On bihypernomials related to balancing and Chebyshev polynomials. (English) Zbl 1535.11026
Based on the authors’ abstract: the paper under review involves an introductory study of balancing and Lucas-balancing bihypernomials as a generalization of bihyperbolic balancing and Lucas-balancing numbers. Furthermore, the authors investigate the properties of some types of Chebyshev bihypernomials and relations between them. They use only elementary techniques together with the well-known identities to prove their results.
Reviewer: Mahadi Ddamulira (Kampala)
MSC:
| 11B39 | Fibonacci and Lucas numbers and polynomials and generalizations |
| 11B83 | Special sequences and polynomials |
| 42C05 | Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis |
References:
| [1] | A. Berczes, K. Liptai, I. Pink, On generalized balancing sequences, Fibonacci Quart, 48(2), 2010, 121-128. · Zbl 1211.11015 |
| [2] | A. Behera, G.K. Panda, On the square roots of triangular numbers, Fibonacci Quart, 37(2), 1999, 98-105. · Zbl 0962.11014 |
| [3] | M. Bilgin, S. Ersoy, Algebraic properties of bihyperbolic numbers, Adv. Appl. Clifford Algebr., 30(13), 2020, https://doi.org/10.1007/s00006-019-1036-2. · Zbl 1442.30049 · doi:10.1007/s00006-019-1036-2 |
| [4] | P. Borwein, T. Erdélyi, Polynomials and Polynomial Inequalities, Springer New York, NY, 1995. · Zbl 0840.26002 |
| [5] | D. Bród, A. Szynal-Liana, I. W loch, On the combinatorial properties of bi-hyperbolic balancing numbers, Tatra Mt. Math. Publ., 77, 2020, 27-38. · Zbl 1478.11015 |
| [6] | D. Bród, A. Szynal-Liana, I. W loch, Bihyperbolic numbers of the Fibonacci type and their idempotent representation, Comment. Math. Univ. Carolin., 62(4), 2021, 409-416. · Zbl 1538.11032 |
| [7] | D. Bród, A. Szynal-Liana, I. W loch, On some combinatorial properties of bihyperbolic numbers of the Fibonacci type, Math. Methods Appl. Sci., 44, 2021, 4607-4615. · Zbl 1471.11054 |
| [8] | G.B. Djordjević, G.V. Milovanović, Special classes of polynomials. University of Niš, Faculty of Technology Leskovac, 2014. |
| [9] | R. Frontczak, On balancing polynomials, Appl. Math. Sci., 13(2), 2019, 57-66. |
| [10] | W. Gautschi, On mean convergence of extended Lagrange interpolation, J. Comp. Appl. Math., 43, 1992, 19-35. · Zbl 0761.41003 |
| [11] | Y. Li, On Chebyshev Polynomials, Fibonacci Polynomials, and Their Deriva-tives, J. Appl Math., 2014, 2014, 1-8. · Zbl 1406.11021 |
| [12] | Y. Li, Some properties of Fibonacci and Chebyshev polynomials, Adv. Differ Equ., 118, 2015, 1-12. · Zbl 1422.11027 |
| [13] | K. Liptai, F. Luca, A. Pintér, L. Szalay, Generalized balancing numbers, Ind. Math. (N.S.), 20, 2009, 87-100. · Zbl 1239.11035 |
| [14] | J.C. Mason, D.C. Handscomb, Chebyshev Polynomials, Chapman & HALL/CRC, Florida, 2003. D. Bród, A. Szynal-Liana · Zbl 1015.33001 |
| [15] | G.K. Panda, Some fascinating properties of balancing numbers, Proc. Eleventh Internat. Conference on Fibonacci Numbers and Their Applica-tions, Cong. Numerantium, 194, 2009, 185-189. · Zbl 1262.11019 |
| [16] | G.K. Panda, R.K. Davala, Perfect Balancing Numbers, Fibonacci Quart, 53(3), 2015, 261-264. · Zbl 1397.11045 |
| [17] | G.K. Panda, P.K. Ray, Cobalancing numbers and cobalancers, Int. J. Math. Math. Sci., 8, 2005, 1189-1200. · Zbl 1085.11017 |
| [18] | B.K. Patel, N. Irmak, P.K. Ray, Incomplete Balancing and Lucas-Balancing Numbers, Math. Reports, 20, 2018, 59-72. · Zbl 1399.11045 |
| [19] | G.M. Phillips, Interpolation and Approximation by Polynomials, Springer, New York, NY, 2003. · Zbl 1023.41002 |
| [20] | P.K. Ray, On the properties of k-balancing numbers, Ain Shams Engineering Journal, 9(3), 2018, 395-402. |
| [21] | A. Szynal-Liana, I. W loch, A study on Fibonacci and Lucas bihypernomials, Discuss. Math. Gen. Algebra Appl., 42(2), 2022, 409-423. · Zbl 1513.11063 |
| [22] | A. Szynal-Liana, I. W loch, M. Liana, On certain bihypernomials related to Pell and Pell-Lucas numbers, Commun. Fac. Sci. Univ. Ank. Sér. A1 Math. Stat., 71(2), 2022, 422-433. |
| [23] | Dorota Bród Faculty of Mathematics and Applied Physics, Rzeszow University of Technology, al. Powstańców Warszawy 12, 35-959 Rzeszów, Poland E-mail: dorotab@prz.edu.pl |
| [24] | Anetta Szynal-Liana Faculty of Mathematics and Applied Physics, Rzeszow University of Technology, al. Powstańców Warszawy 12, 35-959 Rzeszów, Poland E-mail: aszynal@prz.edu.pl |
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