return the hypergeometric dispersion of two polynomials depending on a hypergeometric term
HGDispersion(p, q, x, r)
independent variable, for example, x
list of equations that specifies the tower of hypergeometric extensions
The HGDispersion(p, q, x, r) command returns the hypergeometric dispersion of p and q, that is,
where E: Ex=x+1 is the shift operator and p⁡x and q⁡x are polynomials in K(r), where K is the ground field and r is the tower of hypergeometric extensions. Each ri is specified by a hypergeometric term, that is, Eriri is a rational function over K. The HGDispersion function returns −1 if the hypergeometric dispersion is not defined.
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, t=n!, or specified as a list consisting of the name of the term variable and the consecutive term ratio, for example, t,n+1.
The computation of hypergeometric dispersions is reduced to solving the σ-orbit problem (see OrbitProblemSolution) in the shortened tower of hypergeometric extensions.
p ≔ φ4⁢s2+φ2⁢s+1
q ≔ s2+s+1
ext ≔ s=φx
p ≔ 24⁢s2+22⁢s+1+v
q ≔ 16⁢x+32⁢x+22⁢x+12⁢s2+4⁢x+3⁢x+2⁢x+1⁢s+1+8⁢v
ext ≔ v=2x,s=x!
Abramov, S.A., and Bronstein, M. "Hypergeometric dispersion and the orbit problem." Proc. ISSAC 2000.
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