Higher Order Recurrence Research Articles

Explicit solution of system of two higher-order recurrences

Explicit solution of system of two higher-order recurrences

Simulating time delays and space–time memory interactions: An analytical approach

This study introduces a novel analytical framework to explore the effects of Caputo spatial and temporal memory indices combined with a proportional time delay on (non)linear (1+2)-dimensional evolutionary models. The solution is expressed as a Cauchy product of an absolutely convergent series that effectively captures the dynamics of these parameters. By extending the differential transform method into higher-dimensional fractional space, we reformulate the evolution equation as a (non)linear higher-order recurrence relation, which enables the precise determination of fractional series coefficients. Our findings show that Caputo derivatives and time delay significantly impact the system’s behavior, with graphical analysis revealing a continuous transition from a stationary to an integer state solution. The study also identifies a quantitative analogy between the Caputo-time fractional derivative and proportional time delay that highlights the role of Caputo derivatives as memory indices. This method has proven highly effective in deriving solutions for fractional higher-dimensional extensions of evolutionary equations.

On multiplicative dependent vectors with coordinates from certain higher order recurrences

In this paper, we prove that there are finitely many multiplicative dependent vectors with coordinates from non-degenerate linear recurrence sequences [Formula: see text] of a fixed order [Formula: see text] These sequences satisfy the recurrence relation [Formula: see text] with initials [Formula: see text]. Here, the coefficients of the recurrence relation are positive integers satisfying [Formula: see text] or [Formula: see text] In both these conditions, [Formula: see text]

Principal Solutions of Recurrence Relations and Irrationality Questions in Number Theory

We apply the theory of disconjugate linear recurrence relations to the study of irrational quantities in number theory. In particular, for an irrational number associated with solutions of linear three-term recurrence relations, we show that there exists a four-term linear recurrence relation whose solutions show that the number has an irrational square if and only if the four-term recurrence relation has a principal solution of a certain type. The result is extended to higher-order recurrence relations, and a transcendence criterion can also be formulated in terms of these principal solutions. The method generates new series expansions of positive integer powers of ζ(3) and ζ(2) in terms of Apéry’s now classic sequences.

Higher-order recurrence relations, Sobolev-type inner products and matrix factorizations

It is well known that Sobolev-type orthogonal polynomials with respect to measures supported on the real line satisfy higher-order recurrence relations and these can be expressed as a (2N + 1)-banded symmetric semi-infinite matrix. In this paper, we state the connection between these (2N + 1)-banded matrices and the Jacobi matrices associated with the three-term recurrence relation satisfied by the standard sequence of orthonormal polynomials with respect to the 2-iterated Christoffel transformation of the measure.

On a Study of a Family of Higher-Order Recurrence Relations

In this work, group theory is applied to obtain generators of the Lie algebra for a class of rational difference equations of order k + l where k and l are positive integers. This is followed by deriving invariants and solutions via some linear recurrences and with the introduction of some number-theoretic arithmetical functions. Furthermore, sufficient conditions for the existence of solutions for periods two and four are provided in special cases. The results in this paper generalize certain known results.

Bispectral Jacobi type polynomials

We study the bispectrality of Jacobi type polynomials, which are eigenfunctions of higher-order differential operators and can be defined by taking suitable linear combinations of a fixed number of consecutive Jacobi polynomials. Jacobi type polynomials include, as particular cases, the Krall-Jacobi polynomials. As the main results we prove that the Jacobi type polynomials always satisfy higher-order recurrence relations (i.e., they are bispectral). We also prove that the Krall-Jacobi families are the only Jacobi type polynomials which are orthogonal with respect to a measure on the real line.

Evidence for improved survival with bevacizumab treatment in recurrent high-grade gliomas: a retrospective study with (\u201cpseudo-randomized\u201d) treatment allocation by the health insurance provider

