Some Properties of the Generalized Apostol Type Hermite-Based Polynomials

• Journal title : Kyungpook mathematical journal
• Volume 55, Issue 3,  2015, pp.597-614
• Publisher : Department of Mathematics, Kyungpook National University
• DOI : 10.5666/KMJ.2015.55.3.597
Title & Authors
Some Properties of the Generalized Apostol Type Hermite-Based Polynomials

Abstract
In this paper, we study some properties of the generalized Apostol type Hermite-based polynomials. which extend some known results. We also deduce some properties of the generalized Apostol-Bernoulli polynomials, the generalized Apostol-Euler polynomials and the generalized Apostol-Genocchi polynomials of high order. Numerous properties of these polynomials and some relationships between $\small{F_n{^{({\alpha})}}(x;{\lambda};{\mu};{\nu};c)}$ and $\small{_HF_n{^{({\alpha})}}(x;{\lambda};{\mu};{\nu};c)}$ are established. Some implicit summation formulae and general symmetry identities are derived by using different analytical means and applying generating functions.
Keywords
Hermite polynomials;generalized Apostol type Hermite-based polynomials;summation formulae;
Language
English
Cited by
1.
A new generalization of Apostol type Hermite–Genocchi polynomials and its applications, SpringerPlus, 2016, 5, 1
References
1.
A. Erdelyi, W. Magnus, F. Oberhettinger and F. Tricomi, Higher transcendental functions, vols. 1-3, (1953).

2.
D. Q. Lu and Q. M. Luo, Some properties of the generalized Apostol tpe polynomials, Boundary Value Problems., (2013),2013:64.

3.
E. T. Bell, Exponential polynomials, Ann. of Math., 35(1943), 258-277.

4.
E. R. Hansen, A table of series and products, Printice Hall, Englewood Cliffs, NJ, (1975).

5.
F. Magnus, W. Oberhettinger and R. P. Soni, Some formulas and theorem for the special functions of mathematical physics, Third enlarged edition, Springer-Verlag, New York, (1966).

6.
G. Dattoli, S. Lorenzutt and C. Cesarano, Finite sums and generalized forms of Bernoulli polynomials,Rendiconti di Mathematica, 19(1999), 385-391.

7.
H. M. Srivastava and H. L .Manocha, A treatise on generating functions, Ellis Horwood Limited. Co. New York, (1984).

8.
H. M. Srivastava, M. Garg and S. Choudhary, A new generalization of the Bernoulli and related polynomials, Russian J. Math. Phys, 17(2010), 251-261.

9.
H. M. Srivastava, M. Garg and S. Choudhary, Some new families of generalized Euler and genochhi polynomials, Taiwanese J. Math, 15(1)(2011), 283-305.

10.
H. Yang, An identity of symmetry for the Bernoulli polynomials, Discrete Math. (2007)dol:10:10,16/j.disc 2007.03.030.

11.
L. Comlet, The art of finite and infinite expansions, (Translated from french by J. W. Nilenhuys), Reidel, Dordrecht, 1974.

12.
M. Abramowitz and I. A. Stegun, Handbook of mathematical functions with formulas graphs and mathematical tables, National Bureau of Standards, Washington, DC, 1964.

13.
M. A. Pathan, A new class of generalized Hermite-Bernoulli polynomials, Georgian Mathematical Journal, 19(2012), 559-573.

14.
M. A. Pathan and W. A. Khan, Some implicit summation formulas and symmetric identities for the generalized Hermite-Bernoulli polynomials, Mediterr.J.Math.(2014),DOI 10.1007/s00009-014-0423-0, Springer Basel 2014.

15.
Q. M. Luo, Apostol Euler polynomials of higher order and gaussian hypergeometric functions, Taiwanese J. Math., 10(4)(2006), 917-925.

16.
Q. M. Luo, q-extensions for the Apostol-Genocchi polynomials, Gen. Math., 17(2)(2009), 113-125.

17.
Q. M. Luo, Extensions for the Genocchi polynomials and its fourier expansions and integral representations, Osaka j. Math., 48(2011), 291-310.

18.
Q. M. Luo and H. M. Srivastava Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials, J. Math. Anal. Appl, 308(1)(2005), 290-302.

19.
Q. M. Luo and H. M. Srivastava, Some generalizations of the Apostol Genocchi polynomials and the stirling number of the second kind, Appl. Math. Comput, 217(2011), 5702-5728.

20.
Q. M. Luo and H. M. Srivastava, Some relationships between the Apostol-Bernoulli and Apostol-Euler polynomials, Comput. Math. Appl, 51(2006), 631-642.

21.
S. Khan, M. A. Pathan, N. A. M. Hassan, G. Yasmin , Implicit summation formula for Hermite and related polynomials, J. Math. Anal. Appl, 344(2008), 408-416.

22.
S. L. Yang and Z. K. Qiao, Some symmetry identities for the Euler polynomials, J. Math. Resrch. Exposition, 30(3)(2010), 457-464.

23.
Y. Luke, The special functions and their approximations, vols, 1-2, 1969.

24.
Yu. A. Brychkov, On multiple sums of special functions, Integral Trans. Spec. Func., 21(12)(2010), 877-884.

25.
Z. Zhang and H. Yang, Several identities for the generalized Apostol Bernoulli polynomials, Computers and Mathematics with Applications, 56(2008), 2993-2999.