**Turkish Journal of Analysis and Number Theory:**Latest Articles More >>

## Article

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

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

^{2}Department 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

## 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). | ||

[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. | ||

## Article

# Refinements and Sharpening of some Huygens and Wilker Type Inequalities

^{1}Department of Information Engineering, Weihai Vocational University, Weihai, Shandong, China

^{2}Department of Mathematics, Chongqing Normal University, Chongqing City, China

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

^{4}Department 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

## 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. | ||

[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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[11] | D.-W. Niu, J. Cao, and F. Qi, Generalizations of Jordan's inequality and concerned relations, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 72 (2010), no. 3, 85-98. | ||

[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. | ||

[13] | F. Qi, L.-H. Cui, and S.-L. Xu, Some inequalities constructed by Tchebysheff's integral inequality, Math. Inequal. Appl. 2 (1999), no. 4, 517-528. | ||

[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. | ||

## Article

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

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

^{2}College of Science, Department of Mathematics, University of Hail, Hail, Saudi Arabia

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

^{4}Department 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

## Keywords

## References

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[15] | F. Qi, Z.-L. Wei, and Q. Yang, Generalizations and refinements of Hermite-Hadamard's inequality, Rocky Mountain J. Math. 35 (2005), no. 1, 235-251. | ||

[16] | F. Qi and B.-Y. Xi, Some integral inequalities of Simpson type for GA-"-convex functions, Georgian Math. J. 20 (2013), no. 4, 775-78. | ||

[17] | M. Z. Sarikaya and N. Aktan, On the generalization of some integral inequalities and their applications. | ||

[18] | D.-P. Shi, B.-Y. Xi, and F. Qi, Hermite-Hadamard type inequalities for (m,h1,h2)-convex functions via Riemann-Liouville fractional integrals, Turkish J. Anal. Number Theory 2 (2014), no. 1, 22-27. | ||

[19] | Y. Shuang, Y. Wang, and F. Qi, Some inequalities of Hermite-Hadamard type for functions whose third derivatives are (α;m)-convex, J. Comput. Anal. Appl. 17 (2014), no. 2, 272-279. | ||

[20] | G. Toader, Some generalizations of the convexity, Univ. Cluj-Napoca, Cluj-Napoc. 1985, 329-338. | ||

[21] | S.-H. Wang and F. Qi, Hermite-Hadamard type in-equalities for n-times differentiable and preinvex functions, J. Inequal. Appl. 2014, 2014: 49, 9 pages. | ||

[22] | S.-H. Wang and F. Qi, Inequalities of Hermite-Hadamard type for convex functions which are Turkish Journal of Analysis and Number Theory 8 n-times differentiable, Math. In equal. Appl. 16 (2013), no. 4, 1269-1278. | ||

[23] | S.-H. Wang, B.-Y. Xi, F. Qi, Some new inequalities of Hermite-Hadamard type for n-time differentiable functions which are m-convex, Analysis (Munich) 32 (2012), no. 3, 247-262. | ||

[24] | Y. Wang, B.-Y. Xi, and F. Qi, Hermite-Hadamard type integral inequalities when the power of the absolute value of the first derivative of the integrand is preinvex, Matematiche (Catania) 69 (2014), no. 1, 89-96. | ||

[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. | ||

[29] | B.-Y. Xi and F. Qi, Some inequalities of Hermite-Hadamard type for h-convex functions, Adv. Inequal. Appl. 2 (2013), no. 1, 1-15. | ||

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[31] | B.-Y. Xi, S.-H. Wang, and F. Qi, Properties and inequalities for the h- and (h; m)-logarithmically convex functions, Creat. Math. Inform. 23 (2014), no. 1, 123-130. | ||

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[33] | T.-Y. Zhang and F. Qi, Integral inequalities of Hermite-Hadamard type for m-AH convex func-tions, Turkish J. Anal. Number Theory 2 (2014), no. 3, 60-64. | ||

## Article

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

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

^{2}Department 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

## Keywords

## References

[1] | Alzer, H., “On some inequalities for the gamma and psi functions,” Mathematics of Computation, 66, no. 217, 373-389, 1997. | ||

[2] | Artin, E., The Gamma function, translated by M. Butler, Holt, Rinehart and Winston, New York, 1964. | ||

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[20] | Mishra, V. N. and Mishra, L. N., “Trigonometric Approximation of Signals (Functions) in L_{p}-Norm”, Int. Journal of Contemp. Math. Sciences, Vol. 7, no. 19, 909-918, 2012. | ||

[21] | Mishra, V. N., Khatri, K. and Mishra, L. N., “Using Linear Operators to Approximate Signals of Lip(α,p), (p≥1)-Class, Filomat 27: 2, 353-363, 2013. | ||

[22] | Mortici, C., “Fast convergences towards Euler-Mascheroni constant”, Computational and Appl. Math., Vol. 29, N. 3, 479-491, 2010. | ||

[23] | Mortici, C., “Very accurate estimates of the polygamma functions”, Asympt. Anal. 68, no. 3, 125-134, 2010. | ||

[24] | Mortici, C., “A new representation formula for the factorial function”, Thai Journal of Mathematics, Vol. 8, no. 2, 249-254, 2010. | ||

[25] | Qi, F., Cui, R.-Q., Chen, C.-P. and Guo, B.-N., “Some completely monotonic functions involving polygamma functions and an application”, Journal of Mathematical Analysis and Applications 310, no. 1, 303-308, 2005. | ||

[26] | Qi, F. and Luo, Q.-M., “Bounds for the ratio of two gamma functions--From Wendel's and related inequalities to logarithmically completely monotonic functions”, Banach Journal of Mathematical Analysis 6, no. 2, 132-158, 2012. | ||

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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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[23] | B.-N. Guo and F. Qi, A property of logarithmically absolutely monotonic functions and the logarithmically complete monotonicity of a power-exponential function, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 72 (2010), no. 2, 21-30. | ||

[24] | B.-N. Guo and F. Qi, A simple proof of logarithmic convexity of extended mean values, Numer. Algorithms 52 (2009), no. 1, 89-92. | ||

[25] | B.-N. Guo and F. Qi, Monotonicity and logarithmic convexity relating to the volume of the unit ball, Optim. Lett. 7 (2013), no. 6, 1139-1153. | ||

[26] | B.-N. Guo and F. Qi, On the degree of the weighted geometric mean as a complete Bernstein function, Afr. Mat. 26 (2015), in press. | ||

[27] | B.-N. Guo and F. Qi, Refinements of lower bounds for polygamma functions, Proc. Amer. Math. Soc. 141 (2013), no. 3, 1007-1015. | ||

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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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## 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

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## 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

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[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

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[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. | ||

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## 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. | ||

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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. | ||

[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. | ||