Home On the power sum problem of Lucas polynomials and its divisible property
Article Open Access

On the power sum problem of Lucas polynomials and its divisible property

  • Wang Xiao EMAIL logo
Published/Copyright: June 22, 2018
Download Article (PDF)
Open Mathematics
From the journal Open Mathematics Volume 16 Issue 1

Abstract

The main purpose of this paper is to use the mathematical induction and the properties of Lucas polynomials to study the power sum problem of Lucas polynomials. In the end, we obtain an interesting divisible property.

Keywords: Lucas polynomials; Power sum problem; Mathematical induction; Divisible property
MSC 2010: 11B39

1 Introduction

For any integer n ≥ 0, the famous Fibonacci polynomials {Fn(x)} and Lucas polynomials {Ln(x)} are defined by F0(x) = 0, F1(x) = 1, L0(x) = 2, L1(x) = x and Fn+2(x) = xFn+1(x) + Fn(x), Ln+2(x) = xLn+1(x) + Ln(x). The general terms of Fn(x) and Ln(x) are given by

Fn+1(x)=∑k=0[n2](n−kk)xn−2k

and

Ln(x)=∑k=0[n2]nn−k(n−kk)xn−2k,(1)

where (mn)=m!n!(m−n)!, and [x] denotes the greatest integer ≤ x.

It is easy to prove the identities

Fn(x)=1x2+4[(x+x2+42)n−(x−x2+42)n]

and

Ln(x)=(x+x2+42)n+(x−x2+42)n.(2)

If we take x = 1, then {Fn(x)} becomes Fibonacci sequences {Fn(1)}, and {Ln(x)} becomes Lucas sequences {Ln(1)}.

Since these sequences and polynomials have very important positions in the theory and application of mathematics, many scholars have studied their various properties, and obtained a series of important results, some of which can be found in references [1, 2, 3, 4, 5], and some other related papers can also be found in [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. For example, in a private communication with Curtis Cooper, R. S. Melham suggested that it would be interesting to discover an explicit expansion for

L1L2⋯L2m+1∑k=1nF2k2m+1

as a polynomial in F2n+1. Wiemann and Cooper [1] reported some conjectures of Melham related to the sum ∑k=1nF2k2m+1. Kiyota Ozeki [2] proved that

∑k=1nF2k2m+1=15m∑j=0m(−1)jL2m+1−2j(j2m+1)(F(2m+1−2j)(2n+1)−F2m+1−2j).

Helmut Prodinger [3] studied the more general summation ∑k=0nF2k+δ2m+1+ϵ, where δ, ϵ ∈ {0, 1}, and obtained many interesting identities.

For the sum of powers of Lucas numbers, Helmut Prodinger [3] also obtained some similar conclusions.

R. S. Melham [4] also proposed the following two conjectures:

Conjecture A

Letmbe a positive integer. Then the sum

L1L3L5⋯L2m+1∑k=1nF2k2m+1

can be expressed as (F2n+1 − 1)2P2m−1(F2n+1), whereP2m−1(x) is a polynomial of degree 2m − 1 with integer coefficients.

Conjecture B

Letm ≥ 0 be an integer. Then the sum

L1L3L5⋯L2m+1∑k=1nL2k2m+1

can be expressed as (L2n+1 − 1) Q2m(L2n+1), whereQ2m(x) is a polynomial of degree 2mwith integer coefficients.

Wang Tingting and Zhang Wenpeng [5] studied these problems, and proved several conclusions for ∑k=1hFkn(x)and∑k=1hLkn(x), where h and n are any positive integers.

As some applications of Theorem 2 in reference [5], they deduced the following:

Corollary A

Leth ≥ 1 andn ≥ 0 be two integers. Then the sum

L1(x)L3(x)L5(x)⋯L2n+1(x)∑m=1hL2m2n+1(x)

can be expressed as (L2h+1(x) − x) Q2n(x, L2h+1(x)), whereQ2n(x, y) is a polynomial in two variablesxandywith integer coefficients and degree 2nofy.

Corollary B

Leth ≥ 1 andn ≥ 0 be two integers. Then the sum

L1(x)L3(x)L5(x)⋯L2n+1(x)∑m=1hF2m2n+1(x)

can be expressed as (F2h+1(x) − 1) H2n(x, F2h+1(x)), whereH2n(x, y) is a polynomial in two variablesxandywith integer coefficients and degree 2n of y.

Therefore, Wang Tingting and Zhang Wenpeng [5] solved Conjecture B completely. They also obtained some substantial progress for Conjecture A.

