(p; q)-FIBONACCI AND (p; q)-LUCAS SUMS BY THE DERIVATIVES OF SOME POLYNOMIALS

Year 2019, Volume: 68 Issue: 2, 1733 - 1741, 01.08.2019

B. Demirtürk Bitim , N. Topal

https://doi.org/10.31801/cfsuasmas.551883

Abstract

The main purpose of this paper is to survey several sum formulae of (p,q)-Fibonacci number U_{n} and (p,q)-Lucas number V_{n} by using the first and the second derivatives of the equations

xⁿ=(x²-px+q)(∑U_{j}x^{n-1-j})+U_{n}x-qU_{n-1}

and

2xⁿ⁺¹-pxⁿ=(x²-px+q)(∑V_{j}x^{n-1-j})+V_{n}x-qV_{n-1}.

Keywords

Fibonacci polynomial, Lucas polynomial, sum of products

References

• Amini, A.R., Fibonacci numbers from a long division formula, Mathematical Spectrum, 40 (2008), 59-61.
• Belbachir, H. and Bencherif, F., Sums of products of generalized Fibonacci and Lucas numbers, arXiv: 0708.2347v1, 17.08.2007.
• Benjamin, A. T. and Quinn, J. J., Proofs that really count, the art of combinatorial proofs, Mathematical Association of America, Providence, RI, 2003.
• Čerin, Z., Sums of products of generalized Fibonacci and Lucas numbers, Demonstratio Math., 42(2), (2009), 247-258.
• Čerin, Z., Some alternating sums of Lucas numbers, Cent. Eur. J. Math., 3(1), (2015), 1-13.
• Čerin, Z., Bitim, B. D. and Keskin, R., Sum formulae of generalized Fibonacci and Lucas numbers, Honam Math. J., 40(1), (2018), 199-210.
• Chong, C. Y., Cheah, C. L. and Ho, C. K., Some identities of generalized Fibonacci sequence, AIP Conf. Proc., 1605 (2014), 661-665 .
• Horadam, A. F., Basic properties of certain generalized sequences of numbers, Fibonacci Quart., 3(5), (1965), 161-176.
• Koshy, T., Fibonacci and Lucas numbers with applications, Wiley and Sons, Canada, 2001.
• Long, C. T., Discovering Fibonacci identities, Fibonacci Quart., 24(2), (1986), 160-167.
• Lucas, E., Théorie des fonctions numériques simplement périodiques, Amer. J. Math., 1 (1878), 184-240.
• Melham, R. S., On sums of powers of terms in a linear recurrence, Port. Math., 56(4), (1999), 501-508.
• Shannon, A. G. and Horadam, A. F., Special recurrence relations associated with the sequence {w_{n}(a,b;p,q)}, Fibonacci Quart., 17(4), (1979), 294-299.
• Vajda, S., Fibonacci and Lucas numbers and the golden section, Ellis Horwood Ltd. Publ., England, 1989.