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Research Article

Open Access Peer-reviewed

Burak KURT^{ }

Received September 28, 2019; Revised October 26, 2019; Accepted November 28, 2019

Many kinds of generalizations of these polynomials and numbers have been presented in the literature (see [1,2]). Bayad and Hamahata in [1] introduced and investigated the poly-Bernoulli polynomials and proved some relations for these polynomials. Cenkci and Kamatsu in [3] are defined q-parameter poly-Bernoulli numbers. They proved some relations between these polynomials. Kim et al. in ([4,5]) defined poly-Genocchi polynomials and gave the some properties for the multiple-poly-Bernoulli numbers and multiple-zeta values. We introduce the Hermite-based poly-Genocchi polynomials with a q-parameter. After, we give and investigate some properties and identities for these polynomials. Furthermore, we prove closed formula and two explicit relations.

*As usual, throughout this paper, N denotes the set of natural numbers, **denotes the set of nonnegative integers, Z denotes the set of integer numbers, R denotes the set of real numbers and C denotes the complex numbers.*

*In the usual nota**t**ions, let** ** and ** denote respectively, the classical Bernoulli polynomials, the classical Euler polynomials and the classical Genocchi polynomials defined by the following generating functions;*

(1) |

(2) |

and

(3) |

*Also, let x=0,*

Where and are respectively, the classical Bernoulli numbers, the classical Euler numbers and the classical Genoccchi numbers.

*The 2-variable Hermite-Kampé de Feriét polynomials are defined by (see *^{ 6, 7, 8}*)*

(4) |

*Let ** **the k-th polylogarithm function is defined by (see *^{ 1, 6, 9}*)*

(5) |

when k=1, In the case are the rational functions:

*Further information about poly-logarithm function and polynomials (see *^{ 1, 4}*).*

*Cenkci et al. in *^{ 3}* defined the weighted Stirling numbers of the second kind as*

(6) |

*Duran et al. in *^{ 6}* defined the Hermite-based **𝜆**-Stirling polynomials of the second kind as*

(7) |

*The special values of the (6) are given in *^{ 6}*. Let** ** We define the Hermite-based poly-Genocchi polynomials with a q-parameter by the following generating functions:*

(8) |

*For x=y=0, we get ** which is called a new class of the Hermite-based poly-Genocchi number with a q-parameter. Some special cases of ** are following remarks.*

**Remark.*** For y=0, we have** ** called the Hermite-based poly-Genocchi polynomials with a q-parameter.*

**Remark. ***For q=1, ** **reduces to the Hermite-based poly-Genocchi polynomials.*

**Remark.*** For q=1 and y=0, ** **reduces to the poly-Genocchi polynomials.*

**Remark.*** When q=k=1 and y=0, we obtain the classical Genocchi polynomials.*

*Srivastava and Srivastava et al. in (*^{ 10, 11}*) investigated some properties and proved some theorems for the Bernoulli, Euler and Genocchi polynomials. D. S. Kim et al. in (*^{ 12, 13}*) introduced the poly-Bernoulli polynomials and gave some recurrences relations and idebtities. Cenkci et al. in *^{ 3}* gave the poly-Bernoulli polynomials with a q-parameter. Kurt *^{ 7}* gave the poly-Genocchi polynomials with a q-parameter. Duran et al. in (*^{ 6, 14}*) considered the (p,q)-Hermite polynomials and the (p,q)-Euler polynomials.*

*In this section, we give some basic identites and relations for the Hermite-based poly-Genocchi polynomial with a q-parameter. Further we give closed formula and explicit relation for these polynomials.*

**Theorem. ***The Hermite-based poly-Genocchi polynomials** with a q-parameter satisfy following relation:*

and

*The proof of this theorem is easily obtain from the Hermite-based poly-Genocchi polynomials definition.*

**Theorem. ***The following relation holds true:*

(9) |

*Proof. By (4) and (8), we write as*

*L. H. S of this equation is*

(10) |

*R. H. S of this equation is*

(11) |

*From (10) and (11), we obtain (9).*

**Theorem.*** There is the following relation between the Hermite-based poly-Genocchi polynomials with a q-parameter and the Euler polynomials:*