It is easy to see that in the above Conjecture A and Conjecture B, the subscripts of Fn and Ln in sums ∑k=1nF2k2m+1and∑k=1nL2k2m+1 are even numbers. Inspired by the above researches, we naturally ask that for any positive integers n and h with h ≥ 2, whether there exists a polynomial U(x) with degree > 1 such that the congruence

L1(x)L3(x)⋯L2n+1(x)∑m=0h−1L2m+12n+1(x)≡0modU(x).(3)

If so, what would U(x) look like? How is it different from the form in Conjecture A and Conjecture B ? Obviously, it is hard to guess the exact form of U(x) from (d) of Theorem 1 in reference [5]. Maybe that is why there is no such formal conjecture in [4].

In this paper, we shall use the mathematical induction and the properties of Lucas polynomials to study this problem, and give an exact polynomial U(x) in (3). The result is detailed in the following theorems.

Theorem A

For any positive integersnandhwithh ≥ 2, we have

L1(x)L3(x)⋯L2n+1(x)∑m=0h−1L2m+12n+1(x)≡0modx⋅(L2h(x)−2).

Theorem B

For any positive integersnandhwithh ≥ 2, we have

L1(x)L3(x)⋯L2n+1(x)∑m=0h−1F2m+12n+1(x)≡0modx⋅F2h(x).

Taking x = 1, from Theorem 1 and Theorem 2 we may immediately deduce the following two corollaries:

Corollary A

For any positive integersnandhwithh ≥ 2, we have

L1L3⋯L2n+1∑m=0h−1L2m+12n+1≡0mod(L2h−2).

Corollary B

For any positive integersnandhwithh ≥ 2, we have

L1L3⋯L2n+1∑m=0h−1F2m+12n+1≡0modF2h.

2 Two Lemmas

Lemma A

For any non-negative integernandk, we have the identity

Ln(L2k+1(x))=Ln(2k+1)(x).

Proof

Let α=x+x2+42,β=x−x2+42, and replace x by L2k+1(x) in (2). Since we have α2k+1β2k+1 = −1,

L2k+1(x)+L2k+12(x)+4=α2k+1+β2k+1+(α2k+1+β2k+1)2+4=α2k+1+β2k+1+α2(2k+1)+β2(2k+1)−2+4=α2k+1+β2k+1+(α2k+1−β2k+1)2=2α2k+1

and

L2k+1(x)−L2k+12(x)+4=α2k+1+β2k+1−(α2k+1+β2k+1)2+4=α2k+1+β2k+1−(α2k+1−β2k+1)2=2β2k+1,

from (2) we have the identity

Ln(L2k+1(x))=(L2k+1+L2k+12+42)n+(L2k+1−L2k+12+42)n=αn(2k+1)+βn(2k+1)=Ln(2k+1)(x).

This completes the proof of Lemma 1. □

Lemma B

For any positive integersnandh, we have the congruence

L1(x)(L2h(2n+1)(x)−2)−(2n+1)L2n+1(x)(L2h(x)−2)≡0modx(L2h(x)−2).

Proof

From (1) we know that x ∣ L2n+1(x). Because that L1(x) = x, so in order to prove Lemma 2, we only need to prove the congruence

L2h(2n+1)(x)−2≡0mod(L2h(x)−2).(4)

Next we prove (4) by complete induction. It is clear that (4) is true for n = 0. If n = 1, then apply the identity

L2h3(x)=(α2h+β2h)3=L6h(x)+3L2h(x),

we have

L6h(x)−2=L2h3(x)−3L2h(x)−2=(L2h(x)−2)(L2h(x)+1)2≡0mod(L2h(x)−2).

That is to say, the congruence (4) is true for n = 1.

Suppose that the congruence (4) is true for all integers 0 ≤ n ≤ s. That is,

L2h(2n+1)(x)−2≡0mod(L2h(x)−2)(5)

holds for all integers 0 ≤ n ≤ s.

Then for positive integer n = s + 1, we have the identities

L4h(x)=L2h2(x)−2≡2mod(L2h(x)−2)

and

L4h(x)L2h(2s+1)(x)=L2h(2s+3)(x)+L2h(2s−1)(x),

from (5) we can deduce the congruence equations as follows

L2h(2s+3)(x)−2=L4h(x)L2h(2s+1)(x)−L2h(2s−1)(x)−2≡2L2h(2s+1)(x)−L2h(2s−1)(x)−2≡2(L2h(2s+1)(x)−2)−(L2h(2s−1)(x)−2)≡0mod(L2h(x)−2).

That is to say, the congruence (4) is true for n = s + 1.

