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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 $F\left(x,y\right)$ of x and y, the Zeilberger(F, x, y, Dx) calling sequence constructs for $F\left(x,y\right)$ a Z-pair $L,G$ that consists of a linear differential operator with coefficients that are polynomials of x over the complex number field

$L={a}_{v}\left(x\right){\mathrm{Dx}}^{v}+\mathrm{...}+{a}_{1}\left(x\right)\mathrm{Dx}+{a}_{0}\left(x\right)$

 and a hyperexponential function $G\left(x,y\right)$ of x and y such that

$LoF\left(x,y\right)=\mathrm{Dy}G\left(x,y\right)$

 • Dx and Dy are the differential operators with respect to x, and y, respectively, defined by $\mathrm{Dx}\left(F\left(x,y\right)\right)=\frac{\partial }{\partial x}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}F\left(x,y\right)$, and $\mathrm{Dy}\left(F\left(x,y\right)\right)=\frac{\partial }{\partial y}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}F\left(x,y\right)$.
 • By assigning values to the global variables _MINORDER and _MAXORDER, the algorithm is restricted to finding a Z-pair $L,G$ for $F\left(x,y\right)$ 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 $\left[L,G\right]$ representing the computed Z-pair $L,G$.

Examples

 > $\mathrm{with}\left(\mathrm{DEtools}\right):$
 > $F≔\mathrm{exp}\left(-\frac{{x}^{2}}{{y}^{2}}-{y}^{2}\right)$
 ${F}{≔}{{ⅇ}}^{{-}\frac{{{x}}^{{2}}}{{{y}}^{{2}}}{-}{{y}}^{{2}}}$ (1)
 > $\mathrm{Zpair}≔\mathrm{Zeilberger}\left(F,x,y,\mathrm{Dx}\right):$
 > $L≔\mathrm{Zpair}\left[1\right]$
 ${L}{≔}{{\mathrm{Dx}}}^{{2}}{-}{4}$ (2)
 > $G≔\mathrm{Zpair}\left[2\right]$
 ${G}{≔}\frac{{2}{}{{ⅇ}}^{{-}\frac{{{y}}^{{4}}{+}{{x}}^{{2}}}{{{y}}^{{2}}}}}{{y}}$ (3)

References

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