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Article

Bounds for the Ratio of Two Gamma Functions: from Gautschi’s and Kershaw’s Inequalities to Complete Monotonicity

1Institute of Mathematics, Henan Polytechnic University, Jiaozuo City, Henan Province, China


Turkish Journal of Analysis and Number Theory. 2014, 2(5), 152-164
DOI: 10.12691/tjant-2-5-1
Copyright © 2014 Science and Education Publishing

Cite this paper:
Feng Qi. Bounds for the Ratio of Two Gamma Functions: from Gautschi’s and Kershaw’s Inequalities to Complete Monotonicity. Turkish Journal of Analysis and Number Theory. 2014; 2(5):152-164. doi: 10.12691/tjant-2-5-1.

Correspondence to: Feng  Qi, Institute of Mathematics, Henan Polytechnic University, Jiaozuo City, Henan Province, China. Email: qifeng618@gmail.com, qifeng618@hotmail.com, qifeng618@qq.com

Abstract

In the expository and survey paper, along one of main lines of bounding the ratio of two gamma functions, the author looks back and analyses some inequalities, the complete monotonicity of several functions involving ratios of two gamma or q-gamma functions, the logarithmically complete monotonicity of a function involving the ratio of two gamma functions, some new bounds for the ratio of two gamma functions and divided differences of polygamma functions, and related monotonicity results.

Keywords

References

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Article

On the Simpson’s Inequality for Convex Functions on the Co-Ordinates

1Ataturk University, K.K. Education Faculty, Department of Mathematics, Erzurum, Turkey

2Ağri İbrahim Çeçen University, Faculty of Science and Letters, Department of Mathematics, AĞRI, Turkey

3Yüzüncü Yil University, Education Faculty, Department of Mathematics, Van, Turkey


Turkish Journal of Analysis and Number Theory. 2014, 2(5), 165-169
DOI: 10.12691/tjant-2-5-2
Copyright © 2014 Science and Education Publishing

Cite this paper:
M. EMIN ÖZDEMIR, AHMET OCAK AKDEMIR, HAVVA KAVURMACI. On the Simpson’s Inequality for Convex Functions on the Co-Ordinates. Turkish Journal of Analysis and Number Theory. 2014; 2(5):165-169. doi: 10.12691/tjant-2-5-2.

Correspondence to: AHMET  OCAK AKDEMIR, Ağri İbrahim Çeçen University, Faculty of Science and Letters, Department of Mathematics, AĞRI, Turkey. Email: aocakakdemir@gmail.com

Abstract

In this paper, a new lemma is proved and inequalities of Simpson type are established for convex functions on the co-ordinates and bounded functions.

Keywords

References

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[14]  Sarkaya, M.Z., Set, E. and Özdemir, M.E., On new inequalities of Simpson.s type for s-convex functions, Computers & Mathematics with Applications, 60, 8 (2010).
 
[15]  Liu, B.Z., An inequality of Simpson type, Proc. R. Soc. A, 461 (2005), 2155-2158.
 
[16]  Dragomir, S.S., Agarwal, R.P. and Cerone, P., On Simpson’s inequality and applications, J. of Ineq. and Appl., 5 (2000), 533-579.
 
[17]  Alomari, M., Darus, M. and Dragomir, S.S., New inequalities of Simpson.s type for s-convex functions with applications, RGMIA Res. Rep. Coll., 12 (4) (2009), Article 9.
 
[18]  Ujević, N., Double integral inequalities of Simpson type and applications, J. Appl. Math. and Computing, 14 (2004), no: 1-2, p. 213-223.
 
[19]  Zhongxue, L., On sharp inequalities of Simpson type and Ostrowski type in two independent variables, Comp. and Math. with Appl., 56 (2008), 2043-2047.
 
[20]  Özdemir, M.E., Tunç, M. and Akdemir, A.O., On some new Hadamard-like inequalities for co-ordinated s-convex Functions, Facta Universitatis Series Mathematics and Informatics, Vol 28 No 3 (2013).
 
[21]  Özdemir, M.E., Akdemir, A.O. and Yldz, Ç., On co-ordinated quasi-convex functions, Czechoslovak Mathematical Journal, 62(137) (2012), 889-900.
 