Now Lemma 2 follows from (4) and completes the induction. □

3 Proofs of the theorems

In this section, we shall use mathematical induction to complete the proofs of our theorems. Here we only prove Theorem 1. Similarly, we can also deduce Theorem 2 and thus we omit its proving process here. After replacing x by L2m+1(x) in (1), we obtain the following expression with Lemma 1

L(2n+1)(2m+1)(x)=L2n+1(L2m+1(x))=∑k=0n2n+12n+1−k(2n+1−kk)L2m+12n+1−2k(x)

or

L(2n+1)(2m+1)(x)−(2n+1)L2m+1(x)=∑k=0n−12n+12n+1−k(2n+1−kk)L2m+12n+1−2k(x).(6)

For any positive integer h ≥ 2, we first introduce that the identities

∑m=0h−1L(2n+1)(2m+1)(x)=∑m=0h−1(α(2n+1)(2m+1)+β(2n+1)(2m+1))=α2n+1⋅α2h(2n+1)−1α2(2n+1)−1+β2n+1⋅β2h(2n+1)−1β2(2n+1)−1=α2h(2n+1)−1α2n+1+β2n+1+β2h(2n+1)−1β2n+1+α2n+1=L2h(2n+1)(x)−2L2n+1(x),(7)
∑m=0h−1L2m+1(x)=∑m=0h−1(α2m+1+β2m+1)=L2h(x)−2L1(x).(8)

Then, combining (6), (7) and (8) we have

∑m=0h−1[L(2n+1)(2m+1)(x)−(2n+1)L2m+1(x)]=L2h(2n+1)(x)−2L2n+1(x)−(2n+1)⋅L2h(x)−2L1(x)=∑k=0n−12n+12n+1−k(2n+1−kk)∑m=0h−1L2m+12n+1−2k(x).(9)

Now we apply (9) and mathematical induction to prove the congruence

L1(x)L3(x)⋯L2n+1(x)∑m=0h−1L2m+12n+1(x)≡0modx(L2h(x)−2).(10)

If n = 1, then from (9) we have

L1(x)L3(x)[L6h(x)−2L3(x)−3⋅L2h(x)−2L1(x)]=L1(x)L3(x)∑m=0h−1L2m+13(x).(11)

From Lemma 2 we know that

L1(x)L3(x)[L6h(x)−2L3(x)−3⋅L2h(x)−2L1(x)]≡0modx(L2h(x)−2).(12)

Combining (11) and (12) we know that the congruence (10) is true for n = 1.

Suppose that (10) is true for all 1 ≤ n ≤ s. That is,

L1(x)L3(x)⋯L2n+1(x)∑m=0h−1L2m+12n+1(x)≡0modx(L2h(x)−2)(13)

holds for all 1 ≤ n ≤ s.

Then for n = s + 1, from (9) we have

L2h(2s+3)(x)−2L2s+3(x)−(2s+3)⋅L2h(x)−2L1(x)=∑k=0s2s+32s+3−k(2s+3−kk)∑m=0h−1L2m+12s+3−2k(x)=∑m=0h−1L2m+12s+3(x)+∑k=1s2s+32s+3−k(2s+3−kk)∑m=0h−1L2m+12s+3−2k(x).(14)

Applying Lemma 2 we have the congruence

L1(x)L3(x)⋯L2s+3(x)[L2h(2s+3)(x)−2L2s+3(x)−(2s+3)⋅L2h(x)−2L1(x)]≡0modx(L2h(x)−2).(15)

If 1 ≤ k ≤ s, then 3 ≤ 2s + 3 − 2k ≤ 2s + 1. From the inductive assumption (13) we have

L1(x)L3(x)⋯L2s+1(x)L2s+3(x)∑k=1s2s+32s+3−k(2s+3−kk)∑m=0h−1L2m+12s+3−2k(x)≡0modx(L2h(x)−2).(16)

Combining (14), (15) and (16) we may immediately deduce the congruence

L1(x)L3(x)⋯L2s+1(x)L2s+3(x)∑m=0h−1L2m+12s+3(x)≡0modx(L2h(x)−2).

This completes the proof of our theorem by mathematical induction.

  1. Competing interests: The author declare that they have no competing interests.

Acknowledgement

This work is supported by the N. S. F. (Grant No. 11771351) of P. R. China. The author would like to thank the referees for their very helpful and detailed comments, which have significantly improved the presentation of this paper.