IntroductionDespite a large number of trials, the role of bevacizumab (BEV) in the treatment of recurrent high-grade gliomas is still controversial. Evidence regarding an effect on overall survival in this context is ultimately inconclusive. At the Department of Radiation Oncology at Erlangen, Germany we treated a large cohort of patients with recurrent gliomas where bevacizumab use was determined exclusively by the health care provider’s approval of reimbursement.Methods61 patients (between 06/2008 and 01/2014) with recurrent high-grade gliomas had reimbursement requests for BEV sent to their health insurance. 37 patients out of 61 (60.7%) had their requests approved and therefore received bevacizumab (BEV-arm) as part of their treatment. The remaining 24 (39.3%) patients received standard therapy without bevacizumab (non-BEV-arm). Survival endpoints were defined with reference to the first BEV request to the health insurance provider.ResultsMedian overall survival (OS) for the whole cohort was 7.0 months. OS was significantly better for BEV vs. Non-BEV patients (median, 10.3 vs. 4.2 months, logrank p = 0.023). There was an increased BEV benefit in cases of higher-order recurrences (first order recurrence BEV vs. Non-BEV, 12.5 vs. 10.2 months, p = 0.578) (second or higher order of recurrence, 9.9 vs. 2.6 months, p = 0.010). On multivariate analysis for overall survival the prognostic impact of bevacizumab (HR = 0.43, p = 0.034) remained significant.ConclusionOur results suggest an influence of BEV on overall survival in a heavily pretreated patient population suffering from high-grade gliomas with BEV benefit being greatest in case of second or later recurrence.

Bispectrality of Charlier type polynomials

ABSTRACTGiven a finite set of positive integers G and polynomials , , with degree of equal to g, we associate to them a sequence of Charlier type polynomials defined from the Charlier polynomials by using certain Casoratian determinants (whose entries are the polynomials ). Charlier type polynomials are eigenfunctions of higher order difference operators. When , the polynomials are also orthogonal, and then satisfy a three term recurrente relation. In this paper, we prove that whatever the polynomials 's are, the Charlier type polynomials always satisfy higher order recurrence relations. We also introduce and characterize the algebra of difference operators associated to the recurrence relations satisfied by the sequence . Our characterization is constructive and surprisingly simple. As a consequence, we prove that is, essentially, the unique choice such that the polynomials are orthogonal with respect to a measure.

Behavior and breakdown of higher-order Fermi-Pasta-Ulam-Tsingou recurrences.

We numerically investigate the existence and stability of higher-order recurrences (HoRs), including super-recurrences, super-super-recurrences, etc., in the α and β Fermi-Pasta-Ulam-Tsingou (FPUT) lattices for initial conditions in the fundamental normal mode. Our results represent a considerable extension of the pioneering work of Tuck and Menzel on super-recurrences. For fixed lattice sizes, we observe and study apparent singularities in the periods of these HoRs, speculated to be caused by nonlinear resonances. Interestingly, these singularities depend very sensitively on the initial energy and the respective nonlinear parameters. Furthermore, we compare the mechanisms by which the super-recurrences in the two models breakdown as the initial energy and respective nonlinear parameters are increased. The breakdown of super-recurrences in the β-FPUT lattice is associated with the destruction of the so-called metastable state and thus with relaxation towards equilibrium. For the α-FPUT lattice, we find this is not the case and show that the super-recurrences break down while the lattice is still metastable and far from equilibrium. We close with comments on the generality of our results for different lattice sizes.

Meromorphic solutions of recurrence relations and DRA method for multicomponent master integrals

We formulate a method to find the meromorphic solutions of higher-order recurrence relations in the form of the sum over poles with coefficients defined recursively. Several explicit examples of the application of this technique are given. The main advantage of the described approach is that the analytical properties of the solutions are very clear (the position of poles is explicit, the behavior at infinity can be easily determined). These are exactly the properties that are required for the application of the multiloop calculation method based on dimensional recurrence relations and analyticity (the DRA method).

Higher-order Convolution Identities for Cauchy Numbers

Euler's famous formula written in symbolic notation as $(B_0+B_0)^n=-n B_{n-1}-(n-1)B_n$ was extended to $(B_{l_1}+\cdots+B_{l_m})^n$ for $m\ge 2$ and arbitrary fixed integers $l_1,\dots,l_m\ge 0$. In this paper, we consider the higher-order recurrences for Cauchy numbers $(c_{l_1}+\cdots+c_{l_m})^n$, where the $n$-th Cauchy number $c_n$ ($n\ge 0$) is defined by the generating function $x/\ln(1+x)=\sum_{n=0}^\infty c_n x^n/n!$. In special, we give an explicit expression in the case $l_1=\cdots=l_m=0$ for any integers $n\ge 1$ and $m\ge 2$. We also discuss the case for Cauchy numbers of the second kind $\widehat c_n$ in similar ways.

