perform Zeilberger's algorithm (differential case) - Maple Help

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DEtools[Zeilberger] - perform Zeilberger's algorithm (differential case)

Calling Sequence

Zeilberger(F, x, y, Dx)

Zeilberger(F, x, y, Dx, 'gosper_free')

Parameters

F

-

hyperexponential function in x and y

x

-

name

y

-

name

Dx

-

name; denote the differential operator with respect to x

Description

• 

For a specified hyperexponential function Fx,y of x and y, the Zeilberger(F, x, y, Dx) calling sequence constructs for Fx,y a Z-pair L,G that consists of a linear differential operator with coefficients that are polynomials of x over the complex number field

L=avxDxv+...+a1xDx+a0x

  

and a hyperexponential function Gx,y of x and y such that

LoFx,y=DyGx,y

• 

Dx and Dy are the differential operators with respect to x, and y, respectively, defined by DxFx,y=xFx,y, and DyFx,y=yFx,y.

• 

By assigning values to the global variables _MINORDER and _MAXORDER, the algorithm is restricted to finding a Z-pair L,G for Fx,y such that the order of L is between _MINORDER and _MAXORDER.

• 

The algorithm has two implementations. The default implementation uses a variant of Gosper's algorithm, and another one is based on the universal denominators. With the 'gosper_free' option, Gosper-free implementation is used.

• 

The output from the Zeilberger command is a list of two elements L,G representing the computed Z-pair L,G.

Examples

withDEtools:

F:=ⅇx2y2y2

F:=ⅇx2y2y2

(1)

Zpair:=ZeilbergerF,x,y,Dx:

L:=Zpair1

L:=Dx24

(2)

G:=Zpair2

G:=2ⅇy4+x2y2y

(3)

See Also

SumTools[Hypergeometric][Zeilberger]

References

  

Almkvist, G, and Zeilberger, D. "The method of differentiating under the integral sign." Journal of Symbolic Computation. Vol. 10. (1990): 571-591.


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