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Article

Determinantal Identities of Fibonacci, Fibonacci Like and Lucas Numbers

1Government College, Kannod(M.P.), India

2School of studies in Mathematics

3Vikram University Ujjain, India


Turkish Journal of Analysis and Number Theory. 2014, 2(4), 110-112
DOI: 10.12691/tjant-2-4-1
Copyright © 2014 Science and Education Publishing

Cite this paper:
Sanjay Harne, V.H. Badshah, Sapna Sethiya. Determinantal Identities of Fibonacci, Fibonacci Like and Lucas Numbers. Turkish Journal of Analysis and Number Theory. 2014; 2(4):110-112. doi: 10.12691/tjant-2-4-1.

Correspondence to: Sapna  Sethiya, Vikram University Ujjain, India. Email: sapna.sethiya11@gmail.com

Abstract

Determinants have played a significant part in various areas in mathematics. For instance, they are quite useful in the analysis and solution of system of linear equations. There are different perspectives on the study of determinant. In this paper we present some determinant identities of Fibonacci and Lucas numbers.

Keywords

References

[1]  Benjamin A., Cameron N. and Quinn J.: Fibonacci Determinants- A Combinatorial Approach, Fibonacci Quarterly, 39-55 (1), 2007, Vol 45.
 
[2]  Bicknell-Johnson M. and Spears C. P.:Classes Of Identities For the Generalized Fibonacci number Gn= Gn-1 + Gn-2, n ≥ 2. from Matrices with Constantvalued Determinants, Fibonacci Quarterly, 121-128 (2), 1996, Vol. 34.
 
[3]  B. Singh, O. Sikhwal, and S. Bhatnagar, Fibonacci-Like Sequence and its properties, Int.J. Contemp Math.Sciences, Vol.5, 2010, No.18, 857-868.
 
[4]  Cahill N. and Narayan D.:Fibonacci and Lucas numbers s Tridigonal Matrix Determinants, Fibonacci Quarterly, 216-221 (3), 2004, Vol. 42.
 
[5]  Koshy T.:Fibonacci and Lucas Numbers with Applications, Wiley, 2001.
 
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[6]  Krattenthaler C.: Advanced determinant calculus, Seminaire Lotharingien Combin, Article, b42q, 67, 1999.
 
[7]  Krattenthaler C.: Advanced determinant calculus: A Complement, Liner Algebra Appl., 68-166.
 
[8]  Macfarlane A. J.: Use of Determinants to Present Identities Involving Fiboncci and Related Numbers, Fibonacci Quarterly, 68-7648(1), 2010, Vol. 48.
 
[9]  Spivey M. Z.: Fibonacci Identities via the Determinant sum property, College Mathematics Journal, 286-289 (4), 2006, Vol. 37.
 
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Article

On Some Inequalities for Functions Whose Second Derivatives Absolute Values Are S-Geometrically Convex

1Mustafa Kemal University, Faculty of Science and Arts, Department of Mathematics, Hatay, Turkey

2The Institute for Graduate Studies in Sciences and Engineering, Kilis 7 Aralk University, Kilis, Turkey

3AGRI Ibrahim Çeçen University, Faculty of Science and Arts, Department of Mathematics, AGRI, Turkey


Turkish Journal of Analysis and Number Theory. 2014, 2(4), 113-118
DOI: 10.12691/tjant-2-4-2
Copyright © 2014 Science and Education Publishing

Cite this paper:
MEVLÜT TUNÇ, IBRAHİM KARABAYIR, EBRU YÜKSEL. On Some Inequalities for Functions Whose Second Derivatives Absolute Values Are S-Geometrically Convex. Turkish Journal of Analysis and Number Theory. 2014; 2(4):113-118. doi: 10.12691/tjant-2-4-2.

Correspondence to: MEVLÜT  TUNÇ, Mustafa Kemal University, Faculty of Science and Arts, Department of Mathematics, Hatay, Turkey. Email: mevluttttunc@gmail.com

Abstract

In this paper, the authors achieve some new Hadamard type in- equalities using elementary well known inequalities for functions whose second derivatives absolute values are s-geometrically and geometrically convex. And also they get some applications for special means for positive numbers.

Keywords

References

[1]  M. Alomari, M. Darus, S. S. Dragomir: New inequalities of Hermite-Hadamard type for func-tions whose second derivatives absolute values are quasi-convex, Tamkang J. Math., Vol. 41 No. 4 (2010/12), 353-359.
 
[2]  S.S. Dragomir, R.P. Agarwal: Two inequalities for differentiable mappings and applicationsto special means of real numbers and to trapezoidal formula. Appl Math Lett, Vol. 11 No: 5, (1998) 91.95.
 