(12) |

*Proof. By (2) and (8), we write*

*By using Cauchy product and comparing the **coefficients of** ** we have (12).*

**Theorem.*** There is the following relation between the Hermite-based poly-Genocchi polynomials with a q-parameter and the Bernoulli polynomials:*

(13) |

*Proof. From (1) and (8), we write as*

*By using Cauchy product and comparing the coefficients of** ** we have (13).*

**Theorem.*** The following relation holds true:*

(14) |

*Proof. By (8),*

*From here, we have (14).*

**Theorem.*** There is following relationship between the Hermite-based poly-Genocchi polynomials with a q-parameter and the Stirling numbers of the second kind as:*

(15) |

*Proof. By (16) and (8), we write as:*

*L. H. S. of this equation is*

(16) |

*R. H. S. of this equation*

*By using Cauchy product, we have*

(17) |

*From (16) and (17), we get (15).*

**Theorem. ***The following relation holds true:*

(18) |

*Proof. By (8),*

(19) |

*We replace t by t+u in (19)*

*From this equation, we write as*

(20) |

*In the last equation, we replace x by v, we get*

(21) |

*By (12) and (13), we write*

(22) |

*Now, by applying the following known series identity [*^{ 19}*, p. 52, Eq. 1. 6 (2)]*

(23) |

in the right hand side of (23), we get

(24) |

*Finally, upon first replacing n by n-p and m by m-r by using the Cauchy product in the left hand side of the above equation (24) and comparing the coefficients of ** and ** On both sides of the resulting equation, we have (18).*

**Theorem ***(Closed Formula) The following relation holds true:*

(25) |

*Proof. By replacing k by (-k) in (8), we get*

(26) |

*For **𝜆**=1 and j=2 in (8), we get*

(27) |

*We put the equation (8) and (27) in (26). We have*

*From the last equation, comparing the coefficients of ** and** ** we have (25).*

*The present investigation was supported by the Scientific Research Project Administration of Akdeniz University.*

[1] | A. Bayad and Y. Hamahata, “Polylogarithms and poly-Bernoulli polynomials,” Kyushu. J. Math., 65, pp. 15-34, 2011. | ||

In article | View Article | ||

[2] | R. Sanchez-Peregrino, “Closed formula for Poly-Bernoulli numbers,” Fibonacci Quart., 40, pp. 362-364, 2002. | ||

In article | |||

[3] | M. Cenkci and T. Komatsu, “Poly-Bernoulli numbers and polynomials with a q-parameter,” J. Number Theory, 152, pp. 38-54, 2015. | ||

In article | View Article | ||

[4] | T. Kim, S. Y. Jang and J. J. Seo, “A note on poly-Genocchi numbers and polynomials,” Appl. Math. Sci., 18(96), 4775-4781, 2014. | ||

In article | View Article | ||

[5] | T. Kim, “Multiple zeta values and their application,” Lecture Notes in Number Theory, 31-95, 1996. | ||

In article | |||

[6] | U. Duran, M Acikgoz and S. Araci, “Hermite based poly-Bernoulli polynomials with q-parameter,” Advanced Stud. In Comptem., 28(2), pp. 285-296, 2018. | ||

In article | View Article | ||

[7] | B. Kurt, “Identities and relation on the Poly-Genocchi polynomials with a q-parameter,” J. Inequa. Special Func., 9(1), pp. 1-8, 2018. | ||

In article | |||

[8] | M. A. Ozarslan, Hermite-based unified Apostol-Bernoulli, Euler and Genocchi polynomials,” Advances in Diff. Equa., 2013.2013.1847. | ||

In article | View Article | ||

[9] | Y. Hamahata, “Poly-Euler polynomials and Arakawa-Kaneko type zeta functions,” Functione et. App. Commentarii Mathematica, 51(1), pp. 7-27, 2014. | ||

In article | View Article | ||

[10] | H. M. Srivastava, “Some generalization and basic (or q-) extensition of the Bernoulli, Euler and Genocchi polynomials,” App. Math. Information Sci., 5, pp. 390-444, 2011. | ||

In article | |||

[11] | H. M. Srivastava and J. Choi, “Series associated with the zeta and related functions,” Kluver Academic Pub., Dordrect, Boston and London, 2001. | ||

In article | View Article | ||

[12] | D. S. Kim and T. Kim, “Higher-order Frobenius-Euler and Poly-Bernoulli mixed-type polynomials,” Advances in Diff. Equa., 2013.2013.251. | ||