References

[1] Wiemann, M, Cooper C: Divisibility of an F-L Type Convolution, Applications of Fibonacci Numbers, Vol. 9, Kluwer Acad. Publ., Dordrecht, 2004, 267-28710.1007/978-0-306-48517-6_26Search in Google Scholar

[2] Kiyota Ozeki, On Melham’s sum, The Fibonacci Quarterly, 2008/2009, 46/47, 107–110Search in Google Scholar

[3] Helmut Prodinger, On a sum of Melham and its variants, The Fibonacci Quarterly, 2008/2009, 46/47, 207–215Search in Google Scholar

[4] Melham, R. S: Some conjectures concerning sums of odd powers of Fibonacci and Lucas numbers, The Fibonacci Quarterly, 2008/2009, 46/47, 312-315Search in Google Scholar

[5] Wang, T, Zhang, W: Some identities involving Fibonacci. Lucas polynomials and their applications Bull. Math. Soc. Sci. Math. Roumanie, 2012, 55, 95–103Search in Google Scholar

[6] Li, X: Some identities involving Chebyshev polynomials. Mathematical Problems in Engineering, 2015, 2015, Article ID 95069510.1155/2015/950695Search in Google Scholar

[7] Ma, Y, Lv, X: Several identities involving the reciprocal sums of Chebyshev polynomials. Mathematical Problems in Engineering, 2017, 2017, Article ID 419457910.1155/2017/4194579Search in Google Scholar

[8] Yi, Y, Zhang, W: Some identities involving the Fibonacci polynomials, The Fibonacci Quarterly, 2002, 40, 314–318Search in Google Scholar

[9] Ma, R, Zhang, W: Several identities involving the Fibonacci numbers and Lucas numbers, The Fibonacci Quarterly, 2007, 45, 164–170Search in Google Scholar

[10] Kim, D. S, Kim, T, Lee, S: Some identities for Bernoulli polynomials involving Chebyshev polynomials. J. Comput. Anal. Appl., 2014, 16, 172–180Search in Google Scholar

[11] Kim, D. S, Dolgy, D. V, Kim, T, Rim, S. H: Identities involving Bernoulli and Euler polynomials arising from Chebyshev polynomials. Proc. Jangjeon Math. Soc., 2012, 15, 361–37010.1155/2012/784307Search in Google Scholar

[12] Kim, T, Kim, D. S, Seo, J. J, Dolgy, D. V: Some identities of Chebyshev polynomials arising from non-linear differential equations. J. Comput. Anal. Appl., 2017, 23, 820–832Search in Google Scholar

[13] He, Y: Some new results on products of Apostol-Bernoulli and Apostol-Euler polynomials, J. Math. Anal. Appl., 2015, 431, 34–4610.1016/j.jmaa.2015.05.055Search in Google Scholar

[14] He, Y, Wang, C. P: New symmetric identities involving the Eulerian polynomials, J. Comput. Anal. Appl., 2014, 17, 498–504Search in Google Scholar

[15] He, Y, Kim, D. S: General convolution identities for Apostol-Bernoulli, Euler and Genocchi polynomials, J. Nonlinear Sci. Appl., 2016, 9, 4780–479710.22436/jnsa.009.06.115Search in Google Scholar

[16] Chen, L, Zhang, W: Chebyshev polynomials and their some interesting applications, Advances in Difference Equations, 2017, 2017: 30310.1186/s13662-017-1365-1Search in Google Scholar

