Honam Mathematical Journal (호남수학학술지)

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GENERALIZED PELL SEQUENCES RELATED TO THE EXTENDED GENERALIZED HECKE GROUPS ${\bar{H}}$ 3,q AND AN APPLICATION TO THE GROUP ${\bar{H}}$ 3,3

  • Birol, Furkan (Institute of Sciences, Department of Mathematics, Balikesir University) ;
  • Koruoglu, Ozden (Necatibey Faculty of Education, Department of Mathematics, Balikesir University) ;
  • Sahin, Recep (Faculty of Arts and Sciences, Department of Mathematics, Balikesir University) ;
  • Demir, Bilal (Necatibey Faculty of Education, Department of Mathematics, Balikesir University)
  • Received : 2018.09.06
  • Accepted : 2018.10.29
  • Published : 2019.03.25
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Abstract

We consider the extended generalized Hecke groups ${\bar{H}}_{3,q}$ generated by $X(z)=-(z-1)^{-1}$, $Y(z)=-(z+{\lambda}_q)^{-1}$ with ${\lambda}_q=2\;cos({\frac{\pi}{q}})$ where $q{\geq}3$ an integer. In this work, we study the generalized Pell sequences in ${\bar{H}}_{3,q}$. Also, we show that the entries of the matrix representation of each element in the extended generalized Hecke Group ${\bar{H}}_{3,3}$ can be written by using Pell, Pell-Lucas and modified-Pell numbers.

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References

  1. A. Dasdemir, On the Pell, Pell-Lucas and Modified Pell Numbers By Matrix Method, Applied Mathematical Sciences, Vol. 5, no. 64, (2011), 3173-3181.
  2. A. F. Horadam, Applications of Modified Pell Numbers to Representations, Ulam Quaterly, Volume 3, Number 1, (1994).
  3. A. F. Horadam, Basic Properties of a Certain Generalized Sequence of Numbers, The Fibonacci Quarterly, Vol. 3 (3),(1965), 161-176.
  4. B. Demir, O. Koruoglu, R. Sahin, Conjugacy Classes of Extended Generalized Hecke Groups. Rev. Un. Mat. Argentina 57 , no. 1, (2016), 49-56.
  5. C. E. Serkland, The Pell sequence and some generalizations, Master'sThesis, San Jose State Univ., Aug. 1972.
  6. C. L. May, The real genus of groups of odd order, Rocky Mountain J. Math. 37 (2007), 1251-1269. https://doi.org/10.1216/rmjm/1187453109
  7. D. Singerman, PSL(2, q) as an image of the extended modular group with applications to group actions on surfaces, Proc. Edinburgh Math. Soc. (2), 30 , Groups St. Andrews 1985., (1987), 143-151.
  8. E. G. Karpuz, A. S. Cevik, Grobner-Shirshov bases for extended modular, extended Hecke, and Picard groups, Russian version appears in Mat. Zametki 92, no. 5, 699-706 (2012). Math. Notes 92 , no. 5-6, (2012), 636-642.
  9. E. G. Karpuz, A. S. Cevik, Some decision problems for extended modular groups, Southeast Asian Bull. Math. 35 , no. 5, (2011), 793-804,.
  10. G. A. Jones, J. S. Thornton, Automorphisms and Congruence Subgroups of the Extended Modular Group, Journal of the London Mathematical Society, Vol. s2-34, Issue 1, (1986), 26-40. https://doi.org/10.1112/jlms/s2-34.1.26
  11. J. Ercolano, Matrix generators of Pell sequences, Fibonacci Quart. ,No. 1, (1979), 71-77.
  12. J. Lehner, Uniqueness of a class of Fuchsian groups, III. J. Math. Surveys, 8, A.M.S. Providence, R.L. (1964).
  13. K. Calta and T. A. Schmidt, Infinitely many lattice surfaces with special pseudo-Anosov maps, J. Mod. Dyn. 7, No. 2, (2013), 239-254. https://doi.org/10.3934/jmd.2013.7.239
  14. K. Calta and T. A. Schmidt, Continued fractions for a class of triangle groups, J. Aust. Math. Soc. 93, No. 1-2, (2012), 21-42 . https://doi.org/10.1017/S1446788712000651
  15. M. Bicknell, A primer on the Pell sequence and related sequences, Fibonacci Quart. 13, no. 4, (1975), 345-349, .
  16. N.Y. Ozgur, Generalizations of Fibonacci and Lucas sequences, Note di Matematica 21, n. 1, 2002, (2002), 113-125.
  17. O. Koruoglu and R. Sahin, Generalized Fibonacci sequences related to the extended Hecke groups and an application to the extended modular group. Turkish J. Math. 34 , no. 3, (2010), 325-332.
  18. P. Catarino, On Some Identities and Generating Functions for k- Pell Numbers, Int. Journal of Math. Analysis, Vol. 7, no. 38, (2013,) 1877-1884. https://doi.org/10.12988/ijma.2013.35131
  19. Q. Mushtaq, A. Razaq, Homomorphic images of circuits in PSL(2,Z)-space, Bull. Malays. Math. Sci. Soc. 40 , no. 3, (2017), 1115-1133. https://doi.org/10.1007/s40840-016-0357-8
  20. Q. Mushtaq, U. Hayat, Pell numbers, Pell-Lucas numbers and modular group, Algebra Colloquium 14(1), (2007), 97-102. https://doi.org/10.1142/S1005386707000107
  21. Q. Mushtaq, U. Hayat, Horadam generalized Fibonacci numbers and the modular group Indian Journal of Pure and Applied Mathematics 38(5), (2007).
  22. R. S. Kulkarni, An arithmetic-geometric method in the study of the subgroups of the modular group, Amer. J. Math., 113 ,(1991), pp.1053-1133. https://doi.org/10.2307/2374900
  23. R. Sahin, S. Ikikardes O. Koruoglu, On the power subgroups of the extended modular group, Turkish J. Math., 28, (2004), 143-151.
  24. S.H. Jafari-Petroudia, B. Pirouzb, On some properties of (k,h)-Pell sequence and (k,h)-Pell-Lucas ssequence, Int. J. Adv. Appl. Math. and Mech. 3(1), (2015), 98-101.
  25. S. Ikikardes, Z. S. Demircioglu, R. Sahin, Generalized Pell sequences in some principal congruence subgroups of the Hecke groups, Math. Rep. (Bucur.) 18(68), no. 1, (2016), 129-136.
  26. S. Ikikardes, R. Sahin, Some results on O*-groups, Rev. Un. Mat. Argentina 53, no. 2, (2012), 25-30.
  27. W.P. Hooper, Grid graphs and lattice surfaces, Int. Math. Res. Not., no. 12, IMRN 2013, 2657-2698. https://doi.org/10.1093/imrn/rns124