Habilitationen https://kobra.uni-kassel.de:443/handle/123456789/2017080853216 2021-10-26T09:56:35Z 2021-10-26T09:56:35Z Algorithmic Methods for Mixed Recurrence Equations, Zeros of Classical Orthogonal Polynomials and Classical Orthogonal Polynomial Solutions of Three-Term Recurrence Equations Tcheutia, Daniel Duviol https://kobra.uni-kassel.de:443/handle/123456789/11294 2021-06-23T14:24:14Z 2019-07-01T00:00:00Z Using an algorithmic approach, we derive classes of mixed recurrence equations satisfied by classical orthogonal polynomials. Starting from certain structure relations satisfied by classical orthogonal polynomials or their connection formulae, we show that our mixed recurrence equations are structurally valid. However, they couldn't be easily obtained with classical methods and for this reason, our algorithmic approach is important. The main algorithmic tool used here is an extended version of Zeilberger's algorithm. As application of the mixed recurrence equations: we investigate interlacing properties of zeros of sequences of classical orthogonal polynomials; we prove quasi-orthogonality of certain classes of polynomials and determine the location of the extreme zeros of the quasi-orthogonal polynomials with respect to the end points of the interval of orthogonality of the polynomial sequence, where possible; we find bounds for the extreme zeros of classical orthogonal polynomials. Every orthogonal polynomial system $\{p_n(x)\}_{n\geq 0}$ satisfies a three-term recurrence relation of the type $p_{n+1}(x)=(A_nx+B_n)p_n(x)-C_np_{n-1}(x)~ (n=0,1,2,\ldots, p_{-1}\equiv 0),$ with $C_nA_nA_{n-1}>0$. Moreover, Favard's theorem states that the converse is also true. A general method to derive the coefficients $A_n$, $B_n$, $C_n$ in terms of the polynomial coefficients of the divided-difference equations satisfied by orthogonal polynomials on a quadratic or $q$-quadratic lattice is revisited. The Maple implementations rec2ortho of Koornwinder and Swarttouw (1998) or retode of Koepf and Schmersau (2002) were developed to identify classical orthogonal polynomials knowing their three-term recurrence relations. The two implementations rec2ortho and retode do not handle classical orthogonal polynomials on a quadratic or $q$-quadratic lattice. We extend the Maple implementation retode of Koepf and Schmersau (2002) to cover classical orthogonal polynomials on quadratic or $q$-quadratic lattices and to answer as application an open problem submitted by Alhaidari (2017) during the 14th International Symposium on Orthogonal Polynomials, Special Functions and Applications. 2019-07-01T00:00:00Z Tcheutia, Daniel Duviol Using an algorithmic approach, we derive classes of mixed recurrence equations satisfied by classical orthogonal polynomials. Starting from certain structure relations satisfied by classical orthogonal polynomials or their connection formulae, we show that our mixed recurrence equations are structurally valid. However, they couldn't be easily obtained with classical methods and for this reason, our algorithmic approach is important. The main algorithmic tool used here is an extended version of Zeilberger's algorithm. As application of the mixed recurrence equations: we investigate interlacing properties of zeros of sequences of classical orthogonal polynomials; we prove quasi-orthogonality of certain classes of polynomials and determine the location of the extreme zeros of the quasi-orthogonal polynomials with respect to the end points of the interval of orthogonality of the polynomial sequence, where possible; we find bounds for the extreme zeros of classical orthogonal polynomials. Every orthogonal polynomial system $\{p_n(x)\}_{n\geq 0}$ satisfies a three-term recurrence relation of the type $p_{n+1}(x)=(A_nx+B_n)p_n(x)-C_np_{n-1}(x)~ (n=0,1,2,\ldots, p_{-1}\equiv 0),$ with $C_nA_nA_{n-1}>0$. Moreover, Favard's theorem states that the converse is also true. A general method to derive the coefficients $A_n$, $B_n$, $C_n$ in terms of the polynomial coefficients of the divided-difference equations satisfied by orthogonal polynomials on a quadratic or $q$-quadratic lattice is revisited. The Maple implementations rec2ortho of Koornwinder and Swarttouw (1998) or retode of Koepf and Schmersau (2002) were developed to identify classical orthogonal polynomials knowing their three-term recurrence relations. The two implementations rec2ortho and retode do not handle classical orthogonal polynomials on a quadratic or $q$-quadratic lattice. We extend the Maple implementation retode of