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Article

On an Integral Involving Bessel Polynomials and -Function of Two Variables and Its Application

1Department of Mathematics, Seth Motilal (P.G.) College, Jhunjhunu, Rajasthan, India

2Department of Mathematics,Government College, Kaladera, Jaipur, Rajasthan , India


Turkish Journal of Analysis and Number Theory. 2014, 2(4), 130-133
DOI: 10.12691/tjant-2-4-5
Copyright © 2014 Science and Education Publishing

Cite this paper:
Harmendra Kumar Mandia, Yashwant Singh. On an Integral Involving Bessel Polynomials and -Function of Two Variables and Its Application. Turkish Journal of Analysis and Number Theory. 2014; 2(4):130-133. doi: 10.12691/tjant-2-4-5.

Correspondence to: Yashwant  Singh, Department of Mathematics,Government College, Kaladera, Jaipur, Rajasthan , India. Email: mandiaharmendra@gmail.com; dryashu23@yahoo.in

Abstract

This paper deals with the evaluation of an integral involving product of Bessel polynomials and -function of two variables. By making use of this integral the solution of the time-domain synthesis problem is investigated.

Keywords

References

[1]  Bajpai, S.D. and Al-Hawaj, A.Y.; Application of Bessel polynomials involving generalized hypergeometric functions, J.Indian Acad. Math., vol.13 (1), (1991), 1-5.
 
[2]  Erdelyi, A. et. al.; Higher Transcendental Functions, vol.1, McGraw-Hill, New York, 1953.
 
[3]  Erdelyi, A. et. al.; Tables of Integral Transforms, vol.2, McGraw-Hill, New York, 1954.
 
[4]  Exton, H.; Handbook of Hypergeometric Integrals, ELLIS Harwood Ltd., Chichester, 1978.
 
[5]  Inayat-Hussain, A.A.; New properties of hypergeometric series derivable from Feynman integrals: II A generalization of the H-function, J. Phys. A. Math. Gen. 20 (1987).
 
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[6]  Mathai, A.M. and Saxena, R.K.; Lecture Notes in Maths. 348, Generalized Hypergeometric Functions With Applications in Statistics and Physical Sciences, Springer-Verlag, Berlin, 1973.
 
[7]  Mittal, P.K. and Gupta, K.C.; An integral involving generalized function of two variables. Proc. Indian Acad. Sci. Sect. A( 75), (1961), 67-73.
 
[8]  Singh,Y. and Mandia, H. ; A study of -function of two variables, International Journal of Innovative research in science, engineering and technology,Vol.2,(9),(2013), 4914-4921.
 
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Article

Refinements and Sharpening of some Huygens and Wilker Type Inequalities

1Department of Information Engineering, Weihai Vocational University, Weihai, Shandong, China

2Department of Mathematics, Chongqing Normal University, Chongqing City, China

3College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mongolia Autonomous Region, China

4Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City, China;Institute of Mathematics, Henan Polytechnic University, Jiaozuo City, Henan Province, China


Turkish Journal of Analysis and Number Theory. 2014, 2(4), 134-139
DOI: 10.12691/tjant-2-4-6
Copyright © 2014 Science and Education Publishing

Cite this paper:
Wei-Dong Jiang, Qiu-Ming Luo, Feng Qi. Refinements and Sharpening of some Huygens and Wilker Type Inequalities. Turkish Journal of Analysis and Number Theory. 2014; 2(4):134-139. doi: 10.12691/tjant-2-4-6.

Correspondence to: Feng  Qi, College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mongolia Autonomous Region, China. Email: qifeng618@gmail.com, qifeng618@hotmail.com, qifeng618@qq.com

Abstract

In the article, some Huygens and Wilker type inequalities involving trigonometric and hyperbolic functions are refined and sharpened.

Keywords

References

[1]  M. Abramowitz and I. A. Stegun (Eds), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series 55, 4th printing, with corrections, Washington, 1965.
 
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[3]  H. Alzer and S.-L. Qiu, Monotonicity theorems and inequalities for complete elliptic integrals, J. Comput. Appl. Math. 172 (2004), no. 2, 289-312.
 
[4]  B.-N. Guo, W. Li, and F. Qi, Proofs of Wilker's inequalities involving trigonometric functions, The 7th International Conference on Nonlinear Functional Analysis and Applications, Chinju, South Korea, August 6-10, 2001; Inequality Theory and Applications, Volume 3, Yeol Je Cho, Jong Kyu Kim, and Sever S. Dragomir (Eds), Nova Science Publishers, Hauppauge, NY, pp. 109-112.
 
[5]  B.-N. Guo, B.-M. Qiao, F. Qi, and W. Li, On newproofs of Wilker's inequalities involving trigonometric functions, Math. Inequal. Appl. 6 (2003), no. 1, 19-22.
 
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[6]  Z.-H. Huo, D.-W. Niu, J. Cao, and F. Qi, A generalization of Jordan's inequality and an application, Hacet. J. Math. Stat. 40 (2011), no. 1, 53-61.
 
[7]  W.-D. Jiang, Q.-M. Luo, and F. Qi. Refinements and sharpening of some Huygens and Wilker type inequalities, available online at http://arxiv.org/abs/1201.6477.
 
[8]  C. Mortici, The natural approach of Wilker-Cusa-Huygens inequalities, Math. Inequal. Appl. 14 (2011), no. 3, 535-541.
 
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[12]  D.-W. Niu, Z.-H. Huo, J. Cao, and F. Qi, A general refinement of Jordan's inequality and a refinement of L. Yang's inequality, Integral Transforms Spec. Funct. 19 (2008), no. 3, 157-164.
 