[22]  Özdemir, M.E., Kavurmac, H., Akdemir, A.O. and Avc, M., Inequalities for convex and s-convex functions on Δ = [a b]×[c,d], Journal of Inequalities and Applications, 2012, Published: 1 February 2012.
 
[23]  Özdemir, M.E., Yldz, Ç. and Akdemir, A.O., On some new Hadamard-type inequalities for co-ordinated quasi-convex functions, Hacettepe Journal of Mathematics and Statistics, 41(5) (2012), 697-707.
 
[24]  İşcan, İ., A new generalization of some integral inequalities for (α,m)-convex functions, Mathematical Sciences, 7(1) (2013), 1-8.
 
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Article

Identities of Generalized Fibonacci-Like Sequence

1Department of Mathematical Sciences and Computer application, Bhundelkhand University, Jhansi (U. P.) India

2Department of Mathematics, Mandsaur Institute of Technology, Mandsaur (M. P.) India

3Schools of Studies in Mathematics, Vikram University Ujjain, (M. P.) India


Turkish Journal of Analysis and Number Theory. 2014, 2(5), 170-175
DOI: 10.12691/tjant-2-5-3
Copyright © 2014 Science and Education Publishing

Cite this paper:
Mamta Singh, Omprakash Sikhwal, Yogesh Kumar Gupta. Identities of Generalized Fibonacci-Like Sequence. Turkish Journal of Analysis and Number Theory. 2014; 2(5):170-175. doi: 10.12691/tjant-2-5-3.

Correspondence to: Yogesh  Kumar Gupta, Schools of Studies in Mathematics, Vikram University Ujjain, (M. P.) India. Email: yogeshgupta.880@rediffmail.com

Abstract

The Fibonacci and Lucas sequences are well-known examples of second order recurrence sequences. The Fibonacci sequence, Lucas numbers and their generalization have many interesting properties and applications to almost every field. Fibonacci sequence is defined by the recurrence formula Fn=Fn-1+Fn-2, n≥2 and F0=0, F1=1, where Fn is a nth number of sequence. Many authors have defined Fibonacci pattern based sequences which are popularized and known as Fibonacci-Like sequences. In this paper, Generalized Fibonacci-Like sequence is introduced and defined by the recurrence relation Mn=Mn-1+Mn-2, n≥2, with M0=2, M1=s+1, where s being a fixed integers. Some identities of Generalized Fibonacci-Like sequence are presented by Binet’s formula. Also some determinant identities are discussed.

Keywords

References

[1]  A. F. Horadam: A Generalized Fibonacci sequence, American Mathematical Monthly, Vol. 68. (5), 1961, 455-459.
 
[2]  A. F. Horadam: Basic Properties of a Certain Generalized Sequence of Numbers, The Fibonacci Quarterly, Vol. 3 (3), 1965, 161-176.
 
[3]  A. T. Benjamin and D. Walton, Counting on Chebyshev polynomials, Math. Mag. 82, 2009, 117-126.
 
[4]  B. Singh, O. Sikhwal and S. Bhatnagar: Fibonacci-Like Sequence and its Properties, Int. J. Contemp. Math. Sciences, Vol. 5 (18), 2010, 859-868.
 
[5]  B. Singh, S. Bhatnagar and O. Sikhwal: Fibonacci-Like Polynomials and some Identities, International Journal of Advanced Mathematical Sciences, 1(3),(2013),152-157.
 
Show More References
[6]  B. Singh, S. Bhatnagar and O. Sikhwal: Fibonacci-Like Sequence, International Journal of Advanced Mathematical Sciences, 1 (3) (2013), 145-151.
 
[7]  B. Singh, S. Bhatnagar and O. Sikhwal: Generalized Identties of Companion Fibonacci-Like Sequences, Global Journal of Mathematical Analysis, 1 (3) 2013, 104-109.
 
[8]  D. V. Jaiswal: On a Generalized Fibonacci sequence, Labdev J. Sci. Tech. Part A 7, 1969, 67-71.
 
[9]  M. Edson and O. Yayenie: A New Generalization of Fibonacci sequence and Extended Binet’s Formula, Integers Vol. 9, 2009, 639-654.
 
[10]  M. E. Waddill and L. Sacks: Another Generalized Fibonacci sequence, The Fibonacci Quarterly, Vol. 5 (3), 1967, 209-222.
 