Received: 2017-11-15
Accepted: 2018-05-15
Published Online: 2018-06-22

© 2018 Xiao, published by De Gruyter

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Articles in the same Issue

  1. Regular Articles
  2. Algebraic proofs for shallow water bi–Hamiltonian systems for three cocycle of the semi-direct product of Kac–Moody and Virasoro Lie algebras
  3. On a viscous two-fluid channel flow including evaporation
  4. Generation of pseudo-random numbers with the use of inverse chaotic transformation
  5. Singular Cauchy problem for the general Euler-Poisson-Darboux equation
  6. Ternary and n-ary f-distributive structures
  7. On the fine Simpson moduli spaces of 1-dimensional sheaves supported on plane quartics
  8. Evaluation of integrals with hypergeometric and logarithmic functions
  9. Bounded solutions of self-adjoint second order linear difference equations with periodic coeffients
  10. Oscillation of first order linear differential equations with several non-monotone delays
  11. Existence and regularity of mild solutions in some interpolation spaces for functional partial differential equations with nonlocal initial conditions
  12. The log-concavity of the q-derangement numbers of type B
  13. Generalized state maps and states on pseudo equality algebras
  14. Monotone subsequence via ultrapower
  15. Note on group irregularity strength of disconnected graphs
  16. On the security of the Courtois-Finiasz-Sendrier signature
  17. A further study on ordered regular equivalence relations in ordered semihypergroups
  18. On the structure vector field of a real hypersurface in complex quadric
  19. Rank relations between a {0, 1}-matrix and its complement
  20. Lie n superderivations and generalized Lie n superderivations of superalgebras
  21. Time parallelization scheme with an adaptive time step size for solving stiff initial value problems
  22. Stability problems and numerical integration on the Lie group SO(3) × R3 × R3
  23. On some fixed point results for (s, p, α)-contractive mappings in b-metric-like spaces and applications to integral equations
  24. On algebraic characterization of SSC of the Jahangir’s graph 𝓙n,m
  25. A greedy algorithm for interval greedoids
  26. On nonlinear evolution equation of second order in Banach spaces
  27. A primal-dual approach of weak vector equilibrium problems
  28. On new strong versions of Browder type theorems
  29. A Geršgorin-type eigenvalue localization set with n parameters for stochastic matrices
  30. Restriction conditions on PL(7, 2) codes (3 ≤ |𝓖i| ≤ 7)
  31. Singular integrals with variable kernel and fractional differentiation in homogeneous Morrey-Herz-type Hardy spaces with variable exponents
  32. Introduction to disoriented knot theory
  33. Restricted triangulation on circulant graphs
  34. Boundedness control sets for linear systems on Lie groups
  35. Chen’s inequalities for submanifolds in (κ, μ)-contact space form with a semi-symmetric metric connection
  36. Disjointed sum of products by a novel technique of orthogonalizing ORing
  37. A parametric linearizing approach for quadratically inequality constrained quadratic programs
  38. Generalizations of Steffensen’s inequality via the extension of Montgomery identity
  39. Vector fields satisfying the barycenter property
  40. On the freeness of hypersurface arrangements consisting of hyperplanes and spheres
  41. Biderivations of the higher rank Witt algebra without anti-symmetric condition
  42. Some remarks on spectra of nuclear operators
  43. Recursive interpolating sequences
  44. Involutory biquandles and singular knots and links
  45. Constacyclic codes over 𝔽pm[u1, u2,⋯,uk]/〈 ui2 = ui, uiuj = ujui〉
  46. Topological entropy for positively weak measure expansive shadowable maps
  47. Oscillation and non-oscillation of half-linear differential equations with coeffcients determined by functions having mean values
  48. On 𝓠-regular semigroups
  49. One kind power mean of the hybrid Gauss sums
  50. A reduced space branch and bound algorithm for a class of sum of ratios problems
  51. Some recurrence formulas for the Hermite polynomials and their squares
  52. A relaxed block splitting preconditioner for complex symmetric indefinite linear systems
  53. On f - prime radical in ordered semigroups
  54. Positive solutions of semipositone singular fractional differential systems with a parameter and integral boundary conditions
  55. Disjoint hypercyclicity equals disjoint supercyclicity for families of Taylor-type operators
  56. A stochastic differential game of low carbon technology sharing in collaborative innovation system of superior enterprises and inferior enterprises under uncertain environment
  57. Dynamic behavior analysis of a prey-predator model with ratio-dependent Monod-Haldane functional response
  58. The points and diameters of quantales
  59. Directed colimits of some flatness properties and purity of epimorphisms in S-posets
  60. Super (a, d)-H-antimagic labeling of subdivided graphs
  61. On the power sum problem of Lucas polynomials and its divisible property
  62. Existence of solutions for a shear thickening fluid-particle system with non-Newtonian potential
  63. On generalized P-reducible Finsler manifolds
  64. On Banach and Kuratowski Theorem, K-Lusin sets and strong sequences
  65. On the boundedness of square function generated by the Bessel differential operator in weighted Lebesque Lp,α spaces
  66. On the different kinds of separability of the space of Borel functions
  67. Curves in the Lorentz-Minkowski plane: elasticae, catenaries and grim-reapers
  68. Functional analysis method for the M/G/1 queueing model with single working vacation