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[14]  F. Qi, D.-W. Niu, and B.-N. Guo, Refinements, generalizations, and applications of Jordan's inequality and related problems, J. Inequal. Appl. 2009 (2009), Article ID 271923, 52 pages.
 
[15]  F. Qi and A. Sofo, An alternative and united proof of a double inequality for bounding the arithmeticgeometric mean, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 71 (2009), no. 3, 69-76.
 
[16]  J. S_andor and M. Bencze, On Huygens' trigonometric inequality, RGMIA Res. Rep. Coll. 8 (2005), no. 3, Art. 14; Available online at http://rgmia. org/v8n3.php.
 
[17]  J. S. Sumner, A. A. Jagers, M. Vowe, and J. Anglesio, Inequalities involving trigonometric functions, Amer. Math. Monthly 98 (1991), no. 3, 264-267.
 
[18]  J. B. Wilker, Problem E 3306, Amer. Math. Monthly 96 (1989), no. 1, 55.
 
[19]  S.-H. Wu and H. M. Srivastava, A further refinement of Wilker's inequality, Integral Transforms Spec. Funct. 19 (2008), no. 10, 757-765.
 
[20]  L. Zhu, Some new Wilker-type inequalities for circular and hyperbolic functions, Abstr. Appl. Anal. 2009 (2009), Article ID 485842, 9 pages.
 
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Article

Some Integral Inequalities of Hermite-Hadamard Type for Functions Whose n-times Derivatives are (α,m)-Convex

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

2College of Science, Department of Mathematics, University of Hail, Hail, Saudi Arabia

3Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City, China

4Department of Mathematics, University of Engineering and Technology, Lahore, Pakistan


Turkish Journal of Analysis and Number Theory. 2014, 2(4), 140-146
DOI: 10.12691/tjant-2-4-7
Copyright © 2014 Science and Education Publishing

Cite this paper:
Feng Qi, Muhammad Amer Latif, Wen-Hui Li, Sabir Hussain. Some Integral Inequalities of Hermite-Hadamard Type for Functions Whose n-times Derivatives are (α,m)-Convex. Turkish Journal of Analysis and Number Theory. 2014; 2(4):140-146. doi: 10.12691/tjant-2-4-7.

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

Abstract

In the paper, the authors find some new integral inequalities of Hermite-Hadamard type for functions whose derivatives of the n-th order are (α,m)-convex and deduce some known results. As applications of the newly-established results, the authors also derive some inequalities involving special means of two positive real numbers.

Keywords

References

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[25]  Y. Wang, M.-M. Zheng, and F. Qi, Integral inequalities of Hermite-Hadamard type for functions whose derivatives are (α,m)-preinvex, J. Inequal. Appl. 2014, 2014:97, 10 pages.
 
[26]  B.-Y. Xi, R.-F. Bai, and F. Qi, Hermite-Hadamard type inequalities for the m- and (α,m)-geometrically convex functions, Aequationes Math. 84 (2012), no. 3, 261-269.
 
[27]  B.-Y. Xi, J. Hua, and F. Qi, Hermite-Hadamard type inequalities for extended s-convex functions on the co-ordinates in a rectangle, J. Appl. Anal. 20 (2014), no. 1, 29-39.
 
[28]  B.-Y. Xi and F. Qi, Some Hermite-Hadamard type inequalities for geometrically quasi-convex functions, Proc. Indian Acad. Sci. Math. Sci. (2014), in press.
 
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Article

A New Proof of an Inequality for the Logarithm of the Gamma Function and Its Sharpness

1Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia

2Department of Mathematics, Faculty of Science, Mansoura University, ansoura 35516, Egypt


Turkish Journal of Analysis and Number Theory. 2014, 2(4), 147-151
DOI: 10.12691/tjant-2-4-8
Copyright © 2014 Science and Education Publishing

Cite this paper:
Mansour Mahmoud. A New Proof of an Inequality for the Logarithm of the Gamma Function and Its Sharpness. Turkish Journal of Analysis and Number Theory. 2014; 2(4):147-151. doi: 10.12691/tjant-2-4-8.

Correspondence to: Mansour  Mahmoud, Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia. Email: mansour@mans.edu.eg

Abstract

In the paper, the author shows that the partial sums are alternatively larger and smaller than the generalized Euler’s harmonic numbers with sharp bounds, where γ is the Euler's constant, are the Bernoulli numbers and ψ is the digamma function.

Keywords

References

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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

[1]  Latif, M.A. and Alomari, M., On Hadamard-type inequalities for h-convex functions on the co-ordinates, International Journal of Math. Analysis, 3 (2009), no: 33, 1645-1656.
 
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[8]  Özdemir, M.E., Set, E. and Sarkaya, M.Z., Some new Hadamard.s type inequalities for co-ordinated m-convex and (α, m)-convex functions, Hacettepe J. of. Math. and St., 40, 219-229, (2011).
 
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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

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[11]  M. Singh, Y. Gupta, O. Sikhwal, Generalized Fibonacci-Lucas Sequences its Properties, Global Journal of Mathematical Analysis, 2(3), 2014, 160-168.
 
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[13]  S. Falcon and A. Plaza: On the Fibonacci K- Numbers, Chaos, Solutions & Fractals, Vol. 32 (5), 2007, 1615-1624.
 
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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

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[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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[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.
 
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[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.
 
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[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.
 
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[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.
 
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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

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[5]  B. Singh, O. Sikhwal and Y. K. Panwar, “Generalized Determinantal Identities Involving Lucas Polynomials,” Applied Mathematical Sciences, 3 (8), 377-388, 2009.
 
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[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.
 
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