[11]  M. Singh, Y. Gupta, O. Sikhwal, Generalized Fibonacci-Lucas Sequences its Properties, Global Journal of Mathematical Analysis, 2(3), 2014, 160-168.
 
[12]  O. Sikhwal, Generalization of Fibonacci Sequence: An Intriguing Sequence, Lap Lambert Academic Publishing GmbH & Co. KG, Germany (2012).
 
[13]  S. Falcon and A. Plaza: On the Fibonacci K- Numbers, Chaos, Solutions & Fractals, Vol. 32 (5), 2007, 1615-1624.
 
[14]  S. Vajda, Fibonacci & Lucas Numbers and the Golden Section, Theory and Applications, Ellis Horwood Ltd., Chichester, 1989.
 
[15]  T. Koshy, Fibonacci and Lucas Numbers with Applications, Wiley- Interscience Publication, New York (2001).
 
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Article

Some New Ostrowski Type Inequalities for Co-Ordinated Convex Functions

1Department of Mathematics, Faculty of Science and Arts, Düzce University, Düzce-TURKEY


Turkish Journal of Analysis and Number Theory. 2014, 2(5), 176-182
DOI: 10.12691/tjant-2-5-4
Copyright © 2014 Science and Education Publishing

Cite this paper:
Mehmet Zeki SARIKAYA, Hüseyin BUDAK, Hatice YALDIZ. Some New Ostrowski Type Inequalities for Co-Ordinated Convex Functions. Turkish Journal of Analysis and Number Theory. 2014; 2(5):176-182. doi: 10.12691/tjant-2-5-4.

Correspondence to: Mehmet  Zeki SARIKAYA, Department of Mathematics, Faculty of Science and Arts, Düzce University, Düzce-TURKEY. Email: sarikayamz@gmail.com

Abstract

In this paper, we obtain new identity for function of two variables and apply them to give new Ostrowski type integral inequality for double integrals involving functions whose derivatives are co-ordinates convex function on in R2 with .

Keywords

References

[1]  M. Alomari and M. Darus, Co-ordinated s-convex function in the first sense with some Hadamard-type inequalities, Int. J. Contemp. Math. Sciences, 3 (32) (2008), 1557-1567.
 
[2]  M. Alomari and M. Darus, On the Hadamard's inequality for log -convex functions on the coordinates, J. of Inequal. and Appl, Article ID 283147, (2009), 13 pages.
 
[3]  S.S. Dragomir, On Hadamard's inequality for convex functions on the co-ordinates in a rectangle from the plane, Taiwanese Journal of Mathematics, 4 (2001), 775-788.
 
[4]  M.E. Özdemir, E. Set and M.Z. Sarikaya, New some Hadamard's type inequalities for co-ordinated m-convex and (α,m) -convex functions, RGMIA, Res. Rep. Coll., 13 (2010), Supplement, Article 4.
 
[5]  N. S. Barnett and S. S. Dragomir, An Ostrowski type inequality for double integrals and applications for cubature formulae, Soochow J. Math., 27 (1), (2001), 109-114.
 
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[6]  P. Cerone and S.S. Dragomir, Ostrowski type inequalities for functions whose derivatives satisfy certain convexity assumptions, Demonstratio Math., 37 (2004), no. 2, 299-308.
 
[7]  S. S. Dragomir, Ostrowski type inequalities for functions whose derivatives are h-convex in absolute value, RGMIA Research Report Collection, 16 (2013), Article 71, 15 pp.
 
[8]  M. A. Latif and M. Alomari, Hadamard-type inequalities for product two convex functions on the co-ordinetes, Int. Math. Forum, 4 (47), 2009, 2327-2338.
 
[9]  M. A. Latif and M. Alomari, On the Hadamard-type inequalities for h-convex functions on the co-ordinetes, Int. J. of Math. Analysis, 3 (33), 2009, 1645-1656.
 
[10]  M. A. Latif, S. Hussain and S. S. Dragomir, New Ostrowski type inequalities for co-ordinated convex functions, TJMM, 4 (2012), No. 2, 125-136.
 
[11]  M. A. Latif, S. S. Dragomir, A. E. Matouk, New inequalites of Ostrowski type for co-ordinated s -convex functions via fractional integrals, Journal of Fractional Calculus and Applications,Vol. 4 (1) Jan. 2013, pp. 22-36.
 