  69. Existence of asymptotically periodic solutions for semilinear evolution equations with nonlocal initial conditions
  70. The existence of solutions to certain type of nonlinear difference-differential equations
  71. Domination in 4-regular Knödel graphs
  72. Stepanov-like pseudo almost periodic functions on time scales and applications to dynamic equations with delay
  73. Algebras of right ample semigroups
  74. Random attractors for stochastic retarded reaction-diffusion equations with multiplicative white noise on unbounded domains
  75. Nontrivial periodic solutions to delay difference equations via Morse theory
  76. A note on the three-way generalization of the Jordan canonical form
  77. On some varieties of ai-semirings satisfying xp+1 ≈ x
  78. Abstract-valued Orlicz spaces of range-varying type
  79. On the recursive properties of one kind hybrid power mean involving two-term exponential sums and Gauss sums
  80. Arithmetic of generalized Dedekind sums and their modularity
  81. Multipreconditioned GMRES for simulating stochastic automata networks
  82. Regularization and error estimates for an inverse heat problem under the conformable derivative
  83. Transitivity of the εm-relation on (m-idempotent) hyperrings
  84. Learning Bayesian networks based on bi-velocity discrete particle swarm optimization with mutation operator
  85. Simultaneous prediction in the generalized linear model
  86. Two asymptotic expansions for gamma function developed by Windschitl’s formula
  87. State maps on semihoops
  88. 𝓜𝓝-convergence and lim-inf𝓜-convergence in partially ordered sets
  89. Stability and convergence of a local discontinuous Galerkin finite element method for the general Lax equation
  90. New topology in residuated lattices
  91. Optimality and duality in set-valued optimization utilizing limit sets
  92. An improved Schwarz Lemma at the boundary
  93. Initial layer problem of the Boussinesq system for Rayleigh-Bénard convection with infinite Prandtl number limit
  94. Toeplitz matrices whose elements are coefficients of Bazilevič functions
  95. Epi-mild normality
  96. Nonlinear elastic beam problems with the parameter near resonance
  97. Orlicz difference bodies
  98. The Picard group of Brauer-Severi varieties
  99. Galoisian and qualitative approaches to linear Polyanin-Zaitsev vector fields
  100. Weak group inverse
  101. Infinite growth of solutions of second order complex differential equation
  102. Semi-Hurewicz-Type properties in ditopological texture spaces
  103. Chaos and bifurcation in the controlled chaotic system
  104. Translatability and translatable semigroups
  105. Sharp bounds for partition dimension of generalized Möbius ladders
  106. Uniqueness theorems for L-functions in the extended Selberg class
  107. An effective algorithm for globally solving quadratic programs using parametric linearization technique
  108. Bounds of Strong EMT Strength for certain Subdivision of Star and Bistar
  109. On categorical aspects of S -quantales
  110. On the algebraicity of coefficients of half-integral weight mock modular forms
  111. Dunkl analogue of Szász-mirakjan operators of blending type
  112. Majorization, “useful” Csiszár divergence and “useful” Zipf-Mandelbrot law
  113. Global stability of a distributed delayed viral model with general incidence rate
  114. Analyzing a generalized pest-natural enemy model with nonlinear impulsive control
  115. Boundary value problems of a discrete generalized beam equation via variational methods
  116. Common fixed point theorem of six self-mappings in Menger spaces using (CLRST) property
  117. Periodic and subharmonic solutions for a 2nth-order p-Laplacian difference equation containing both advances and retardations
  118. Spectrum of free-form Sudoku graphs
  119. Regularity of fuzzy convergence spaces
  120. The well-posedness of solution to a compressible non-Newtonian fluid with self-gravitational potential
  121. On further refinements for Young inequalities
  122. Pretty good state transfer on 1-sum of star graphs
  123. On a conjecture about generalized Q-recurrence
  124. Univariate approximating schemes and their non-tensor product generalization
  125. Multi-term fractional differential equations with nonlocal boundary conditions
  126. Homoclinic and heteroclinic solutions to a hepatitis C evolution model
  127. Regularity of one-sided multilinear fractional maximal functions
  128. Galois connections between sets of paths and closure operators in simple graphs
  129. KGSA: A Gravitational Search Algorithm for Multimodal Optimization based on K-Means Niching Technique and a Novel Elitism Strategy
  130. θ-type Calderón-Zygmund Operators and Commutators in Variable Exponents Herz space
  131. An integral that counts the zeros of a function
  132. On rough sets induced by fuzzy relations approach in semigroups
  133. Computational uncertainty quantification for random non-autonomous second order linear differential equations via adapted gPC: a comparative case study with random Fröbenius method and Monte Carlo simulation
  134. The fourth order strongly noncanonical operators
  135. Topical Issue on Cyber-security Mathematics
  136. Review of Cryptographic Schemes applied to Remote Electronic Voting systems: remaining challenges and the upcoming post-quantum paradigm
  137. Linearity in decimation-based generators: an improved cryptanalysis on the shrinking generator
  138. On dynamic network security: A random decentering algorithm on graphs
https://doi.org/10.1515/math-2018-0063
Keywords for this article
Lucas polynomials; Power sum problem; Mathematical induction; Divisible property
Creative Commons
BY-NC-ND 4.0

