Back
 AM  Vol.6 No.9 , August 2015
New Extension of Unified Family Apostol-Type of Polynomials and Numbers
Abstract: The purpose of this paper is to introduce and investigate new unification of unified family of Apostol-type polynomials and numbers based on results given in [1] [2]. Also, we derive some properties for these polynomials and obtain some relationships between the Jacobi polynomials, Laguerre polynomials, Hermite polynomials, Stirling numbers and some other types of generalized polynomials.
Cite this paper: El-Desouky, B. and Gomaa, R. (2015) New Extension of Unified Family Apostol-Type of Polynomials and Numbers. Applied Mathematics, 6, 1495-1505. doi: 10.4236/am.2015.69134.
References

[1]   Srivastava, H.M., Garg, M. and Choudhary, S. (2010) A New Generalization of the Bernoulli and Related Polynomials, Russian. Journal of Mathematical Physics, 17, 251-261.
http://dx.doi.org/10.1134/S1061920810020093

[2]   Srivastava, H.M., Garg, M. and Choudhary, S. (2011) Some New Families of Generalized Euler and Genocchi Polynomials. Taiwanese Journal of Mathematics, 15, 283-305.

[3]   Srivastava, H.M. and Pintér, á. (2004) Remarks on Some Relationships between the Bernoulli and Euler Polynomials. Applied Mathematics Letters, 17, 375-380.
http://dx.doi.org/10.1016/S0893-9659(04)90077-8

[4]   Luo, Q.-M. and Srivastava, H.M. (2005) Some Generalizations of the Apostol Bernoulli and Apostol Euler Polynomials. Journal of Mathematical Analysis and Applications, 308, 290-302.
http://dx.doi.org/10.1016/j.jmaa.2005.01.020

[5]   Luo, Q.-M. (2006) Apostol-Euler Polynomials of Higher Order and Gaussian Hypergeometric Functions. Taiwanese Journal of Mathematics, 10, 917-925.

[6]   Natalini, P. and Bernardini, A. (2003) A Generalization of the Bernoulli Polynomials. Journal of Applied Mathematics, 153-163.
http://dx.doi.org/10.1155/s1110757x03204101

[7]   Tremblay, R., Gaboury, S. and Fugère, B.-J. (2011) A New Class of Generalized Apostol-Bernoulli Polynomials and Some Analogues of the Srivastava-Pintér Addition Theorem. Applied Mathematics Letters, 24, 1888-1893.
http://dx.doi.org/10.1016/j.aml.2011.05.012

[8]   Luo, Q.-M., Guo, B.-N., Qui, F. and Debnath, L. (2003) Generalizations of Bernoulli Numbers and Polynomials. International Journal of Mathematics and Mathematical Sciences, 59, 3769-3776.
http://dx.doi.org/10.1155/S0161171203112070

[9]   El-Desouky, B.S. and Gomaa, R.S. (2014) A New Unified Family of Generalized Apostol-Euler, Bernoulli and Genocchi Polynomials. Applied Mathematics and Computation, 247, 695-702.
http://dx.doi.org/10.1016/j.amc.2014.09.002

[10]   Kurt, B. (2010) A Further Generalization of Bernoulli Polynomials and on 2D-Bernoulli Polynomials . Applied Mathematical Sciences, 47, 2315-2322.

[11]   Kurt, B. (2013) Some Relationships between the Generalized Apostol-Bernoulli and Apostol-Euler Polynomials. Turkish Journal of Analysis and Number Theory, 1, 54-58.

[12]   Ozden, H. and Simsek, Y. (2014) Modification and Unification of the Apostol-Type Numbers and Polynomials and Their Applications. Applied Mathematics and Computation, 235, 338-351.
http://dx.doi.org/10.1016/j.amc.2014.03.004

[13]   Apostol, T.M. (1951) On the Lerch Zeta Function. Pacific Journal of Mathematics, 1, 161-167.
http://dx.doi.org/10.2140/pjm.1951.1.161

[14]   Dere, R., Simsek, Y. and Srivastava, H.M. (2013) A Unified Presentation of Three Families of Generalized Apostol Type Polynomials Based upon the Theory of the Umbral Calculus and the Umbral Algebra. Journal of Number Theory, 133, 3245-3263.
http://dx.doi.org/10.1016/j.jnt.2013.03.004

[15]   Karande, B.K. and Thakare, N.K. (1975) On the Unification of Bernoulli and Euler Polynomials. Indian Journal of Pure and Applied Mathematics, 6, 98-107.

[16]   Luo, Q.-M. (2004) On the Apostol Bernoulli Polynomials. Central European Journal of Mathematics, 2, 509-515.
http://dx.doi.org/10.2478/BF02475959

[17]   Nörlund, N.E. (1924) Vörlesunge über differezerechnung. Springer-Verlag, Berlin.
http://dx.doi.org/10.1007/978-3-642-50824-0

[18]   Carlitz, L. (1962) Some Generalized Multiplication Formulae for the Bernoulli Polynomials and Related Functions. Monatshefte für Mathematik, 66, 1-8.

[19]   Comtet, L. (1972) Nombers de Stirling generaux et fonctions symetriques. Comptes Rendus de l’Académie des Sciences (Series A), 275, 747-750.

[20]   Gould, H.W. (1960) Stirling Number Representation Problems. Proceedings of the American Mathematical Society, 11, 447-451.
http://dx.doi.org/10.1090/S0002-9939-1960-0114767-8

[21]   Srivastava, H.M. and Choi, J. (2001) Series Associated with the Zeta and Related Functions. Kluwer Academic, Dordrecht.
http://dx.doi.org/10.1007/978-94-015-9672-5

[22]   Charalambides, C.A. (2005) Generalized Stirling and Lah Numbers. In: Charalambides, C.A., Ed., Combinatorial Methods in Discrete Distributions, John Wiley & Sons, Inc., Hoboken, 121-158.

[23]   özarslan, M.A. and Bozer, M. (2013) Unified Bernstein and Bleimann-Butzer-Hahn Basis and Its Properties. Advances in Difference Equations, 2013, 55.
http://dx.doi.org/10.1186/1687-1847-2013-55

 
 
Top