Koepf and Schmersau (2002) to cover classical orthogonal polynomials on quadratic or $q$-quadratic lattices and to answer as application an open problem submitted by Alhaidari (2017) during the 14th International Symposium on Orthogonal Polynomials, Special Functions and Applications. Applicability of ordinal-array-based indicators to strange nonchaotic attractors Eyebe Fouda, Jean Sire Armand https://kobra.uni-kassel.de:443/handle/123456789/2017081053247 2021-06-23T14:24:23Z 2017-06-12T00:00:00Z Time series are useful for modeling systems behavior, for predicting some events (catastrophes, epidemics, weather, . . . ) or for classification purposes (pattern recognition, pattern analysis). Among the existing data analysis algorithms, ordinal pattern based algorithms have been shown effective when dealing with simulation data. However, when applied to quasi-periodically forced systems, they fail to detect SNA and tori as regular dynamics. In this work we address this concern by defining ordinal array (OA) based indicators, namely the OA complexity (OAC) and three OA asymptotic growth indices: the periodicity, the quasi-periodicity and the non-regularity index. OA growth indices allow to clearly distinguish between periodic and quasi-periodic dynamics, which is not possible with the existing ordinal pattern-based entropy and complexity measures. They clearly output integer values for periodic dynamics and non-integer values for SNA and quasi-periodic dynamics. SNA and quasi-periodic dynamics are distinguished from weakly chaotic dynamics by the sign of the non-regularity index: it is positive for chaotic data and negative for regular dynamics. A further test based on the dependence of the OA growth indices on the time series length allows us to distinguish between tori and SNA. Moreover, by defining the upper limits of the OA growth indices for purely random data, a classification between deterministic and stochastic data is achieved. The non-regularity index may also be used as a complexity measure for non-regular dynamics by considering large time series length, but the OAC still provides a better estimate of the complexity for moderate data length. So, OA growth indices are useful for determining the nature of the data series (periodic, quasi-periodic, chaotic or stochastic), while the OAC allows us to estimate the corresponding complexity. The four indicators thus defined constitute a complete tool for nonlinear data analysis applicable to any type of time series. 2017-06-12T00:00:00Z Eyebe Fouda, Jean Sire Armand Time series are useful for modeling systems behavior, for predicting some events (catastrophes, epidemics, weather, . . . ) or for classification purposes (pattern recognition, pattern analysis). Among the existing data analysis algorithms, ordinal pattern based algorithms have been shown effective when dealing with simulation data. However, when applied to quasi-periodically forced systems, they fail to detect SNA and tori as regular dynamics. In this work we address this concern by defining ordinal array (OA) based indicators, namely the OA complexity (OAC) and three OA asymptotic growth indices: the periodicity, the quasi-periodicity and the non-regularity index. OA growth indices allow to clearly distinguish between periodic and quasi-periodic dynamics, which is not possible with the existing ordinal pattern-based entropy and complexity measures. They clearly output integer values for periodic dynamics and non-integer values for SNA and quasi-periodic dynamics. SNA and quasi-periodic dynamics are distinguished from weakly chaotic dynamics by the sign of the non-regularity index: it is positive for chaotic data and negative for regular dynamics. A further test based on the dependence of the OA growth indices on the time series length allows us to distinguish between tori and SNA. Moreover, by defining the upper limits of the OA growth indices for purely random data, a classification between deterministic and stochastic data is achieved. The non-regularity index may also be used as a complexity measure for non-regular dynamics by considering large time series length, but the OAC still provides a better estimate of the complexity for moderate data length. So, OA growth indices are useful for determining the nature of the data series (periodic, quasi-periodic, chaotic or stochastic), while the OAC allows us to estimate the corresponding complexity. The four indicators thus defined constitute a complete tool for nonlinear data analysis applicable to any type of time series.