Matematik Bölümünde 18.01.2016 tarihli seminer



Fen Fakültesi toplantı salonunda 18 Ocak 2016 Pazartesi günü saat 11.00’de Bingöl Üniversitesi’nden Dr. Zafer Şiar tarafından “Some New Identities Concerning Horadam and its Companion Sequence by Matrix Method“ başlıklı İngilizce Matematik semineri verilmiştir.


Some New ·Identities concerning Horadam and its Companion Sequence by Matrix Method

Zafer ¸Siar

Bingöl University, Mathematics Department, Bingöl/TURKEY

January 11, 2016


In this study we will give a general relation between Horadam sequence

and the nxn matrices satisfying the relation X2 = PX + QI: Then we

will obtain some new identities between Horadam sequence fWng and its

companion sequence fXng using the 22 matrices X satisfying the rela-

tion X2 = PX +QI: Moreover, we will give some sum formulae including

the sequences fWng ; fXng ; fUng ; and fVng : Lastly, we we will give an

application of the sequences fWng and fXng to trigonometric functions,

which is some new angle addition formulas such as:

sin r sin(m + n + r) = sin(m + r) sin(n + r) 􀀀 sinm sin n;

cos r cos(m + n + r) = cos(m + r) cos(n + r) 􀀀 sinm sin n;


cos r sin(m + n) = cos(n + r) sinm + cos(m 􀀀 r) sin n:


[1] Gamaliel Cerda-Morales, Matrix methods in Horadam Sequences, Hacettepe

journal of Mathematics and Statistics, 42 (2) (2013), 173-179.

[2] T.-X. He and P. J. S. Shiue, On Sequences of Numbers and Polynomials

De.ned By Linear Recurrence Relations of Order 2, International Journal

of Mathematics and Mathematical Sciences, Volume 2009 (2009), 21 page.

[3] A. F. Horadam, Basic Properties of Certain Generalized Sequence of Num-

bers, Fibonacci Quart., 3.3 (1965), 161.176.

[4] A. F. Horadam, Tschebysche¤ and Other Functions Associated with the

Sequence fWn(a; b; p; qg ; The Fibonacci Quarterly, 7(1) (1969), 14-22.


[5] D. Kalman and R. Mena, The Fibonacci Numbers-Exposed, Mathematics

Magazine 76 (2003), 167-181.

[6] R. Keskin and B. Demirtürk, Some New Fibonacci and Lucas Identities

by Matrix Methods, International Journal of Mathematical Education in

Science and Technology, 2009, 1-9.

[7] E. Lucas, Th·eorie des Nombres, Paris, 1961, Chapter 18.

[8] R. S. Melham and A. G. Shannon, Some Congruence Properties of Gen-

eralized Second-Order Integer Sequences, The Fibonacci Quarterly, 32.5

(1994), 424-428.

[9] R. S. Melham, Certain Classes on Finite Sums that Involve Generalized

Fibonacci and Lucas Numbers, The Fibonacci Quarterly, 42.1 (2004), 47-


[10] S. Rabinowitz, Algorithmic Manipulation of Second Order Linear Recur-

rences, Fibonacci Quart., 37(2) (1999), 162.177.

[11] A. G. Shannon and A. F. Horadam, Special Recurrence Relations Associated

with the fW(a; b; p; q)g ; The Fibonacci Quarterly, 17.4 (1979), 294-299.

[12] Z. ¸Siar and R. Keskin, Some new identities concerning generalized Fibonacci

and Lucas numbers, Hacettepe journal of Mathematics and Statistics, 42(3) (2013), 211-222.