Special Numbers on Analytic Functions
Author(s)
Yilmaz Simsek
Affiliation(s)
Department of Mathematics, Faculty of Science, University of Akdeniz, Antalya, Turkey.
Department of Mathematics, Faculty of Science, University of Akdeniz, Antalya, Turkey.
ABSTRACT
The aim of this paper is to give some analytic functions which are related to the generating functions for the central factorial numbers. By using these functions and p-adic Volkenborn integral, we derive many new identities associated with the Bernoulli and Euler numbers, the central factorial numbers and the Stirling numbers. We also give some remarks and comments on these analytic functions, which are related to the generating functions for the special numbers.
KEYWORDS
Bernoulli Numbers, Euler Numbers, The Central Factorial Numbers, Array Polynomials, Stirling Numbers of the First Kind and the Second Kind, Generating Function, Functional Equation, Analytic Functions
Bernoulli Numbers, Euler Numbers, The Central Factorial Numbers, Array Polynomials, Stirling Numbers of the First Kind and the Second Kind, Generating Function, Functional Equation, Analytic Functions
Cite this paper
Simsek, Y. (2014) Special Numbers on Analytic Functions. Applied Mathematics, 5, 1091-1098. doi: 10.4236/am.2014.57102.
Simsek, Y. (2014) Special Numbers on Analytic Functions. Applied Mathematics, 5, 1091-1098. doi: 10.4236/am.2014.57102.
References
[1] Chang, C.-H. and Ha, C.-W. (2006) A Multiplication Theorem for the Lerch Zeta Function and Explicit Representations of the Bernoulli and Euler polynomials. Journal of Mathematical Analysis and Applications, 315, 758-767.
http://dx.doi.org/10.1016/j.jmaa.2005.08.013 [2] Cigler, J. Fibonacci Polynomials and Central Factorial Numbers. Preprint.
http://homepage.univie.ac.at/johann.cigler/preprints/central-factorial.pdf [3] Comtet, L. (1974) Advanced Combinatorics: The Art of Finite and Infinite Expansions. Reidel, Dordrecht and Boston, (Translated from the French by J. W. Nienhuys). [4] Kim, T. (2002) q-Volkenborn integration. Russian Journal of Mathematical Physics, 19, 288-299. [5] Kim, T. (2006) q-Euler Numbers and Polynomials Associated with p-Adic q-Integral and Basic q-zeta Function. Trends in International Mathematics and Science Study, 9, 7-12. [6] Jang, L.C. and Kim, T. (2008) A New Approach to q-Euler Numbers and Polynomials. Journal of Concrete and Applicable Mathematics, 6, 159-168. [7] Kim, D.S. and Kim, T. (2013) Daehee Numbers and Polynomials. Applied Mathematical Sciences, 7, 5969-5976. [8] Kim, D.S., Kim, T. and Seo, J. (2013) A Note on Changhee Numbers and Polynomials. Advanced Studies in Theoretical Physics, 7, 993-1003. [9] Rainville, E.D. (1960) Special Functions. The Macmillan Company, New York. [10] Simsek, Y. (2010) On q-Deformed Stirling numbers. International Journal of Computer Mathematics, 15, 70-80. [11] Simsek, Y. (2010) Complete sum of products of (h,q)-Extension of Euler Polynomials and Numbers. Journal of Difference Equations and Applications, 16, 1331-1348.
http://dx.doi.org/10.1080/10236190902813967 [12] Simsek, Y. (2013) Identities Associated with Generalized Stirling Type Numbers and Eulerian Type Polynomials. Mathematical and Computational Applications, 18, 251-263. [13] Simsek, Y. (2013) Generating Functions for Generalized Stirling type Numbers, Array Type Polynomials, Eulerian Type Polynomials and Their Applications. Fixed Point Theory and Applications, 87, 343-1355. [14] Schikhof, W.H. (1984) Ultrametric Calculus: An Introduction to p-Adic Analysis. Cambridge Studies in Advanced Mathematics 4, Cambridge University Press, Cambridge. [15] Srivastava, H.M. (2011) Some Generalizations and Basic (or q-) Extensions of the Bernoulli, Euler and Genocchi Polynomials. Applied Mathematics & Information Sciences, 5, 390-444. [16] Srivastava, H.M., Ozarslan, M.A. and Kaanoglu, C. (2010) Some Families of Generating Functions for a Certain Class of Three-Variable Polynomials. Integral Transforms and Special Functions, 21, 885-896.