Articles in the same Issue

  1. Regular Articles
  2. Algebraic proofs for shallow water bi–Hamiltonian systems for three cocycle of the semi-direct product of Kac–Moody and Virasoro Lie algebras
  3. On a viscous two-fluid channel flow including evaporation
  4. Generation of pseudo-random numbers with the use of inverse chaotic transformation
  5. Singular Cauchy problem for the general Euler-Poisson-Darboux equation
  6. Ternary and n-ary f-distributive structures
  7. On the fine Simpson moduli spaces of 1-dimensional sheaves supported on plane quartics
  8. Evaluation of integrals with hypergeometric and logarithmic functions
  9. Bounded solutions of self-adjoint second order linear difference equations with periodic coeffients
  10. Oscillation of first order linear differential equations with several non-monotone delays
  11. Existence and regularity of mild solutions in some interpolation spaces for functional partial differential equations with nonlocal initial conditions
  12. The log-concavity of the q-derangement numbers of type B
  13. Generalized state maps and states on pseudo equality algebras
  14. Monotone subsequence via ultrapower
  15. Note on group irregularity strength of disconnected graphs
  16. On the security of the Courtois-Finiasz-Sendrier signature
  17. A further study on ordered regular equivalence relations in ordered semihypergroups
  18. On the structure vector field of a real hypersurface in complex quadric
  19. Rank relations between a {0, 1}-matrix and its complement
  20. Lie n superderivations and generalized Lie n superderivations of superalgebras
  21. Time parallelization scheme with an adaptive time step size for solving stiff initial value problems
  22. Stability problems and numerical integration on the Lie group SO(3) × R3 × R3
  23. On some fixed point results for (s, p, α)-contractive mappings in b-metric-like spaces and applications to integral equations
  24. On algebraic characterization of SSC of the Jahangir’s graph 𝓙n,m
  25. A greedy algorithm for interval greedoids
  26. On nonlinear evolution equation of second order in Banach spaces
  27. A primal-dual approach of weak vector equilibrium problems
  28. On new strong versions of Browder type theorems
  29. A Geršgorin-type eigenvalue localization set with n parameters for stochastic matrices
  30. Restriction conditions on PL(7, 2) codes (3 ≤ |𝓖i| ≤ 7)
  31. Singular integrals with variable kernel and fractional differentiation in homogeneous Morrey-Herz-type Hardy spaces with variable exponents
  32. Introduction to disoriented knot theory
  33. Restricted triangulation on circulant graphs
  34. Boundedness control sets for linear systems on Lie groups
  35. Chen’s inequalities for submanifolds in (κ, μ)-contact space form with a semi-symmetric metric connection
  36. Disjointed sum of products by a novel technique of orthogonalizing ORing
  37. A parametric linearizing approach for quadratically inequality constrained quadratic programs
  38. Generalizations of Steffensen’s inequality via the extension of Montgomery identity
  39. Vector fields satisfying the barycenter property
  40. On the freeness of hypersurface arrangements consisting of hyperplanes and spheres
  41. Biderivations of the higher rank Witt algebra without anti-symmetric condition
  42. Some remarks on spectra of nuclear operators
  43. Recursive interpolating sequences
  44. Involutory biquandles and singular knots and links
  45. Constacyclic codes over 𝔽pm[u1, u2,⋯,uk]/〈 ui2 = ui, uiuj = ujui〉
  46. Topological entropy for positively weak measure expansive shadowable maps
  47. Oscillation and non-oscillation of half-linear differential equations with coeffcients determined by functions having mean values
  48. On 𝓠-regular semigroups
  49. One kind power mean of the hybrid Gauss sums
  50. A reduced space branch and bound algorithm for a class of sum of ratios problems
  51. Some recurrence formulas for the Hermite polynomials and their squares
  52. A relaxed block splitting preconditioner for complex symmetric indefinite linear systems
  53. On f - prime radical in ordered semigroups
  54. Positive solutions of semipositone singular fractional differential systems with a parameter and integral boundary conditions
  55. Disjoint hypercyclicity equals disjoint supercyclicity for families of Taylor-type operators
  56. A stochastic differential game of low carbon technology sharing in collaborative innovation system of superior enterprises and inferior enterprises under uncertain environment
  57. Dynamic behavior analysis of a prey-predator model with ratio-dependent Monod-Haldane functional response
  58. The points and diameters of quantales
  59. Directed colimits of some flatness properties and purity of epimorphisms in S-posets
  60. Super (a, d)-H-antimagic labeling of subdivided graphs
  61. On the power sum problem of Lucas polynomials and its divisible property
  62. Existence of solutions for a shear thickening fluid-particle system with non-Newtonian potential
  63. On generalized P-reducible Finsler manifolds
  64. On Banach and Kuratowski Theorem, K-Lusin sets and strong sequences
  65. On the boundedness of square function generated by the Bessel differential operator in weighted Lebesque Lp,α spaces
  66. On the different kinds of separability of the space of Borel functions
  67. Curves in the Lorentz-Minkowski plane: elasticae, catenaries and grim-reapers
  68. Functional analysis method for the M/G/1 queueing model with single working vacation
  69. Existence of asymptotically periodic solutions for semilinear evolution equations with nonlocal initial conditions