In article | View Article | ||

[13] | D. S. Kim and T. Kim, “Higher order Bernoulli and poly-Bernoulli mixed type polynomials,” Georgian Math. J., 22, pp. 265-272, 2015. | ||

In article | View Article | ||

[14] | U. Duran and M. Acikgoz, “On (p,q)-Euler numbers and polynomials associated with (p,q)-Volkenborn integral,” Int. J. of Number Theory, 14(1), pp. 241-253, 2018. | ||

In article | View Article | ||

[15] | A. Bayad and T. Kim, “Higher recurrences for Apostol-Bernoulli-Euler polynomials,” Russ. J. of Math. Phys., 19(1), pp. 1-10, 2012. | ||

In article | View Article | ||

[16] | U. Duran, M Acikgoz, A. Esi and S. Araci, “A note on the (p,q)-Hermite polynomials,” App. Math. And Information Sciences, 12, pp. 227-231, 2018. | ||

In article | View Article | ||

[17] | H. Jolany, R. B. Corcino and T. Komatsu, “More properties on multi-Poly-Euler polynomials,” Bull. Soc. Math. Mex., 21, pp. 149-162, 2015. | ||

In article | View Article | ||

[18] | D. S. Kim and T. Kim, Hermite and Poly-Bernoulli mixed type polynomials,” Advances in Diff. Equa., 2013. 2013. 343. | ||

In article | View Article | ||

[19] | D. S. Kim, T. Kim, D. V. Dolgy and S. H. Rim, “Some new identities of Bernoulli, Euler and Hermite polynomials arising from umbral calculus,” Advances in Diff. Equa., 2013. 2013. 73. | ||

In article | View Article | ||

[20] | D. S. Kim and T. Kim, “Some identities of Frobenius-Euler polynomials arising from umbral calculus,” Advances in Diff. Equa., 2012. 2012. 196. | ||

In article | View Article | ||

[21] | D. S. Kim and T. Kim, “A note on poly-Bernoulli and higher order poly-Bernoulli polynomials,” Russ. J. Math., 22(1), pp. 26-33, 2015. | ||

In article | View Article | ||

[22] | B. Kurt and Y. Simsek, “On the generalized Apostol-type Frobenius-Euler polynomials,” Advances in Diff. Equa., 2013. 2013. 1. | ||

In article | View Article | ||

[23] | H. M. Srivastava and H. L. Manocho, “A Treatise on generating functions,” Halsted Press, John Willey and sums, New York, Chichester, Brisbane and Toronto, 1984. | ||

In article | |||

Published with license by Science and Education Publishing, Copyright © 2019 Burak KURT

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

Burak KURT. On the Hermite-Based Poly-Genocchi Polynomials with a q-Parameter. *Turkish Journal of Analysis and Number Theory*. Vol. 7, No. 6, 2019, pp 145-149. http://pubs.sciepub.com/tjant/7/6/2

KURT, Burak. "On the Hermite-Based Poly-Genocchi Polynomials with a q-Parameter." *Turkish Journal of Analysis and Number Theory* 7.6 (2019): 145-149.

KURT, B. (2019). On the Hermite-Based Poly-Genocchi Polynomials with a q-Parameter. *Turkish Journal of Analysis and Number Theory*, *7*(6), 145-149.

KURT, Burak. "On the Hermite-Based Poly-Genocchi Polynomials with a q-Parameter." *Turkish Journal of Analysis and Number Theory* 7, no. 6 (2019): 145-149.

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[1] | A. Bayad and Y. Hamahata, “Polylogarithms and poly-Bernoulli polynomials,” Kyushu. J. Math., 65, pp. 15-34, 2011. | ||

In article | View Article | ||

[2] | R. Sanchez-Peregrino, “Closed formula for Poly-Bernoulli numbers,” Fibonacci Quart., 40, pp. 362-364, 2002. | ||

In article | |||

[3] | M. Cenkci and T. Komatsu, “Poly-Bernoulli numbers and polynomials with a q-parameter,” J. Number Theory, 152, pp. 38-54, 2015. | ||

In article | View Article | ||

[4] | T. Kim, S. Y. Jang and J. J. Seo, “A note on poly-Genocchi numbers and polynomials,” Appl. Math. Sci., 18(96), 4775-4781, 2014. | ||