[3]  Hermite-Hadamard in-equalities and applications, RGMIA monographs, Victoria University, 2000. [Online: http://www.staff.vu.edu.au/RGMIA/monographs/hermite-hadamard.html].
 
[4]  J. Hadamard: Étude sur les propriétés des fonctions entières et en particulier d.une function considerée par Riemann, J. Math Pures Appl., 58, (1893) 171. 215.
 
[5]  H. Hudzik and L. Maligranda: Some remarks on s-convex functions, Aequationes Math., Vol. 48 (1994), 100-111.
 
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[6]  İ. İscan, Some New Hermite-Hadamard Type Inequalities for Geometrically Convex Functions, Mathematics and Statistics, 1 (2): 86-91, 2013.
 
[7]  İ. İscan, On Some New Hermite-Hadamard type inequalities for s-geometrically convex functions, International Journal of Mathematics and Mathematical Sciences, Volume 2014 (2014), Article ID 163901, 8 pages.
 
[8]  D. S. Mitrinovi´c, J. Peµcari´c and A. M. Fink: Classical and new inequalities in analysis, Kluwer Academic, Dordrecht, 1993.
 
[9]  J. E. Peµcari´c, F. Proschan and Y. L. Tong: Convex Functions, Partial Orderings, and Statistical Applications, Academic Press Inc., 1992.
 
[10]  M. Tunç: On some new inequalities for convex fonctions, Turk. J. Math. 36 (2012), 245-251.
 
[11]  B.-Y. Xi, R.-F. Bai and F. Qi: Hermite-Hadamard type inequalities for the m- and (α;m)-geometrically convex functions. Aequationes Math.
 
[12]  T.-Y. Zhang, A.-P. Ji and F. Qi: On Integral nequalities of Hermite-Hadamard Type for s-Geometrically Convex Functions. Abstract and Applied Analysis.
 
[13]  T.-Y. Zhang, M. Tunç, A.-P. Ji, B.-Y. Xi: Erratum to. On Integral Inequalities of Hermite-Hadamard Type for s-Geometrically Convex Functions. Abstract and Applied Analysis. Volume 2014, Article ID 294739, 5 pages.
 
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Article

Some Properties of k-Jacobsthal Numbers with Arithmetic Indexes

1School of Studies in Mathematics, Vikram University, Ujjain (India)

2College of Horticulture, Mandsaur (M.P.)


Turkish Journal of Analysis and Number Theory. 2014, 2(4), 119-124
DOI: 10.12691/tjant-2-4-3
Copyright © 2014 Science and Education Publishing

Cite this paper:
Deepika Jhala, G.P.S. Rathore, Kiran Sisodiya. Some Properties of k-Jacobsthal Numbers with Arithmetic Indexes. Turkish Journal of Analysis and Number Theory. 2014; 2(4):119-124. doi: 10.12691/tjant-2-4-3.

Correspondence to: Deepika  Jhala, School of Studies in Mathematics, Vikram University, Ujjain (India). Email: jhala.deepika28@gmail.com

Abstract

In this paper, we derive various formulae for the sum of k-Jacobsthal numbers with indexes in an arithmetic sequence, say an+r for fixed integers a and r Also, we describe generating function and the alternated sum formula for k-Jacobsthal numbers with indexes in an arithmetic sequence.

Keywords

References

[1]  Bolat C., Kose H., On the Properties of k-Fibonacci Numbers. Int. J.Contemp. Math. Sciences 2010, 22 (5), 1097-1105.
 
[2]  Campos H., Catarino P., Aires A.P., Vasco P. and Borges A., On Some Identities of k-Jacobsthal-Lucas Numbers, Int. Journal of Math. Analysis, 2014, 8 (10), 489-494.
 
[3]  Catarino P., On Some Identities and Generating Functions for k-Pell Numbers. Int. Journal of Math. Analysis 2013, 7 (38), 1877-1884.
 
[4]  Catarino P., Vasco P., Modified k-Pell Sequence: Some Identities and Ordinary Generating Function. Applied Mathematical Sciences 2013, 7 (121), 6031-6037.
 
[5]  Catarino P., Vasco P., On some Identities and Generating Functions for k-Pell-Lucas Sequence. Applied Mathematical Sciences 2013, 7 (98), 4867-4873.
 
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[6]  Catarino P., On Some Identities for k-Fibonacci Sequence. International Journal of Contemporary mathematical Sciences 2014, 9 (1), 37-42.
 