More on the infinite sum of reciprocal Fibonacci, Pell and higher order recurrences

Recently [Z. Wenpeng, W. Tingting, Applied Mathematics and Computation 218 (10) (2012) 6164–6167; T. Komatsu, V. Laohakosol, Journal of Integer Sequences 13 (5) (2010) Article 10.5.8.] computed partial infinite sums including reciprocal usual Fibonacci, Pell and generalized order-k Fibonacci numbers. In this paper we will present generalizations of earlier results by considering more generalized higher order recursive sequences with additional one coefficient parameter.

Higher-order recurrences for Bernoulli numbers

Euler's well-known nonlinear relation for Bernoulli numbers, which can be written in symbolic notation as ( B 0 + B 0 ) n = − n B n − 1 − ( n − 1 ) B n , is extended to ( B k 1 + ⋯ + B k m ) n for m ⩾ 2 and arbitrary fixed integers k 1 , … , k m ⩾ 0 . In the general case we prove an existence theorem for Euler-type formulas, and for m = 3 we obtain explicit expressions. This extends the authors' previous work for m = 2 .

Higher-Order Three-Term Recurrences and Asymptotics of Multiple Orthogonal Polynomials

The asymptotic theory is developed for polynomial sequences that are generated by the three-term higher-order recurrence $$Q_{n+1}=zQ_{n}-a_{n-p+1}Q_{n-p},\quad p\in\mathbb{N},n\geq p,$$ where z is a complex variable and the coefficients ak are positive and satisfy the perturbation condition ∑n=1∞|an−a|<∞. Our results generalize known results for p=1, that is, for orthogonal polynomial sequences on the real line that belong to the Blumenthal–Nevai class. As is known, for p≥2, the role of the interval is replaced by a starlike set S of p+1 rays emanating from the origin on which the Qn satisfy a multiple orthogonality condition involving p measures. Here we obtain strong asymptotics for the Qn in the complex plane outside the common support of these measures as well as on the (finite) open rays of their support. In so doing, we obtain an extension of Weyl’s famous theorem dealing with compact perturbations of bounded self-adjoint operators. Furthermore, we derive generalizations of the classical Szegő functions, and we show that there is an underlying Nikishin system hierarchy for the orthogonality measures that is related to the Weyl functions. Our results also have application to Hermite–Pade approximants as well as to vector continued fractions.

Powers of sequences and recurrence

We study recurrence, and multiple recurrence, properties along the $k$-th powers of a given set of integers. We show that the property of recurrence for some given values of $k$ does not give any constraint on the recurrence for the other powers. This is motivated by similar results in number theory concerning additive basis of natural numbers. Moreover, motivated by a result of Kamae and Mend\`es-France, that links single recurrence with uniform distribution properties of sequences, we look for an analogous result dealing with higher order recurrence and make a related conjecture.

A generalization of Poincaré's theorem for recurrence systems

A new proof and a genuine generalization to systems of first order equations is given from Poincaré classical theorem on ratio asymptotics of solutions of higher order recurrence equations. The asymptotic behavior of a fundamental system of solutions is obtained.

ELLIPTIC CURVES AND QUADRATIC RECURRENCE SEQUENCES

The explicit solution of a general three-term bilinear recurrence relation of fourth order is constructed here in terms of the Weierstrass sigma function. The construction of the elliptic curve associated to the Somos 4 sequence is presented as an example. An interpretation via the theory of integrable systems is provided, leading to a conjecture relating certain higher-order recurrences with hyperelliptic curves of higher genus.

Automated higher-order complexity analysis

This paper describes the automated complexity analysis (ACA) system for automated higher-order complexity analysis of functional programs synthesized with the N UPRL proof development system. We introduce a general framework for defining models of computational complexity for functional programs based on an annotation of a given operational language semantics. Within this framework, we use type decomposition and polynomialization to express the complexity of higher-order terms. Symbolic interpretation of open terms automates complexity analysis, which involves generating and solving higher-order recurrence equations. Finally, the use of the ACA system is demonstrated by analyzing three different implementations of the pigeonhole principle.

Some Spectral Properties of Infinite Band Matrices

For operators generated by a certain class of infinite band matrices we establish a characterization of the resolvent set in terms of polynomial solutions of the underlying higher order recurrence relations. This enables us to describe some asymptotic behaviour of the corresponding systems of vector orthogonal polynomials. Finally, we provide some new convergence results for matrix Pade approximants.