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Published in: Programming and Computer Software 2/2020

01-03-2020

Algorithm for Constructing an Analogue of the Binet Formula

Authors: V. I. Kuzovatov, A. A. Kytmanov, O. I. Kuzovatova

Published in: Programming and Computer Software | Issue 2/2020

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Abstract

In this paper, we describe an algorithm for constructing an analogue of the Binet formula, which is essential in deriving a functional relation to the classical Riemann zeta-function. The algorithm is implemented in the Maple computer algebra system. An example that illustrates the operation of the algorithm is presented.

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Literature
1.
go back to reference von zur Gathen, J. and Gerhard, J., Modern Computer Algebra, Cambridge University Press, 2013, 3rd ed.CrossRef von zur Gathen, J. and Gerhard, J., Modern Computer Algebra, Cambridge University Press, 2013, 3rd ed.CrossRef
2.
go back to reference Bykov, V.I., Kytmanov, A.M., and Lazman, M.Z., Elimination Methods in Polynomial Computer Algebra, Kluwer, 1998.CrossRef Bykov, V.I., Kytmanov, A.M., and Lazman, M.Z., Elimination Methods in Polynomial Computer Algebra, Kluwer, 1998.CrossRef
3.
go back to reference Aizenberg, L.A., On a formula of generalized multidimensional logarithmic residue and solution of systems of nonlinear equations, Dokl. Akad. Nauk SSSR, 1977, vol. 234, no. 3, pp. 505–508.MathSciNet Aizenberg, L.A., On a formula of generalized multidimensional logarithmic residue and solution of systems of nonlinear equations, Dokl. Akad. Nauk SSSR, 1977, vol. 234, no. 3, pp. 505–508.MathSciNet
4.
go back to reference Kytmanov, A.A., Analogs of recurrent Newton formulas, Russian Mathematics, 2009, vol. 53, no. 10, pp. 34–44. Kytmanov, A.A., Analogs of recurrent Newton formulas, Russian Mathematics, 2009, vol. 53, no. 10, pp. 34–44.
5.
go back to reference Kytmanov, A.A., An algorithm for calculating power sums of roots for a class of systems of nonlinear equations, Program. Comput. Software, 2010, vol. 36, no. 2, pp. 55–63.MathSciNetCrossRef Kytmanov, A.A., An algorithm for calculating power sums of roots for a class of systems of nonlinear equations, Program. Comput. Software, 2010, vol. 36, no. 2, pp. 55–63.MathSciNetCrossRef
6.
go back to reference Kytmanov, A.A., Kytmanov, A.M., and Myshkina, E.K., Finding residue integrals for systems of non-algebraic equations in \({{\mathbb{C}}^{n}}\), J. Symbolic Comput., 2015, vol. 66, pp. 98–110.MathSciNetCrossRef Kytmanov, A.A., Kytmanov, A.M., and Myshkina, E.K., Finding residue integrals for systems of non-algebraic equations in \({{\mathbb{C}}^{n}}\), J. Symbolic Comput., 2015, vol. 66, pp. 98–110.MathSciNetCrossRef
7.
go back to reference Elkadi, M. and Yger, A., Residue calculus and applications, Publ. Res. Inst. Math. Sci., 2007, vol. 43, no. 1, pp. 55–73.MathSciNetCrossRef Elkadi, M. and Yger, A., Residue calculus and applications, Publ. Res. Inst. Math. Sci., 2007, vol. 43, no. 1, pp. 55–73.MathSciNetCrossRef
8.
go back to reference Kytmanov, A.M. and Naprienko, Y.M., An approach to define the resultant of two entire functions, J. Complex Var. Elliptic Eqs., 2017, vol. 62, no. 2, pp. 269–286.MathSciNetCrossRef Kytmanov, A.M. and Naprienko, Y.M., An approach to define the resultant of two entire functions, J. Complex Var. Elliptic Eqs., 2017, vol. 62, no. 2, pp. 269–286.MathSciNetCrossRef
9.
go back to reference Kytmanov, A.M. and Myshkina, E.K., On some approach for finding the resultant of two entire functions, J. Sib. Fed. Univ. Math. Phys., 2019, vol. 12, no. 4, pp. 434–438.MathSciNetCrossRef Kytmanov, A.M. and Myshkina, E.K., On some approach for finding the resultant of two entire functions, J. Sib. Fed. Univ. Math. Phys., 2019, vol. 12, no. 4, pp. 434–438.MathSciNetCrossRef
10.
go back to reference Kuzovatov, V.I. and Kytmanov, A.A., Algorithm for constructing an analog of Plan’s formula, Program. Comput. Software, 2018, vol. 44, pp. 100–104.MathSciNetCrossRef Kuzovatov, V.I. and Kytmanov, A.A., Algorithm for constructing an analog of Plan’s formula, Program. Comput. Software, 2018, vol. 44, pp. 100–104.MathSciNetCrossRef
11.
go back to reference Whittaker, E.T. and Watson, G.N., A Course of Modern Analysis, Cambridge University Press, 1927.MATH Whittaker, E.T. and Watson, G.N., A Course of Modern Analysis, Cambridge University Press, 1927.MATH
12.
go back to reference Kuzovatov, V.I. and Kytmanov, A.M., On an analog of Plan’s formula, J. Contemp. Math. Anal., 2018, vol. 53, no. 3, pp. 139–146.MathSciNetMATH Kuzovatov, V.I. and Kytmanov, A.M., On an analog of Plan’s formula, J. Contemp. Math. Anal., 2018, vol. 53, no. 3, pp. 139–146.MathSciNetMATH
