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## Article

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

^{1}Institute 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

*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

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

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

^{3}Yüzüncü 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

## 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. | ||

## Article

# Identities of Generalized Fibonacci-Like Sequence

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

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

^{3}Schools 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

_{n}=F

_{n-1}+F

_{n-2}, n≥2 and F

_{0}=0, F

_{1}=1, where F

_{n}is a n

^{th}

^{ }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 M

_{n}=M

_{n-1}+M

_{n-2}, n≥2, with M

_{0}=2, M

_{1}=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. | ||

[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). | ||

## Article

# Some New Ostrowski Type Inequalities for Co-Ordinated Convex Functions

^{1}Department 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

^{2}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. | ||

[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). | ||

## Article

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

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

^{2}Department 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

## 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. | ||

[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. | ||

## Article

# Fibonacci Polynomials and Determinant Identities

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

^{2}Department 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

## 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. | ||

[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 G_{n}=G_{n-1}+G_{n-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. | ||

## Article

# Generalized Fibonacci-Lucas Sequence

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

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

^{3}School 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

_{0}=0, F

_{1}=1, where F

_{n}is a n

^{th }number of sequence. The Lucas Sequence is defined by the recurrence formula and L

_{0}=2, L

_{1}=1, where L

_{n}is a n

^{th }number of sequence. In this paper, Generalized Fibonacci-Lucas sequence is introduced and defined by the recurrence relation with B

_{0}= 2b, B

_{1}= 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. | ||

[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). | ||

## Article

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

^{1}Central 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

## 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 L_{p}-norm, J. Math. Anal. Appl, 275(2002), 13-676. | ||

[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. | ||

## Article

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

^{1}Department 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

## Keywords

## References

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

[2] | A. Skowronski, On some properties of J-convex stochastic processes, Aequationes Mathematicae 44 (1992) 249-258. | ||

[3] | A. Skowronski, On wright-convex stochastic processes, Annales Mathematicae Silesianne 9(1995) 29-32. | ||

[4] | D. Kotrys, Hermite-hadamart inequality for convex stochastic processes, Aequationes Mathematicae 83 (2012) 143-151. | ||

[5] | S.S. Dragomir and S. Fitzpatrick, The Hadamard's inequality for s-convex functions in the second sense, Demonstratio Math., 32 (4) (1999), 687-696. | ||

[6] | H. Hudzik and L. Maligranda, Some remarks on s-convex functions, Aequationes Mathematicae, 48 (1994), 100-111. | ||

[7] | S. Maden, M. Tomar and E. Set, s-convex stochastic processes in the first sense, Pure and Applied Mathematics Letters, in press. | ||

[8] | D. Kotrys, Hermite-hadamart inequality for convex stochastic processes, Aequationes Mathematicae 83 (2012) 143-151. | ||

## Article

# Birth of Compound Numbers

^{1}Department 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

## Keywords

## References

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