http://dx.doi.org/10.1080/10652469.2010.481439 [17] Srivastava, H.M. and Choi, J. (2012) Zeta and q-Zeta Functions and Associated Series and Integrals. Elsevier Science Publishers, Amsterdam, London and New York. [18] Srivastava, H.M., Kim, T. and Simsek, Y. (2005) q-Bernoulli Numbers and Polynomials Associated with Multiple q-Zeta Functions and Basic L-Series. Russian Journal of Mathematical Physics, 12, 241-268. [19] Srivastava, H.M. and Liu, G.-D. (2009) Some Identities and Congruences Involving a Certain Family of Numbers. Russian Journal of Mathematical Physics, 16, 536-542.
http://dx.doi.org/10.1134/S1061920809040086
[1] Chang, C.-H. and Ha, C.-W. (2006) A Multiplication Theorem for the Lerch Zeta Function and Explicit Representations of the Bernoulli and Euler polynomials. Journal of Mathematical Analysis and Applications, 315, 758-767.
http://dx.doi.org/10.1016/j.jmaa.2005.08.013 [2] Cigler, J. Fibonacci Polynomials and Central Factorial Numbers. Preprint.
http://homepage.univie.ac.at/johann.cigler/preprints/central-factorial.pdf [3] Comtet, L. (1974) Advanced Combinatorics: The Art of Finite and Infinite Expansions. Reidel, Dordrecht and Boston, (Translated from the French by J. W. Nienhuys). [4] Kim, T. (2002) q-Volkenborn integration. Russian Journal of Mathematical Physics, 19, 288-299. [5] Kim, T. (2006) q-Euler Numbers and Polynomials Associated with p-Adic q-Integral and Basic q-zeta Function. Trends in International Mathematics and Science Study, 9, 7-12. [6] Jang, L.C. and Kim, T. (2008) A New Approach to q-Euler Numbers and Polynomials. Journal of Concrete and Applicable Mathematics, 6, 159-168. [7] Kim, D.S. and Kim, T. (2013) Daehee Numbers and Polynomials. Applied Mathematical Sciences, 7, 5969-5976. [8] Kim, D.S., Kim, T. and Seo, J. (2013) A Note on Changhee Numbers and Polynomials. Advanced Studies in Theoretical Physics, 7, 993-1003. [9] Rainville, E.D. (1960) Special Functions. The Macmillan Company, New York. [10] Simsek, Y. (2010) On q-Deformed Stirling numbers. International Journal of Computer Mathematics, 15, 70-80. [11] Simsek, Y. (2010) Complete sum of products of (h,q)-Extension of Euler Polynomials and Numbers. Journal of Difference Equations and Applications, 16, 1331-1348.
http://dx.doi.org/10.1080/10236190902813967 [12] Simsek, Y. (2013) Identities Associated with Generalized Stirling Type Numbers and Eulerian Type Polynomials. Mathematical and Computational Applications, 18, 251-263. [13] Simsek, Y. (2013) Generating Functions for Generalized Stirling type Numbers, Array Type Polynomials, Eulerian Type Polynomials and Their Applications. Fixed Point Theory and Applications, 87, 343-1355. [14] Schikhof, W.H. (1984) Ultrametric Calculus: An Introduction to p-Adic Analysis. Cambridge Studies in Advanced Mathematics 4, Cambridge University Press, Cambridge. [15] Srivastava, H.M. (2011) Some Generalizations and Basic (or q-) Extensions of the Bernoulli, Euler and Genocchi Polynomials. Applied Mathematics & Information Sciences, 5, 390-444. [16] Srivastava, H.M., Ozarslan, M.A. and Kaanoglu, C. (2010) Some Families of Generating Functions for a Certain Class of Three-Variable Polynomials. Integral Transforms and Special Functions, 21, 885-896.
http://dx.doi.org/10.1080/10652469.2010.481439 [17] Srivastava, H.M. and Choi, J. (2012) Zeta and q-Zeta Functions and Associated Series and Integrals. Elsevier Science Publishers, Amsterdam, London and New York. [18] Srivastava, H.M., Kim, T. and Simsek, Y. (2005) q-Bernoulli Numbers and Polynomials Associated with Multiple q-Zeta Functions and Basic L-Series. Russian Journal of Mathematical Physics, 12, 241-268. [19] Srivastava, H.M. and Liu, G.-D. (2009) Some Identities and Congruences Involving a Certain Family of Numbers. Russian Journal of Mathematical Physics, 16, 536-542.
http://dx.doi.org/10.1134/S1061920809040086