[12]  M. A. Latif and S. S. Dragomir, New Ostrowski type inequalites for co-ordinated S-convex functions in the second sense,Le Matematiche Vol. LXVII (2012), pp. 57-72.
 
[13]  A. M. Ostrowski, Über die absolutabweichung einer differentiebaren funktion von ihrem integralmitelwert, Comment. Math. Helv. 10 (1938), 226-227.
 
[14]  B. G. Pachpatte, On an inequality of Ostrowski type in three independent variables, J. Math. Anal. Appl., 249 (2000), 583-591.
 
[15]  M. Z. Sarikaya, On the Ostrowski type integral inequality, Acta Math. Univ. Comenianae, Vol. LXXIX, 1 (2010), pp. 129-134.
 
[16]  M. Z. Sarikaya On the Ostrowski type integral inequality for double integrals, Demonstratio Mathematica, Vol. XLV No 3 2012.
 
[17]  M. Z. Sarikaya and H. Ogunmez, On the weighted Ostrowski type integral inequality for double integrals, The Arabian Journal for Science and Engineering (AJSE)-Mathematics, (2011) 36: 1153-1160.
 
[18]  M. Z. Sarikaya, E. Set, M. E. Ozdemir and S. S. Dragomir, New some Hadamard's type inequalities for co-ordinated convex functions, Tamsui Oxford Journal of Information and Mathematical Sciences, 28 (2) (2012) 137-152.
 
[19]  M. Z. Sarikaya and H. Yaldiz, On the Hadamard's type inequalities for L-Lipschitzian mapping, Konuralp Journal of Mathematics, Volume 1, No. 2, pp. 33-40 (2013).
 
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Article

Generalizations of Hermite-Hadamard-Fejer Type Inequalities for Functions Whose Derivatives are s-Convex Via Fractional Integrals

1Department of Mathematics, Faculty of Arts and Sciences, Ordu University, Ordu, Turkey

2Department of Mathematics, Faculty of Arts and Sciences, Giresun University, Giresun, Turkey


Turkish Journal of Analysis and Number Theory. 2014, 2(5), 183-188
DOI: 10.12691/tjant-2-5-5
Copyright © 2014 Science and Education Publishing

Cite this paper:
ERHAN SET, IMDAT ISCAN, ILKER MUMCU. Generalizations of Hermite-Hadamard-Fejer Type Inequalities for Functions Whose Derivatives are s-Convex Via Fractional Integrals. Turkish Journal of Analysis and Number Theory. 2014; 2(5):183-188. doi: 10.12691/tjant-2-5-5.

Correspondence to: ERHAN  SET, Department of Mathematics, Faculty of Arts and Sciences, Ordu University, Ordu, Turkey. Email: erhanset@yahoo.com

Abstract

In this work, the new results related to right hand side of Hermite-Hadamard-Fejer inequality for s-convex functions in the second sense via fractionals integrals are obtained. This results are generalization of the results obtained by Işcan in [17].

Keywords

References

[1]  G. Anastassiou, M.R. Hooshmandasl, A. Ghasemi and F. Moftakharzadeh, Montogomery identities for fractional integrals and related fractional inequalities, J. Ineq. Pure and Appl. Math., 10 (4) (2009), Art. 97.
 
[2]  S. Belarbi and Z. Dahmani, On some new fractional integral inequalities, J. Ineq. Pure and Appl. Math., 10 (3) (2009), Art. 86.
 
[3]  Z. Dahmani, New inequalities in fractional integrals, International Journal of Nonlinear Scinece, 9 (4) (2010), 493-497.
 
[4]  Z. Dahmani, On Minkowski and Hermite-Hadamard integral inequalities via fractional integration, Ann. Funct. Anal. 1 (1) (2010), 51-58.
 
[5]  Z. Dahmani, L. Tabharit, S. Taf, Some fractional integral inequalities, Nonl. Sci. Lett. A, 1 (2) (2010), 155-160.
 
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[6]  Z. Dahmani, L. Tabharit, S. Taf, New generalizations of Gruss inequality usin Riemann-Liouville fractional integrals, Bull. Math. Anal. Appl., 2 (3) (2010), 93-99.
 