[7]  Falcon S. and Plaza A., On k-Fibonacci Numbers of Arithmetic Indexes, Applied Mathematics and Computation, 2009, 208, 180-185.
 
[8]  Falcon S., On the k-Lucas Numbers, International Journal of Contemporary mathematical Sciences 2011, 6 (21), 1039-1050.
 
[9]  Falcon S., On the k-Lucas Numbers of Arithmetic Indexes, Applied Mathematics, 2012, 3, 1202-1206.
 
[10]  Horadam A. F., A Generalized Fibonacci Sequence, The American Mathematical Monthly, 1961, 68, 455-459.
 
[11]  Horadam A.F., Jacobsthal Representation Numbers. Fibonacci Quarterly, 1996, 34 (1), 40-54.
 
[12]  Jaiswal D. V., On a Generalized Fibonacci Sequence, Labdev Journal of Science and Technology, Part A, 1969, 7, 67-71.
 
[13]  Jhala D., Sisodiya K., Rathore G.P.S., On Some Identities for k-Jacobsthal Numbers. Int. Journal of Math. Analysis 2013, 7 (2), 551-556.
 
[14]  Lee G. Y., Lee S. G., Kim J. S., Shin H. K., The Binet Formula and the Representation of k-Generalized Fibonacci Numbers, Fibonacci Quarterly, 2001, 39 (2), 158-164.
 
[15]  Lee G. Y., Lee S. G., Shin H. K., On the k-Ggeneralized Fibonacci Matrix Qk, Linear Algebra Applications, 1997, 251, 73-88.
 
[16]  Slone NJA, The On-Line Encyclopedia of Integer Sequences, (2006), www.research.att.com/~njas/sequences/.
 
[17]  Walton J. E. and Horadam A. F., Some Further Identities for the Generalized Fibonacci sequence, Fibonacci Quarterly, 1974, 12 (3), 272-280.
 
[18]  Zai L. J. and Sheng L. J., Some Properties of the Generalization of the Fibonacci Sequence, Fibonacci Quarterly, 1987, 25 (2), 111-117.
 
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Article

Generating Function for M(m,n)

1Senior Lecturer, Department of Mathematics, Raozan University College, Bangladesh

2Premier University, Chittagong, Bangladesh


Turkish Journal of Analysis and Number Theory. 2014, 2(4), 125-129
DOI: 10.12691/tjant-2-4-4
Copyright © 2014 Science and Education Publishing

Cite this paper:
Sabuj Das, Haradhan Kumar Mohajan. Generating Function for M(m,n). Turkish Journal of Analysis and Number Theory. 2014; 2(4):125-129. doi: 10.12691/tjant-2-4-4.

Correspondence to: Haradhan  Kumar Mohajan, Premier University, Chittagong, Bangladesh. Email: haradhan_km@yahoo.com

Abstract

This paper shows that the coefficient of x in the right hand side of the equation is an algebraic relation in terms of z. The exponent of z represents the crank of partitions of a positive integral value of n and also shows that the sum of weights of corresponding partitions of n is the sum of ordinary partitions of n and it is equal to the number of partitions of n with crank m. This paper shows how to prove the Theorem “The number of partitions π of n with crank C(π)=m is M(m,n) for all n>1.”

Keywords

References

[1]  Andrews, G.E., The Theory of Partitions, Encyclopedia of Mathematics and its Application, vol. 2 (G-c, Rotaed) Addison-Wesley, Reading, mass, 1976 (Reissued, Cambridge University, Press, London and New York 1985). 1985.
 
[2]  Andrews, G.E. and Garvan, F.G., Dyson’s Crank of a Partition, Bulletin (New series) of the American Mathematical Society, 18(2): 167-171. 1988.
 
[3]  Atkin, A.O.L. and Swinnerton-Dyer, P., Some Properties of Partitions, Proc. London Math. Soc. 3(4): 84-106. 1954.
 
[4]  Garvan, F.G., Ramanujan Revisited, Proceeding of the Centenary Conference, University of Illinois, Urban-Champion. 1988.
 
[5]  Garvan, F.G.. Dyson’s Rank Function and Andrews’ spt-function, University of Florida, Seminar Paper Presented in the University of Newcastle on 20 August 2013. 2013.
 

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

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[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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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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[5]  J. Hua, B.-Y. Xi, and F. Qi, Hermite-Hadamard type inequalities for geometric-arithmetically s-convex functions, Commun. Korean Math. Soc. 29 (2014), no. 1, 51-63.
 
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[6]  J. Hua, B.-Y. Xi, and F. Qi, Some new inequalities of Simpson type for strongly s-convex functions, Afrika Mat. (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

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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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