  70. The existence of solutions to certain type of nonlinear difference-differential equations
  71. Domination in 4-regular Knödel graphs
  72. Stepanov-like pseudo almost periodic functions on time scales and applications to dynamic equations with delay
  73. Algebras of right ample semigroups
  74. Random attractors for stochastic retarded reaction-diffusion equations with multiplicative white noise on unbounded domains
  75. Nontrivial periodic solutions to delay difference equations via Morse theory
  76. A note on the three-way generalization of the Jordan canonical form
  77. On some varieties of ai-semirings satisfying xp+1 ≈ x
  78. Abstract-valued Orlicz spaces of range-varying type
  79. On the recursive properties of one kind hybrid power mean involving two-term exponential sums and Gauss sums
  80. Arithmetic of generalized Dedekind sums and their modularity
  81. Multipreconditioned GMRES for simulating stochastic automata networks
  82. Regularization and error estimates for an inverse heat problem under the conformable derivative
  83. Transitivity of the εm-relation on (m-idempotent) hyperrings
  84. Learning Bayesian networks based on bi-velocity discrete particle swarm optimization with mutation operator
  85. Simultaneous prediction in the generalized linear model
  86. Two asymptotic expansions for gamma function developed by Windschitl’s formula
  87. State maps on semihoops
  88. 𝓜𝓝-convergence and lim-inf𝓜-convergence in partially ordered sets
  89. Stability and convergence of a local discontinuous Galerkin finite element method for the general Lax equation
  90. New topology in residuated lattices
  91. Optimality and duality in set-valued optimization utilizing limit sets
  92. An improved Schwarz Lemma at the boundary
  93. Initial layer problem of the Boussinesq system for Rayleigh-Bénard convection with infinite Prandtl number limit
  94. Toeplitz matrices whose elements are coefficients of Bazilevič functions
  95. Epi-mild normality
  96. Nonlinear elastic beam problems with the parameter near resonance
  97. Orlicz difference bodies
  98. The Picard group of Brauer-Severi varieties
  99. Galoisian and qualitative approaches to linear Polyanin-Zaitsev vector fields
  100. Weak group inverse
  101. Infinite growth of solutions of second order complex differential equation
  102. Semi-Hurewicz-Type properties in ditopological texture spaces
  103. Chaos and bifurcation in the controlled chaotic system
  104. Translatability and translatable semigroups
  105. Sharp bounds for partition dimension of generalized Möbius ladders
  106. Uniqueness theorems for L-functions in the extended Selberg class
  107. An effective algorithm for globally solving quadratic programs using parametric linearization technique
  108. Bounds of Strong EMT Strength for certain Subdivision of Star and Bistar
  109. On categorical aspects of S -quantales
  110. On the algebraicity of coefficients of half-integral weight mock modular forms
  111. Dunkl analogue of Szász-mirakjan operators of blending type
  112. Majorization, “useful” Csiszár divergence and “useful” Zipf-Mandelbrot law
  113. Global stability of a distributed delayed viral model with general incidence rate
  114. Analyzing a generalized pest-natural enemy model with nonlinear impulsive control
  115. Boundary value problems of a discrete generalized beam equation via variational methods
  116. Common fixed point theorem of six self-mappings in Menger spaces using (CLRST) property
  117. Periodic and subharmonic solutions for a 2nth-order p-Laplacian difference equation containing both advances and retardations
  118. Spectrum of free-form Sudoku graphs
  119. Regularity of fuzzy convergence spaces
  120. The well-posedness of solution to a compressible non-Newtonian fluid with self-gravitational potential
  121. On further refinements for Young inequalities
  122. Pretty good state transfer on 1-sum of star graphs
  123. On a conjecture about generalized Q-recurrence
  124. Univariate approximating schemes and their non-tensor product generalization
  125. Multi-term fractional differential equations with nonlocal boundary conditions
  126. Homoclinic and heteroclinic solutions to a hepatitis C evolution model
  127. Regularity of one-sided multilinear fractional maximal functions
  128. Galois connections between sets of paths and closure operators in simple graphs
  129. KGSA: A Gravitational Search Algorithm for Multimodal Optimization based on K-Means Niching Technique and a Novel Elitism Strategy
  130. θ-type Calderón-Zygmund Operators and Commutators in Variable Exponents Herz space
  131. An integral that counts the zeros of a function
  132. On rough sets induced by fuzzy relations approach in semigroups
  133. Computational uncertainty quantification for random non-autonomous second order linear differential equations via adapted gPC: a comparative case study with random Fröbenius method and Monte Carlo simulation
  134. The fourth order strongly noncanonical operators
  135. Topical Issue on Cyber-security Mathematics
  136. Review of Cryptographic Schemes applied to Remote Electronic Voting systems: remaining challenges and the upcoming post-quantum paradigm
  137. Linearity in decimation-based generators: an improved cryptanalysis on the shrinking generator
  138. On dynamic network security: A random decentering algorithm on graphs
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 6.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/math-2018-0063/html
Scroll to top button