Certain Properties of Apostol-Type Hermite-Based- Frobenius-Genocchi Polynomials

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DOI: 10.46793/KgJMat2106.859K


This paper is well designed to set-up some new identities related to generalized Apostol-type Hermite-based-Frobenius-Genocchi polynomials and by applying the generating functions, we derive some implicit summation formulae and symmetric identities. Further a relationship between Array-type polynomials, Apostol-type Bernoulli polynomials and generalized Apostol-type Frobenius-Genocchi polynomials is also established.


Hermite polynomials, Frobenius-Genocchi polynomials, Apostol-type Hermite-based Genocchi polynomials.


[1]   E. T. Bell, Exponential polynomials, Ann. of Math. 35 (1934), 258–277.

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

[3]   B. Kurt and Y. Simsek, On the generalized Apostol-type Frobenius-Euler polynomials, Adv. Difference Equ. 2013(1) (2013), 1–9.

[4]   D. S. Kim and T. Kim, Some identities of degenerate special polynomials, Open Math. 13 (2015), 380–389.

[5]   W. A. Khan, S. Araci, M. Acikgoz and H. Haroon, A new class of partially degenerate Hermite-Genocchi polynomials, J. Nonlinear Sci. Appl. 10 (2017), 5072–5081.

[6]   W. A. Khan, Some properties of the generalized Apostol-type Hermite-based polynomials, Kyungpook Math. J. 55 (2015), 597–614.

[7]   W. A. Khan and H. Haroon, Some symmetric identities for the generalized Bernoulli, Euler and Genocchi polynomials associated with Hermite polynomials, Springerplus 5(1) (2016), 1–21.

[8]   Q. M. Luo, B. N. Guo, F. Qi and L. Debnath, Generalization of Bernoulli numbers and polynomials, Int. J. Math. Math. Sci. 59 (2003), 3769–3776.

[9]   Q. M. Luo, B. N. Guo, F. Qi and L. Debnath, Generalization of Euler numbers and polynomials, Int. J. Math. Math. Sci. 61 (2003), 3893–3901.

[10]   M. A. Pathan and W. A. Khan, Some implicit summation formulas and symmetric identities for the generalized Hermite-Bernoulli polynomials, Mediterr. J. Math. 12 (2015), 679–695.

[11]   M. A. Pathan and W. A. Khan, A new class of generalized polynomials associated with Hermite and Euler polynomials, Mediterr. J. Math. 13(3) (2016), 913–928.

[12]   M. A. Pathan and W. A. Khan, Some new classes of generalized Hermite-based Apostol-Euler and Apostol-Genocchi polynomials, Fasc. Math. 55(1) (2015), 153–170.

[13]   Y. Simsek, Generating functions for generalized Striling type numbers, Array type polynomials, Eulerian type polynomials and their applications, Fixed Point Theory Appl. (2013), DOI 1186/1687-1812-2013-87.

[14]   Y. Simsek, Generating Functions for q-Apostol type Frobenius-Euler numbers and polynomials, Axioms 1 (2012), 395–403.

[15]   H. M. Srivastava and H. L. Manocha, A Treatise on Generating Functions, Ellis Horwood Limited. Co. New York, 1984.

[16]   H. M. Srivastava, M. Garg and S. A. Choudhari, New generalization of the Bernoulli and related polynomials, Russ. J. Math. Phy. 17 (2010), 251–261.

[17]   B. Y. Yaşar and M. A. Özarslan, Frobenius-Euler and Frobenius-Genocchi polynomials and their differential equations, New Trends Math. Sci. 3(2) (2015), 172–180.