[7]  R. Goren.o, F. Mainardi, Fractional calculus: integral and differential equations of fractional order, Springer Verlag, Wien (1997), 223-276.
 
[8]  I. Işcan, Generalization of different type integral inequalitiesfor s -convex functions via fractional integrals, Applicable Analysis: An Int. J., 93 (9) (2014), 1846.1862.
 
[9]  I. Işcan, New general integral inequalities for quasi-geometrically convex functions via fractional integrals, J. Inequal. Appl., 2013 (491) (2013), 15 pages.
 
[10]  I. I¸scan, On generalization of different type integral inequalities for s -convex functions via fractional integrals, Mathematical Sciences and Applications E-Notes, 2 (1) (2014), 55-67.
 
[11]  S. Miller and B. Ross, An introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, USA, 1993, p. 2.
 
[12]  I. Podlubni, Fractional Differential Equations, Academic Press, San Diego, 1999.
 
[13]  M.Z. Sarkaya, E. Set, H. Yaldz and N. Başak, Hermite-Hadamard’s inequalities for fractional integrals and related fractional inequalities, Mathematical and Computer Modelling, 57 (9) (2013), 2403-2407.
 
[14]  M.Z. Sarikaya and H. Ogunmez, On new inequalities via Riemann-Liouville fractional integration, Abstract an Applied Analysis, 2012 (2012) 10 pages, Article ID 428983.
 
[15]  E. Set, New inequalities of Ostrowski type for mapping whose derivatives are s -convex in the second sense via fractional integrals, Computers and Math. with Appl. 63 (2012), 1147-1154.
 
[16]  M. Bombardelli and S. Varosanec, Properties of h-convex functions related to the Hermite Hadamard Fejer inequalities, Comp. Math. Appl., 58 (2009), 1869-1877.
 
[17]  I. I¸scan, Hermite-Hadamrd-Fejer type inequalities for convex function via fractional integrals, 2014, arXiv: 1404. 7722v1.
 
[18]  M.Z. Sarkaya, On new Hermite Hadamard Fejer type integral inequalities, Stud. Univ. Babe¸ s-Bolyai Math. 57 (3) (2012), 377-386.
 
[19]  M.Z. Sarikaya and S. Erden, On The Hermite- Hadamard-Fejér Type Integral Inequality for Convex Function, Turkish Journal of Analysis and Number Theory, 2014, Vol. 2, No. 3, 85-89.
 
[20]  M.Z. Sarikaya and S. Erden, On the Weighted Integral Inequalities for Convex Functions, RGMIA Research Report Collection, 17 (2014), Article 70, 12 pp.
 
[21]  E. Set, I. I¸scan, M.E. Özdemir and M.Z. Sarkaya, Hermite-Hadamard-Fejer type inequalities for s-Convex functions in the second sense via fractional integrals, submited.
 
[22]  L. Fejér, Über die Fourierreihen, II, Math. Naturwiss, Anz. Ungar. Akad. Wiss., 24 (1906), 369.390. (In Hungarian).
 
[23]  H. Hudzik, L. Maligranda, Some remarks on s-convex functions, Aequationes Math. 48 (1994) 100.111.
 
[24]  S.S. Dragomir, S. Fitzpatrick, The Hadamard.s inequality for s-convex functions in the second sense, Demonstratio Math. 32 (4) (1999) 687.696.
 
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Article

Fibonacci Polynomials and Determinant Identities

1Department of Mathematics, Mandsaur Institute of Technology, Mandsaur (M. P.), India

2Department of Mathematics, Shri Harak Chand Chordia College, Bhanpura (M. P.), India


Turkish Journal of Analysis and Number Theory. 2014, 2(5), 189-192
DOI: 10.12691/tjant-2-5-6
Copyright © 2014 Science and Education Publishing

Cite this paper:
Omprakash Sikhwal, Yashwant Vyas. Fibonacci Polynomials and Determinant Identities. Turkish Journal of Analysis and Number Theory. 2014; 2(5):189-192. doi: 10.12691/tjant-2-5-6.

