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 Mat. Sb., 2007, Volume 198, Number 9, Pages 59–80 (Mi msb3775)

Bases in the solution space of the Mellin system

b Siberian Federal University

Abstract: We consider algebraic functions $z$ satisfying equations of the following form:
\begin{equation*} a_0 z^m+a_1z^{m_1}+a_2 z^{m_2}+…+a_nz^{m_n}+a_{n+1}=0. \tag{1} \end{equation*}
Here $m>m_1>…>m_n>0$, $m,m_i\in\mathbb N$, and $z=z(a_0,…,a_{n+1})$ is a function of the complex variables $a_0,…,a_{n+1}$. Solutions of such algebraic equations are known to satisfy holonomic systems of linear differential equations with polynomial coefficients. In this paper we investigate one such system, which was introduced by Mellin. The holonomic rank of this system of equations and the dimension of the linear space of its algebraic solutions are computed. An explicit base in the solution space of the Mellin system is constructed in terms of roots of (1) and their logarithms. The monodromy of the Mellin system is shown to be always reducible and several results on the factorization of the Mellin operator in the one-variable case are presented.
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DOI: https://doi.org/10.4213/sm3775

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English version:
Sbornik: Mathematics, 2007, 198:9, 1277–1298

Bibliographic databases:

UDC: 517.554+517.588+517.953
MSC: Primary 35G05; Secondary 33C05, 35C10

Citation: A. Dickenstein, T. M. Sadykov, “Bases in the solution space of the Mellin system”, Mat. Sb., 198:9 (2007), 59–80; Sb. Math., 198:9 (2007), 1277–1298

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb3775
• https://doi.org/10.4213/sm3775
• http://mi.mathnet.ru/eng/msb/v198/i9/p59

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This publication is cited in the following articles:
1. Dickenstein A., “Hypergeometric functions and binomials”, Revista de la Unión Matemática Argentina, 49:2 (2008), 97–110
2. Dickenstein A., Matusevich L.F., Miller E., “Binomial $D$-modules”, Duke Math. J., 151:3 (2010), 385–429
3. E. N. Mikhalkin, “The monodromy of a general algebraic function”, Siberian Math. J., 56:2 (2015), 330–338
4. Bogdanov D.V., Kytmanov A.A., Sadykov T.M., “Algorithmic Computation of Polynomial Amoebas”, Computer Algebra in Scientific Computing, Lecture Notes in Computer Science, 9890, eds. Gerdt V., Koepf W., Seiler W., Vorozhtsov E., Springer Int Publishing Ag, 2016, 87–100
5. Kytmanov A.A., Lyapin A.P., Sadykov T.M., “Evaluating the rational generating function for the solution of the Cauchy problem for a two-dimensional difference equation with constant coefficients”, Program. Comput. Softw., 43:2 (2017), 105–111
6. T. M. Sadykov, “On the Analytic Complexity of Hypergeometric Functions”, Proc. Steklov Inst. Math., 298 (2017), 248–255
7. V. R. Kulikov, “A criterion for the convergence of the Mellin–Barnes integral for solutions to simultaneous algebraic equations”, Siberian Math. J., 58:3 (2017), 493–499
8. Sadykov T.M., “Computational Problems of Multivariate Hypergeometric Theory”, Program. Comput. Softw., 44:2 (2018), 131–137
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