perform Zeilberger's algorithm (differential case)
Zeilberger(F, x, y, Dx)
Zeilberger(F, x, y, Dx, 'gosper_free')
hyperexponential function in x and y
name; denote the differential operator with respect to x
For a specified hyperexponential function F⁡x,y of x and y, the Zeilberger(F, x, y, Dx) calling sequence constructs for F⁡x,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
and a hyperexponential function G⁡x,y of x and y such that
Dx and Dy are the differential operators with respect to x, and y, respectively, defined by Dx⁡F⁡x,y=∂∂xF⁡x,y, and Dy⁡F⁡x,y=∂∂yF⁡x,y.
By assigning values to the global variables _MINORDER and _MAXORDER, the algorithm is restricted to finding a Z-pair L,G for F⁡x,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.
F ≔ ⅇ−x2y2−y2
Zpair ≔ Zeilberger⁡F,x,y,Dx:
L ≔ Zpair1
G ≔ Zpair2
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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