Correspondence to: Omprakash  Sikhwal, Department of Mathematics, Mandsaur Institute of Technology, Mandsaur (M. P.), India. Email: opbhsikhwal@rediffmail.com

Abstract

The Fibonacci polynomials and Lucas polynomials are famous for possessing wonderful and amazing properties and identities. In this paper, some determinant identities of Fibonacci polynomials are describe. Entries of determinants are satisfying the recurrence relations of Fibonacci polynomials and Lucas polynomials.

Keywords

References

[1]  A. Lupas, “A Guide of Fibonacci and Lucas Polynomials,” Octagon Math. Mag., 7 (1), 2-12, 1999.
 
[2]  A. Benjamin, N. Cameron and J. Quinn, “Fibonacci Determinants- A Combinatorial Approach,” Fibonacci Quarterly, 45 (1), 39-55, 2007.
 
[3]  B. Singh, O. Sikhwal and S. Bhatnagar, “Fibonacci-Like Sequence,” International Journal of Advanced Mathematical Sciences, 1 (3), 145-151, 2013.
 
[4]  B. Singh, O. Sikhwal and S. Bhatnagar, “Generalized Fibonacci Sequence and its Properties,” Open Journal of Mathematical Modeling, 1 (6), 194-202, 2013.
 
[5]  B. Singh, O. Sikhwal and Y. K. Panwar, “Generalized Determinantal Identities Involving Lucas Polynomials,” Applied Mathematical Sciences, 3 (8), 377-388, 2009.
 
Show More References
[6]  Beverage David, “A Polynomial Representation of Fibonacci Numbers,” Fibonacci Quarterly, 9, 541-544, 1971.
 
[7]  Krattenthaler, “Advanced determinant calculus,” Seminaire Lotharingien Combin, Article, b42q, 67, 1999.
 
[8]  C. Krattenthaler, “Advanced determinant calculus: A Complement,” Liner Algebra Appl., 411, 68-166, 2005.
 
[9]  E. Weisstein et al., “Fibonacci number from MathWorld- A Wolfram Web Resource,” http://mathworld.wolfram.com/FibonacciNumber.html
 
[10]  J.M. Patel, “Problem H-635,” Fibonacci Quarterly, 44 (1), 91, 2006.
 
[11]  M. Bicknell-Johnson and C. Spears, “Classes of Identities for the Generalized Fibonacci number Gn=Gn-1+Gn-2 from Matrices with Constant valued Determinants,” Fibonacci Quarterly, 34, 121-128, 1996.
 
[12]  N. Cahill and D. Narayan, “Fibonacci and Lucas numbers Tridigonal Matrix Determinants,” Fibonacci Quarterly, 42, 216-221, 2004.
 
[13]  O. Sikhwal, Generalization of Fibonacci Sequence: An Intriguing Sequence, Lap Lambert Academic Publishing GmbH & Co. KG, Germany, 2012.
 
[14]  S. Basir and V. Hoggatt, Jr., “A Primer on the Fibonacci Sequence Part II,” Fibonacci Quarterly, 1, 61-68, 1963.
 
[15]  S. L. Basin, “The appearance of Fibonacci Numbers and the Q Matrix in Electrical Network Theory,” Mathematics Magazine, 36 (2), 84-97, 1963.
 
[16]  T. Koshy, Fibonacci and Lucas Numbers With Applications, John Wiley and Sons, New York, 2001.
 
[17]  V.N. Mishra, H.H. Khan, K. Khatri and L. N. Mishra, “Hypergeometric Representation for Baskakov-Durrmeyer-Stancu Type Operators,” Bulletin of Mathematical Analysis and Applications, 5 (3), 18-26, 2013.
 
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Article

Generalized Fibonacci-Lucas Sequence

1School of Studies in Mathematics, Vikram University, Ujjain-456010 (M. P.), India

2Department of Mathematics, Mandsaur Institute of Technology, Mandsaur (M. P.), India

3School of Studies in Mathematics, Vikram University, Ujjain, (M. P.), India


Turkish Journal of Analysis and Number Theory. 2014, 2(6), 193-197
DOI: 10.12691/tjant-2-6-1
Copyright © 2014 Science and Education Publishing

Cite this paper:
Bijendra Singh, Omprakash Sikhwal, Yogesh Kumar Gupta. Generalized Fibonacci-Lucas Sequence. Turkish Journal of Analysis and Number Theory. 2014; 2(6):193-197. doi: 10.12691/tjant-2-6-1.

