Abstract
In recent years, Karr’s difference field theory has been extended to the so-called \(\mathrm {R}{\Pi \Sigma }\)-extensions in which one can represent not only indefinite nested sums and products that can be expressed by transcendental ring extensions, but one can also handle algebraic products of the form \(\alpha ^n\) where \(\alpha \) is a root of unity. In this article we supplement this summation theory substantially by the following building block. We provide new algorithms that represent a finite number of hypergeometric or mixed \(({q}_{1},\dots ,{q}_{e})\)-multibasic hypergeometric products in such a difference ring. This new insight provides a complete summation machinery that enables one to formulate such products and indefinite nested sums defined over such products in \(\mathrm {R}{\Pi \Sigma }\)-extensions fully automatically. As a side-product, one obtains compactified expressions where the products are algebraically independent among each other, and one can solve the zero-recognition problem for such products.
Dedicated to Sergei A. Abramov on the occasion of his 70th birthday
Supported by the Austrian Science Fund (FWF) grant SFB F50 (F5009-N15)
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Notes
- 1.
Their elements are considered as expressions that can be evaluated for sufficiently large \(n\in \mathbb {N}\).
- 2.
This is the case if \(\mathbb {K}\) is strongly \(\sigma \)-computable, or if \(\mathbb {K}\) is a rational function field over a strongly \(\sigma \)-computable field.
- 3.
\(\zeta \) lies on the unity circle. However, not every algebraic number on the unit circle is a root of unity: Take for instance \(\frac{1-\sqrt{3}}{2}+\frac{3^{\frac{1}{4}}}{\sqrt{2}}\,\iota \) and its complex conjugate; they are on the unit circle, but they are roots of the polynomial \(x^{4}-2\,x^{3}-2\,x+1\) which is irreducible in \(\mathbb {Q}[x]\) and which is not a cyclotomic polynomial. For details on number fields containing such numbers see [16].
- 4.
It would suffice to require that the \(f_i\in K[\kappa _1,\ldots ,\kappa _u]\setminus K\) are monic and pairwise co-prime. For practical reasons we require in addition that the \(f_i\) are irreducible. For instance, suppose we have to deal with \((\kappa (\kappa +1))^n\). Then we could take \(f_1=\kappa (\kappa +1)\) and can adjoin the \(\mathrm {\Pi }\)-monomial \(\sigma (t)=f_1\,t\) to model the product. However, if in a later step also the unforeseen products \(\kappa ^n\) and \((\kappa +1)^n\) arise, one has to split t into two monomials, say \(t_1,t_2\), with \(\sigma (t_1)=\kappa \,t_1\) and \(\sigma (t_2)=(\kappa +1)\,t_2\). Requiring that the \(f_i\) are irreducible avoids such undesirable redesigns of an already constructed \(\mathrm {R}\Pi \)-extension.
- 5.
- 6.
Instead of irreducibility it would suffice to require that the \(p_i\in \mathbb {K}[x]\setminus \mathbb {K}\) satisfy property (29). However, suppose that one takes, e.g., \(p_1=x(2\,x+1)\) leading to the \(\mathrm {\Pi }\)-monomial t with \(\sigma (t)=x\,(2x+1)\). Further, assume that later one has to introduce unexpectedly also x and \(2\,x+1\). Then one has to split t to the \(\mathrm {\Pi }\)-monomials \(t_1,t_2\) with \(\sigma (t_1)=x\,t_1\) and \(\sigma (t_2)=(2x+1)\,t_2\), i.e., one has to redesign the already constructed \(\mathrm {R}\Pi \)-extension. In short, irreducible polynomials provide an \(\mathrm {R}{\Pi \Sigma }\)-extension which most probably need not be redesigned if other products have to be considered.
- 7.
We remark that this representation is related to the normal form given in [8].
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Acknowledgements
We would like to thank Michael Karr for his valuable remarks to improve the article. Supported by the Austrian Science Fund (FWF) grant SFB F50 (F5009-N15).
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Ocansey, E.D., Schneider, C. (2018). Representing (\(\varvec{q}\)–)Hypergeometric Products and Mixed Versions in Difference Rings. In: Schneider, C., Zima, E. (eds) Advances in Computer Algebra. WWCA 2016. Springer Proceedings in Mathematics & Statistics, vol 226. Springer, Cham. https://doi.org/10.1007/978-3-319-73232-9_8
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