*Turkish Journal of Analysis and Number Theory*.

**2015**, 3(5), 120-125

DOI: 10.12691/tjant-3-5-2

### A Note on Hermite poly-Bernoulli Numbers and Polynomials of the Second Kind

**Waseem A. Khan**^{1, }, **N. U. Khan**^{2} and **Sarvat Zia**^{1}

^{1}Department of Mathematics, Integral University, Lucknow-226026, India

^{2}Department of Applied Mathematics, Faculty of Engineering and Technology, Aligarh Muslim University, Aligarh, India

**Cite this paper:**

Waseem A. Khan, N. U. Khan and Sarvat Zia. A Note on Hermite poly-Bernoulli Numbers and Polynomials of the Second Kind. *Turkish Journal of Analysis and Number Theory*. 2015; 3(5):120-125. doi: 10.12691/tjant-3-5-2

### Abstract

In the paper, we introduce a new concept of poly-Bernoulli numbers and polynomials of the second kind which is called Hermite poly-Bernoulli numbers and polynomials of the second kind. We also investigate and analyse its applications in number theory, combinatorics and other fields of mathematics. The results derived here are a generalization of some known summation formulae earlier studied by Jolany et al. [17,18], Dattoli et al [14] and Pathan et al [29,30].**Keywords:**Hermite polynomials poly-Bernoulli polynomials of the second kind Hermite poly-Bernoulli polynomials of the second kind summation formulae symmetric identities

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/

### References:

[1] | Andrews, L.C: Special functions for engineers and mathematicians, Macmillan. Co.New York, 1985. | ||

[2] | Apostol, T.M: On the Lerch zeta function, Pacific J.Math. 1(1951), 161-167. | ||

[3] | Araci, S: Novel identities for q-Genocchi numbers and polynomials, J. Funct.Spaces Appl. 2012 (2012) 13p (Article ID 214961). | ||

[4] | Araci, S: Novel identities involving Genocchi numbers and polynomials arisingfrom applications of umbral calculus, Appl.Math. and Comput. 233(2014), 599-607. | ||

[5] | Araci, S, M. Acikgoz, M, Seo, J.J: Explicit formulas involving q-Euler numbers and polynomials, Abstr. Appl. Anal. 2012 (2012) 11p (Article ID 298531). | ||

[6] | Araci, S, Erdal, D and Seo, J.J: A study on the fermionic p-adic q-integral representation on Zp associated with weighted q-Bernstein and q-Genocchi polynomials, Abstr. Appl. Anal. 2011 (2011) 10p (Article ID 649248). | ||

[7] | Araci, S, Acikgoz, M, Jolany, H, Seo, J.J: A unified generating function of the q-Genocchi polynomials with their interpolation functions, Proc. Jangjeon Math. Soc. 15 (2) (2012), 227-233. | ||

[8] | Araci, Sen, E, Acikgoz, M: A note on the modified q-Dedekind sums, Notes Number Theory Discrete Math. 19 (3) (2013), 60-65. | ||

[9] | Araci, A, Acikgoz, S M, Bagdasaryan, A, Sen, E: The Legendre polynomials associated with Bernoulli, Euler, Hermite and Bernstein polynomials, Turkish J.Anal. Number Theory (1) (2013) 13. | ||

[10] | Acikgoz, M, Araci, S, Cangul, I.N: A note on the modified q-Bernstein polynomials for functions of several variables and their reflections on q-Volkenborn integration, Appl. Math. Comput. 218 (3) (2011), 707-712. | ||

[11] | Araci, Sen, E, Acikgoz, M: Theorems on Genocchi polynomials of higher order arising from Genocchi basis, Taiwanese. J.Math. 18(2014), 473-482. | ||

[12] | Araci, Acikgoz, M, Sen, E: On the von Staudt-Clausen’s theorem associated with q-Genocchi numbers, Appl.Math. and Comput. 247(2014), 780-785. | ||

[13] | Bell, E.T: Exponential polynomials, Ann. of Math. 35(1934), 258-277. | ||

[14] | Dattoli, G, Lorenzutta, S and Cesarano, C: Finite sums and generalized forms of Bernoulli polynomials Rendiconti di Mathematica, 19(1999), 385-391. | ||

[15] | Hamahata, Y, Masubuch, H: Special Multi-Poly- Bernoulli numbers, Journal of Integer sequences,10 (2007), 1-6. | ||

