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NEW CONNECTION FORMULAE BETWEEN CHEBYSHEV AND LUCAS POLYNOMIALS: NEW EXPRESSIONS INVOLVING LUCAS NUMBERS VIA HYPERGEOMETRIC FUNCTIONS

W.M. ABD- ELHAMEED|Y.H. YOUSSRI
장전수학회 (Jangjeon Mathematical Society)|2018.07|Advanced Studies in Contemporary Mathematics|제28권 제3호|pp. 357-368|DOI http://dx.doi.org/10.17777/ascm2017.28.3.357

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참고문헌

키워드

recurrence relationsChebyshev polynomialshypergeometric functionsLucas polynomials and numbersconnection problem
주제분류수학

초록

This article deals with the solutions of the connection problems be- tween Lucas polynomials and the first and second kinds of Chebyshev polynomials. We show that the connection coefficients involve hypergeometric functions of the type 2F1 of certain arguments. Thanks to the new derived connection formulae, new formulae of the famous Lucas numbers and their derivatives sequences are presented. As another application of the derived formulae, some new definite integrals involving Lucas and Chebyshev polynomials are evaluated.

Abstract

This article deals with the solutions of the connection problems be- tween Lucas polynomials and the first and second kinds of Chebyshev polynomials. We show that the connection coefficients involve hypergeometric functions of the type 2F1 of certain arguments. Thanks to the new derived connection formulae, new formulae of the famous Lucas numbers and their derivatives sequences are presented. As another application of the derived formulae, some new definite integrals involving Lucas and Chebyshev polynomials are evaluated.

참고문헌 (28)

  1. W.M. Abd-Elhameed, "New product and linearization formulae of Jacobi polynomials of certain parameters", Integral Transforms Spec. Funct., 26(8), pp.586~599 (2015)[DOI]
  2. W.M. Abd-Elhameed, "Linearization formulae for certain Jacobi polynomials", Ramanujan J., 39(1), pp.155~168 (2016)[DOI]
  3. W.M. Abd-Elhameed, "New Tchebyshev-Galerkin operational matrix method for solving linear and nonlinear hyperbolic telegraph type equations", Numer. Methods Partial Differential Equations, 32(6), pp.1553~1571 (2016)[DOI]
  4. W.M. Abd-Elhameed, "New hypergeometric connection formulae between Fibonacci and Chebyshev polynomials", Ramanujan J., pp.1~15 (2015)
  5. I. Area, "Solving connection and linearization problems within the askey scheme and its q-analogue via inversion formulas", J. Cornput. Appl. Math., 133(1), pp.151~162 (2001)[DOI]
  6. S.B. Bozkurt, "On the complex factorization of the Lucas sequence", Appl. Math. Lett., 24(8), pp.1317~1321 (2011)[DOI]
  7. K. Dilcher, "Hypergeometric functions and Fibonacci numbers", Fibonacci Quart., 38(4), pp.342~362 (2000)
  8. E.H. Doha, "On the connection coefficients and recurrence relations arising from expansions in series of Laguerre polynomials", J. Phys. A: Math. Gen., 36(20), pp.5449~5462 (2003)[DOI]
  9. E.H. Doha, "On the construction of recurrence relations for the expansion and connection coefficients in series of Jacobi polynomials", J. Phys. A: Math. Gen., 37(3), pp.657~675 (2004)[DOI]
  10. E.H. Doha, "New linearization formulae for the products of Chebyshev polynomials of third and fourth kind", Rocky Mt. J. Math., 46(2), pp.443~460 (2016)[DOI]
  11. E. H. Doha, "On the coefficients of integrated expansions and integrals of Chebyshev polynomials of third and fourth kinds", Bull. Malays. Math. Sci. Soc., 37(2), pp.383~398 (2014)
  12. E.H. Doha, "Second kind Chebyshev operational matrix algorithm for solving differential equations of Lane-Emden type", New Astron., 23, pp.113~117 (2013)
  13. E.H. Doha, "Recurrences and explicit formulae for the expansion and connection coefficients in series of Bessel polynomials", J. Phys. A: Math. Gen., 37(33), pp.8045~8063 (2004)[DOI]
  14. M. El-Mikkawy, "A new family of k-Fibonacci numbers", Appl. Math. Comput., 215(12), pp.4456~4461 (2010)
  15. S. Falcon, "On the Fibonacci k-numbers", Chaos Soliton Fract., 32(5), pp.1615~1624 (2007)[DOI]
  16. H.H. Gulec, "A new approach to generalized Fibonacci and Lucas numbers with binomial coefficients", Appl. Math. Comput., 220, pp.482~486 (2013)
  17. R. Keskin, "Fibonacci and Lucas congruences and their applications", Acta Math. Sin., 27(4), pp.725~736 (2011)[DOI]
  18. T. Koshy, "Fibonacci and Lucas numbers with applications, volume 51" (2011), John Wiley & Sons
  19. K. Maleknejad, "Numerical solution of nonlinear Volterra integral equations of the second kind by using Chebyshev polynomials", Appl. Math Comput., 188(1), pp.123~128 (2007)
  20. P. Maroni, "Connection coefficients between orthogonal polynomials and the canonical sequence: an approach based on symbolic computation", Numer. Algorithms, 47(3), pp.291~314 (2008)[DOI]
  21. J.C. Mason, "Chebyshev Polynomials" (2010), Chapman and Hall;CRC
  22. J. Sanchez-Ruiz, "Some connection and linearization problems for polynomials in and beyond the askey scheme", J. Comput. Appl. Math., 133(1), pp.579~591 (2001)[DOI]
  23. R. Szwarc, "Linearization and connection coefficients of orthogonal polynomials", Mh. Math., 113(4), pp.319~329 (1992)[DOI]
  24. N. Taskara, "On the properties or Lucas numbers with binomial coefficients", Appl. Math. Lett., 23(1), pp.68~72 (2010)[DOI]
  25. W. Tingting, "Some identities involving Fibonacci, Lucas polynomials and their applications", Bull. Math. Soc. Sci. Math. Roum., 55(1), pp.95~103 (2012)
  26. A. WIoch, "Some identities for the generalized Fibonacci numbers and the generalized Lucas numbers", Appl. Math. Comput., 219(10), pp.5561~5568 (2013)
  27. Yasin Y. Yazlik, "A note on generalized-Horadam sequence", Comput. Math. Appl., 63(1), pp.36~41 (2012)[DOI]
  28. W. Zhang, "On Chebyshev polynomials and Fibonacci numbers", Fibonacci Quart., 40(5), pp.424~428 (2002)