13.
go back to reference Kuzovatov, V.I., Generalization of the Plana Formula, Russian Mathematics, 2018, vol. 62, no. 5, pp. 34–43. Kuzovatov, V.I., Generalization of the Plana Formula, Russian Mathematics, 2018, vol. 62, no. 5, pp. 34–43.
14.
go back to reference Saaryan, A.A., Towards an Abel-Plana Summing Formula, Soviet Journal of Contemporary Physics, 1986, vol. 21, no. 5, pp. 32–36. Saaryan, A.A., Towards an Abel-Plana Summing Formula, Soviet Journal of Contemporary Physics, 1986, vol. 21, no. 5, pp. 32–36.
15.
go back to reference Kuzovatov, V.I. and Kytmanov, A.M., On an analogue of the Binet integral representation, Siberian Electronic Mathematical Reports, 2020, vol. 17, pp. 1–14. Kuzovatov, V.I. and Kytmanov, A.M., On an analogue of the Binet integral representation, Siberian Electronic Mathematical Reports, 2020, vol. 17, pp. 1–14.
16.
go back to reference Kytmanov, A.M. and Myslivets, S.G., On the zeta-function of systems of nonlinear equations, Sib. Math. J., 2007, vol. 48, no. 5, pp. 863–870.MathSciNetCrossRef Kytmanov, A.M. and Myslivets, S.G., On the zeta-function of systems of nonlinear equations, Sib. Math. J., 2007, vol. 48, no. 5, pp. 863–870.MathSciNetCrossRef
17.
go back to reference Titchmarsh, E.C., The Theory of the Riemann Zeta-Function, Oxford University Press, 1951.MATH Titchmarsh, E.C., The Theory of the Riemann Zeta-Function, Oxford University Press, 1951.MATH
18.
go back to reference Gel'fand, I.M. and Levitan, B.M., On a simple identity for the eigenvalues of a second-order differential operator, Dokl. Akad. Nauk SSSR, 1953, vol. 88, no. 4, pp. 593–596. Gel'fand, I.M. and Levitan, B.M., On a simple identity for the eigenvalues of a second-order differential operator, Dokl. Akad. Nauk SSSR, 1953, vol. 88, no. 4, pp. 593–596.
19.
go back to reference Dikii, L.A., Zeta function of an ordinary differential equation on a finite interval, Izv. Akad. Nauk SSSR, Ser. Mat., 1955, vol. 19, no. 4. Dikii, L.A., Zeta function of an ordinary differential equation on a finite interval, Izv. Akad. Nauk SSSR, Ser. Mat., 1955, vol. 19, no. 4.
20.
go back to reference Dikii, L.A., Trace formulas for Sturm–Liouville differential operators, Usp. Mat. Nauk, 1958, vol. 13, no. 3, pp. 111–143.MathSciNet Dikii, L.A., Trace formulas for Sturm–Liouville differential operators, Usp. Mat. Nauk, 1958, vol. 13, no. 3, pp. 111–143.MathSciNet
21.
go back to reference Lidskii, V.B. and Sadovnichii, V.A., Regularized sums of zeros of a class of entire functions, Funct. Anal. Appl., 1967, vol. 1, no. 2, pp. 133–139.CrossRef Lidskii, V.B. and Sadovnichii, V.A., Regularized sums of zeros of a class of entire functions, Funct. Anal. Appl., 1967, vol. 1, no. 2, pp. 133–139.CrossRef
22.
go back to reference Smagin, S.A. and Shubin, M.A., On the zeta-function of a transversally elliptic operator, Russ. Math. Surv., 1984, vol. 39, no. 2, pp. 201–202.MathSciNetCrossRef Smagin, S.A. and Shubin, M.A., On the zeta-function of a transversally elliptic operator, Russ. Math. Surv., 1984, vol. 39, no. 2, pp. 201–202.MathSciNetCrossRef
23.
go back to reference Kuzovatov, V.I. and Kytmanov, A.A., On the zeta-function of zeros of some class of entire functions, J. Sib. Fed. Univ. Math. Phys, 2014, vol. 7, no. 4, pp. 489–499.CrossRef Kuzovatov, V.I. and Kytmanov, A.A., On the zeta-function of zeros of some class of entire functions, J. Sib. Fed. Univ. Math. Phys, 2014, vol. 7, no. 4, pp. 489–499.CrossRef
25.
go back to reference Tsikh, A.K., Multidimensional residues and their applications, Amer. Math. Soc., 1992. Tsikh, A.K., Multidimensional residues and their applications, Amer. Math. Soc., 1992.
26.
go back to reference Egorushkin, O.I., Kolbasina, I.V., and Safonov, K.V., On application of multidimensional complex analysis in formal language and grammar theory, Appl. Discrete Math., 2017, vol. 37, pp. 76–89.CrossRef Egorushkin, O.I., Kolbasina, I.V., and Safonov, K.V., On application of multidimensional complex analysis in formal language and grammar theory, Appl. Discrete Math., 2017, vol. 37, pp. 76–89.CrossRef
Metadata
Title
Algorithm for Constructing an Analogue of the Binet Formula
Authors
V. I. Kuzovatov
A. A. Kytmanov
O. I. Kuzovatova
Publication date
01-03-2020
Publisher
Pleiades Publishing
Published in
Programming and Computer Software / Issue 2/2020
Print ISSN: 0361-7688
Electronic ISSN: 1608-3261
DOI
https://doi.org/10.1134/S0361768820020061

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