Correspondence to: Omprakash  Sikhwal, Department of Mathematics, Mandsaur Institute of Technology, Mandsaur (M. P.), India. Email: opbhsikhwal@rediffmail.com

Abstract

The Fibonacci sequence is a source of many nice and interesting identities. A similar interpretation exists for Lucas sequence. The Fibonacci sequence, Lucas numbers and their generalization have many interesting properties and applications to almost every field. Fibonacci sequence is defined by the recurrence formula and F0=0, F1=1, where Fn is a nth number of sequence. The Lucas Sequence is defined by the recurrence formula and L0=2, L1=1, where Ln is a nth number of sequence. In this paper, Generalized Fibonacci-Lucas sequence is introduced and defined by the recurrence relation with B0 = 2b, B1 = s, where b and s are integers. We present some standard identities and determinant identities of generalized Fibonacci-Lucas sequences by Binet’s formula and other simple methods.

Keywords

References

[1]  A. F. Horadam: A Generalized Fibonacci Sequence, American Mathematical Monthly, Vol. 68. (5), 1961, 455-459.
 
[2]  A. F. Horadam: Basic Properties of a Certain Generalized Sequence of Numbers, The Fibonacci Quarterly, Vol. 3 (3), 1965, 161-176.
 
[3]  A.T. Benjamin and D. Walton: Counting on Chebyshev polynomials, Math. Mag. 82, 2009, 117-126.
 
[4]  B. Singh, O. Sikhwal and S. Bhatnagar: Fibonacci-Like Sequence and its Properties, Int. J. Contemp. Math. Sciences, Vol. 5 (18), 2010, 859-868.
 
[5]  B. Singh, S. Bhatnagar and O. Sikhwal: Fibonacci-Like Polynomials and some Identities, International Journal of Advanced Mathematical Sciences, 1 (3), (2013) 152-157.
 
Show More References
[6]  B. Singh, S. Bhatnagar and O. Sikhwal: Fibonacci-Like Sequence, International Journal of Advanced Mathematical Sciences, 1 (3) (2013) 145-151.
 
[7]  B. Singh, S. Bhatnagar and O. Sikhwal: Generalized Identties of Companion Fibonacci-Like Sequences, Global Journal of Mathematical Analysis, 1 (3), 2013, 104-109.
 
[8]  D. V. Jaiswal: On a Generalized Fibonacci Sequence, Labdev J. Sci. Tech. Part A 7, 1969, 67-71.
 
[9]  M. Edson and O. Yayenie: A New Generalization of Fibonacci sequence and Extended Binet’s Formula, Integers Vol. 9, 2009, 639-654.
 
[10]  M. E. Waddill and L. Sacks: Another Generalized Fibonacci Sequence, The Fibonacci Quarterly, Vol. 5 (3), 1967, 209-222.
 
[11]  S. Falcon and A. Plaza: On the Fibonacci K- Numbers, Chaos, Solutions & Fractals, Vol. 32 (5), 2007, 1615-1624,
 
[12]  S. Vajda, Fibonacci & Lucas Numbers, and the Golden Section, Theory and Applications, Ellis Horwood Ltd., Chichester, 1989.
 
[13]  T. Koshy, Fibonacci and Lucas Numbers with Applications, Wiley-Interscience Publication, New York (2001).
 
Show Less References

Article

Using the Matrix Summability Method to Approximate the Lip (ξ(t), p) Class Functions

1Central Department of Education (Mathematics), Tribhuvan University, Nepal


Turkish Journal of Analysis and Number Theory. 2014, 2(6), 198-201
DOI: 10.12691/tjant-2-6-2
Copyright © 2014 Science and Education Publishing

Cite this paper:
Binod Prasad Dhakal. Using the Matrix Summability Method to Approximate the Lip (ξ(t), p) Class Functions. Turkish Journal of Analysis and Number Theory. 2014; 2(6):198-201. doi: 10.12691/tjant-2-6-2.

Correspondence to: Binod  Prasad Dhakal, Central Department of Education (Mathematics), Tribhuvan University, Nepal. Email: binod_dhakal2004@yahoo.com

Abstract

Most of the summability methods are derived from the matrix means. In this paper, author has been determined the degree of approximation of certain trigonometric functions belonging to the Lip (ξ(t), p) class by matrix method.

