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IsDesingularizable

  

test for desingularizable linear recurrence equations

 

Calling Sequence

Parameters

Description

Options

Examples

References

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 x1<x2<...<xk of its trailing coefficient are called t-singularities. For all integer roots z1<z2<...<zp of its leading coefficient, zi&plus;d are called l-singularities where d is an order of the operator P.

• 

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 x1<x2<...<xm where m<k 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 zr+d<...<zp+d, where 1<r 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 x1<=x2<=...<=xm t-singularities and zr+d<=...<=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

withLREtools&colon;

pnn12vn&plus;n5n2vn&plus;1

p:=nn12vn&plus;n5n2vn&plus;1

(1)

 

IsDesingularizable&lsqb;&apos;trailing&apos;&rsqb;p&comma;vn&comma;&apos;desingularization&apos;&equals;&apos;T1&apos;&comma;&apos;remaining_singularities&apos;&equals;&apos;S1&apos;

true

(2)

 

T1&comma;S1

nvn&plus;152n2&plus;13n&plus;10vn&plus;1&plus;12n2&plus;5n37vn&plus;2&plus;136n21318n&plus;13vn&plus;3&plus;11440n217480n11360vn&plus;4&plus;11440n1720vn&plus;5&comma;0

(3)

 

IsDesingularizable&lsqb;&apos;leading&apos;&rsqb;p&comma;vn&comma;&apos;desingularization&apos;&equals;&apos;T2&apos;&comma;&apos;remaining_singularities&apos;&equals;&apos;S2&apos;

true

(4)

 

T2&comma;S2

3600n2&plus;2640nvn&plus;1680n&plus;26400vn&plus;1&plus;480n2&plus;2400n&plus;6720vn&plus;2&plus;27n2&plus;695n318vn&plus;3&plus;29n&plus;17vn&plus;4&plus;2vn&plus;5&comma;

(5)

 

IsDesingularizable&lsqb;&apos;both&apos;&rsqb;p&comma;vn&comma;&apos;desingularization&apos;&equals;&apos;T3&apos;

true

(6)

 

T3

1440nvn&plus;5173200n26547680n1368000vn&plus;1&plus;720n22412000n&plus;35543520vn&plus;2&plus;691160n2&plus;4837360n&plus;13824480vn&plus;3&plus;38879n2&plus;1078509n&plus;581716vn&plus;4&plus;41759n&plus;66238vn&plus;5&plus;2880vn&plus;6

(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.

See Also

LREtools

 


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