({n}{+}{4}\right){+}{2}{}{v}{}\left({n}{+}{5}\right){,}\left\{{}\right\}$ (5)

 > $\mathrm{IsDesingularizable}['\mathrm{both}']\left(p,v\left(n\right),'\mathrm{desingularization}'='\mathrm{T3}'\right)$
 ${\mathrm{true}}$ (6)

 > $\mathrm{T3}$
 ${1440}{}{n}{}{v}{}\left({n}\right){+}\left({-}{5173200}{}{{n}}^{{2}}{-}{6547680}{}{n}{-}{1368000}\right){}{v}{}\left({n}{+}{1}\right){+}\left({-}{720}{}{{n}}^{{2}}{-}{2412000}{}{n}{+}{35543520}\right){}{v}{}\left({n}{+}{2}\right){+}\left({691160}{}{{n}}^{{2}}{+}{4837360}{}{n}{+}{13824480}\right){}{v}{}\left({n}{+}{3}\right){+}\left({38879}{}{{n}}^{{2}}{+}{1078509}{}{n}{+}{581716}\right){}{v}{}\left({n}{+}{4}\right){+}\left({41759}{}{n}{+}{66238}\right){}{v}{}\left({n}{+}{5}\right){+}{2880}{}{v}{}\left({n}{+}{6}\right)$ (7)

References

 Abramov, S.A., and van Hoeij, M. "Desingularization of Linear Difference Operators with Polynomial Coefficients." Proceedings ISSAC'99, pp. 269-275. 1999.
 Abramov, S.A.; Barkatou, M.A.; and van Hoeij, M. "Apparent Singularities of Linear Difference  Equations with Polynomial Coefficients", http://arXiv.org/abs/math.CA/0409508.
 Mitichkina, A.M. "On an Implementation of Desingularization of Linear Recurrence Operators with Polynomial Coefficients." CAAP-2001, pp. 212-221. 2001.