Keywords

References

[1]  A. Zygmund, Trigonometric series, Cambridge University Press, 1959.
 
[2]  E. C. Titchmarsh, Theory of functions, Oxford University Press, 1939.
 
[3]  M. L Mittal, B. E. Rhoades, V. N. Mishra and U. Shing, Using infinite matrices to functions of class Lip (α,p) using trigonometric polynomials, J. Math. Anal. Appl, 326(2007), 667-676.
 
[4]  O.Töeplitz, Über allgemeine lineare Mittelbildungen, Prace mat. - fiz., 22(1913), 113-119.
 
[5]  P. Chanrda, Trigonometric approximation of function in Lp-norm, J. Math. Anal. Appl, 275(2002), 13-676.
 
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[6]  S. Lal and B. P. Dhakal, On Approximation of functions belonging to Lipschitz class by triangular matrix method of Fourier series, Int. Journal of Math. Analysis, 4(21), 2010, 1041-1047.
 
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Article

Hermite-Hadamard Type Inequalities for s-Convex Stochastic Processes in the Second Sense

1Department of Mathematics, Faculty of Science and Arts, Ordu University, Ordu, Turkey


Turkish Journal of Analysis and Number Theory. 2014, 2(6), 202-207
DOI: 10.12691/tjant-2-6-3
Copyright © 2014 Science and Education Publishing

Cite this paper:
ERHAN SET, MUHARREM TOMAR, SELAHATTIN MADEN. Hermite-Hadamard Type Inequalities for s-Convex Stochastic Processes in the Second Sense. Turkish Journal of Analysis and Number Theory. 2014; 2(6):202-207. doi: 10.12691/tjant-2-6-3.

Correspondence to: MUHARREM  TOMAR, Department of Mathematics, Faculty of Science and Arts, Ordu University, Ordu, Turkey. Email: muharremtomar@odu.edu.tr

Abstract

In this study, s-convex stochastic processes in the second sense are presented and some well-known results concerning s-convex functions are extended to s-convex stochastic processes in the second sense. Also, we investigate relation between s-convex stochastic processes in the second sense and convex stochastic processes.

Keywords

References

[1]  K. Nikodem, on convex stochastic processes, Aequationes Mathematicae 20 (1980) 184-197.
 
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Article

Birth of Compound Numbers

1Department of Computer Science, Jamia Hamdard University, Hamdard Nagar, New Delhi, INDIA


Turkish Journal of Analysis and Number Theory. 2014, 2(6), 208-219
DOI: 10.12691/tjant-2-6-4
Copyright © 2014 Science and Education Publishing

Cite this paper:
Ranjit Biswas. Birth of Compound Numbers. Turkish Journal of Analysis and Number Theory. 2014; 2(6):208-219. doi: 10.12691/tjant-2-6-4.

Correspondence to: Ranjit  Biswas, Department of Computer Science, Jamia Hamdard University, Hamdard Nagar, New Delhi, INDIA. Email: ranjitbiswas@yahoo.com

Abstract

In this paper the author introduces a new kind of numbers called by ‘Compound Numbers’. A region R may or may not have imaginary object. A region even may have more than one imaginary objects too. Corresponding to an imaginary object (if exists) of a region R, we get compound objects for the region R. Imaginary objects and compound objects of a region R are not members of R and so they are called imaginary with respect to the region R only (i.e. it is a local characteristics property with respect to the region concerned), as they could be core members of another region. Every region has its own set of imaginary numbers (if exist). As a particular instance, the compound objects of the set of real numbers are the complex numbers (of existing concept). In this paper the author discovers imaginary objects of the region C (the set of complex numbers). The compound objects of C are called by ‘compound numbers’. Collection of all compound numbers is denoted by the set E. This work just reports the birth of compound numbers, not further details at this stage. It is claimed that “Theory of Numbers” will get a new direction by the birth of compound numbers. A new “Theory of Objects’ and the classical “Theory of Numbers” as a special case of it were also studied in . In this paper we say that every complete region has its own ‘Theory of Numbers’, where the classical ‘theory of numbers’ is just a special instance corresponding to a particular complete region RR. Consequently, we also introduce a new field called by “Object Geometry” of a complete region, being a generalization of our classical geometry of the existing style, from elementary to the higher level.

Keywords

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