관련 논문AI추천

New connection formulae between ultraspherical polynomials and Chebyshev polynomials of third and fourth kindsE. H. Doha장전수학회 · Advanced Studies in Contemporary Mathematics2015 · 제25권 제4호REPRESENTING SUMS OF FINITE PRODUCTS OF CHEBYSHEV POLYNOMIALS OF THE SECOND KIND AND FIBONACCI POLYNOMIALS IN TERMS OF CHEBYSHEV POLYNOMIALS김대산, 김태균, Dmitry V. Dolgy장전수학회 · Advanced Studies in Contemporary Mathematics2018 · 제28권 제3호IDENTITIES FOR SPECIAL NUMBERS AND POLYNOMIALS INVOLVING FIBONACCI-TYPE POLYNOMIALS AND CHEBYSHEV POLYNOMIALSNeslihan Kilar, Yilmaz Simsek장전수학회 · Advanced Studies in Contemporary Mathematics2020 · 제30권 제4호On the computations of contiguous relations for _{2}F_{1} hypergeometric seriesAdel K. Ibrahim, Arjun K. Rathie, Medhat A. Rakha대한수학회 · 대한수학회논문집2009 · 제24권 제2호ADDITIONAL CLOSE LINKS BETWEEN BALANCING AND LUCAS-BALANCING POLYNOMIALSROBERT FRONTCZAK, TARAS GOY장전수학회 · Advanced Studies in Contemporary Mathematics2021 · 제31권 제3호A study of new class of integrals associated with generalized Struve function and PolynomialsAbdul Hakim Khan, Kottakkaran Sooppy Nisar, Sirazul Haq대한수학회 · 대한수학회논문집2019 · 제34권 제1호

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