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LREtools[HypergeometricTerm][HGDispersion] - return the hypergeometric dispersion of two polynomials depending on a hypergeometric term
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Calling Sequence
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HGDispersion(p, q, x, r)
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Parameters
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p
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first polynomial
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q
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second polynomial
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x
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independent variable, for example, x
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r
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list of equations that specifies the tower of hypergeometric extensions
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Description
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The HGDispersion(p, q, x, r) command returns the hypergeometric dispersion of p and q, that is,
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where E: Ex=x+1 is the shift operator and  and  are polynomials in K(r), where K is the ground field and r is the tower of hypergeometric extensions. Each ![r[i]](/support/helpjp/helpview.aspx?si=6368/file06027/math100.png) is specified by a hypergeometric term, that is, ![Er[i]/r[i]](/support/helpjp/helpview.aspx?si=6368/file06027/math102.png) is a rational function over K. The HGDispersion function returns  if the hypergeometric dispersion is not defined.
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The polynomials can contain hypergeometric terms in their coefficients. These terms are defined in the formal parameter r. Each hypergeometric term in the list is specified by a name, for example, t. It can be specified directly in the form of an equation, for example,  , or specified as a list consisting of the name of the term variable and the consecutive term ratio, for example, ![[t, n+1]](/support/helpjp/helpview.aspx?si=6368/file06027/math118.png) .
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The computation of hypergeometric dispersions is reduced to solving the  -orbit problem (see OrbitProblemSolution) in the shortened tower of hypergeometric extensions.
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Examples
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![with(LREtools[HypergeometricTerm])](/support/helpjp/helpview.aspx?si=6368/file06027/math134.png)
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![ext := [s = phi^x]](/support/helpjp/helpview.aspx?si=6368/file06027/math156.png)
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![ext := [s = (3+(4/5)*RootOf(_Z^2+1))^x]](/support/helpjp/helpview.aspx?si=6368/file06027/math159.png)
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![ext := [s = phi^x]](/support/helpjp/helpview.aspx?si=6368/file06027/math188.png)
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![ext := [s = RootOf(_Z^3-5)^x]](/support/helpjp/helpview.aspx?si=6368/file06027/math191.png)
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![ext := [v = 2^x, s = factorial(x)]](/support/helpjp/helpview.aspx?si=6368/file06027/math216.png)
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![ext := [v = 2^x, s = factorial(x)]](/support/helpjp/helpview.aspx?si=6368/file06027/math219.png)
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References
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Abramov, S.A., and Bronstein, M. "Hypergeometric dispersion and the orbit problem." Proc. ISSAC 2000.
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