In article | View Article | ||

[5] | T. Kim, “Multiple zeta values and their application,” Lecture Notes in Number Theory, 31-95, 1996. | ||

In article | |||

[6] | U. Duran, M Acikgoz and S. Araci, “Hermite based poly-Bernoulli polynomials with q-parameter,” Advanced Stud. In Comptem., 28(2), pp. 285-296, 2018. | ||

In article | View Article | ||

[7] | B. Kurt, “Identities and relation on the Poly-Genocchi polynomials with a q-parameter,” J. Inequa. Special Func., 9(1), pp. 1-8, 2018. | ||

In article | |||

[8] | M. A. Ozarslan, Hermite-based unified Apostol-Bernoulli, Euler and Genocchi polynomials,” Advances in Diff. Equa., 2013.2013.1847. | ||

In article | View Article | ||

[9] | Y. Hamahata, “Poly-Euler polynomials and Arakawa-Kaneko type zeta functions,” Functione et. App. Commentarii Mathematica, 51(1), pp. 7-27, 2014. | ||

In article | View Article | ||

[10] | H. M. Srivastava, “Some generalization and basic (or q-) extensition of the Bernoulli, Euler and Genocchi polynomials,” App. Math. Information Sci., 5, pp. 390-444, 2011. | ||

In article | |||

[11] | H. M. Srivastava and J. Choi, “Series associated with the zeta and related functions,” Kluver Academic Pub., Dordrect, Boston and London, 2001. | ||

In article | View Article | ||

[12] | D. S. Kim and T. Kim, “Higher-order Frobenius-Euler and Poly-Bernoulli mixed-type polynomials,” Advances in Diff. Equa., 2013.2013.251. | ||

In article | View Article | ||

[13] | D. S. Kim and T. Kim, “Higher order Bernoulli and poly-Bernoulli mixed type polynomials,” Georgian Math. J., 22, pp. 265-272, 2015. | ||

In article | View Article | ||

[14] | U. Duran and M. Acikgoz, “On (p,q)-Euler numbers and polynomials associated with (p,q)-Volkenborn integral,” Int. J. of Number Theory, 14(1), pp. 241-253, 2018. | ||

In article | View Article | ||

[15] | A. Bayad and T. Kim, “Higher recurrences for Apostol-Bernoulli-Euler polynomials,” Russ. J. of Math. Phys., 19(1), pp. 1-10, 2012. | ||

In article | View Article | ||

[16] | U. Duran, M Acikgoz, A. Esi and S. Araci, “A note on the (p,q)-Hermite polynomials,” App. Math. And Information Sciences, 12, pp. 227-231, 2018. | ||

In article | View Article | ||

[17] | H. Jolany, R. B. Corcino and T. Komatsu, “More properties on multi-Poly-Euler polynomials,” Bull. Soc. Math. Mex., 21, pp. 149-162, 2015. | ||

In article | View Article | ||

[18] | D. S. Kim and T. Kim, Hermite and Poly-Bernoulli mixed type polynomials,” Advances in Diff. Equa., 2013. 2013. 343. | ||

In article | View Article | ||

[19] | D. S. Kim, T. Kim, D. V. Dolgy and S. H. Rim, “Some new identities of Bernoulli, Euler and Hermite polynomials arising from umbral calculus,” Advances in Diff. Equa., 2013. 2013. 73. | ||

In article | View Article | ||

[20] | D. S. Kim and T. Kim, “Some identities of Frobenius-Euler polynomials arising from umbral calculus,” Advances in Diff. Equa., 2012. 2012. 196. | ||

In article | View Article | ||

[21] | D. S. Kim and T. Kim, “A note on poly-Bernoulli and higher order poly-Bernoulli polynomials,” Russ. J. Math., 22(1), pp. 26-33, 2015. | ||

In article | View Article | ||

[22] | B. Kurt and Y. Simsek, “On the generalized Apostol-type Frobenius-Euler polynomials,” Advances in Diff. Equa., 2013. 2013. 1. | ||

In article | View Article | ||

[23] | H. M. Srivastava and H. L. Manocho, “A Treatise on generating functions,” Halsted Press, John Willey and sums, New York, Chichester, Brisbane and Toronto, 1984. | ||

In article | |||