[16] | Hamahata, Y, Masubuchi, H: Recurrence formulae for Multi-poly-Bernoulli numbers, Elec. J.Comb.Num.Theo. 7(2007), A-46. | ||

[17] | Jolany, H, Darafsheh, M.R, Alikelaye,R.E: Generalizations of Poly-Bernoulli Numbers and Polynomials, Int. J. Math. Comb. 2 (2010), A07-14. | ||

[18] | Jolany, H, Corcino, R.B: Explicit formula for generalization of Poly-Bernoulli numbers and polynomials with a,b,c parameters, Journal of Classical Analysis, 6(2015), 119-135. | ||

[19] | Jolany, H, Aliabadi, M, Corcino, R. B and Darafsheh, M.R: A Note on Multi Poly-Euler Numbers and Bernoulli Polynomials, General Mathematics, 20(2-3) (2012), 122-134. | ||

[20] | Jolany, H and Corcino, R.B: More properties on Multi-Euler polynomials, arXiv;1401.627IvI[math NT] 24 Jan 2014. | ||

[21] | Khan, S, Pathan, M.A, Hassan, Nader Ali Makhboul , Yasmin, G: Implicit summation formula for Hermite and related polynomials, J.Math.Anal.Appl. 344(2008), 408-416. | ||

[22] | Kaneko, M: Poly-Bernoulli numbers, J.de Theorie de Nombres 9 (1997), 221-228. | ||

[23] | Khan W.A: Some properties of the generalized Apostol type Hermite-Based polynomials, Kyungpook Math. J., 55(2015), 597-614. | ||

[24] | Khan W.A, A note on Hermite-based poly-Euler and multi poly-Euler polynomials, Palestine J. Math. vol 5(1) (2016), 17-26. | ||

[25] | Khan W.A: A new class of Hermite poly-Genocchi polynomials, J.Anal. and Number. Theory, 4(1) (2016), 1-8. | ||

[26] | Kim, T, Kwaon, H.I, Lee, S.H and Seo, J.J: A note on poly-Bernoulli numbers and polynomials of the second kind, Advances in Defference Equations, (2014), 2014:219. | ||

[27] | Kim, D.S, Kim, T Mansour, T and Dolgy, D.V: On poly-Bernoulli polynomials of this second kind with umbral calculus view point, Advances Diffrences Equations, (2015), 2015:27. | ||

[28] | Kim, T, Kwon, H.I, Seo, J.J: On λ-Bernoulli polynomials of the second kind, Appl.Math.Scien., vol.9 (2015), 5275-5281. | ||

[29] | Pathan, M.A and Khan, W.A: Some implicit summation formulas and symmetric identities for the generalized Hermite based- polynomials, Acta Universitatis Apulensis, 39(2014), 113-136. | ||

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

[31] | Pathan, M.A and Khan, W.A: A new class of generalized polynomials associated with Hermite and Euler polynomials , To appear in Mediterr. J. Math. Springer Basel 2015. | ||

[32] | Pathan, M.A and Khan, W.A:Some implicit summation formulas and symmetric identities for the generalized Hermite-Euler polynomials, East-West J.Maths. 16(1) (2014), 92-109. | ||

[33] | Pathan, M.A and Khan, W.A: A new class of generalized polynomials associated with Hermite and Bernoulli polynomials, LE MATEMATICHE, Vol. LXX (2015), 53-70 | ||

[34] | Pathan, M.A and Khan, W.A: Some new classes of generalized Hermite-based Apostol-Euler and Apostol-Genocchi polynomials, Fasciculli.Math. Vol.55(2015), In Press. | ||

[35] | Park,J.W and Rim, S.H: On the modified q-Bernoulli polynomials with weight, Proc.Jangjeon Math. Soc., 17 (2014), no. 2, 231-236. | ||

[36] | Qi, F and Guo, B.N: Explicit formulas for special values of the Bell polynomials of the second kind and for the Euler numbers and polynomials, September 2014. | ||

[37] | Qi, F: Explicit formulas for computing Bernoulli numbers of the second kind and stirling numbers of the first kind, Filomat, 28:2(2014), 319-327. | ||

[38] | Qi, F: A new formula for the Bernoulli numbers of the second kind in terms of the stirling numbers of the first kind, Publications De L’Institute Mathematique. | ||

[39] | Qi, F. Kim, D.S, Kim, T and Dolgy, D.V: Multiple poly-Bernoulli polynomials of the second kind, Advanced Studies in Contemporary Mathematics, 25(2015), 1-7. | ||

[40] | Sandor, J and Crisci: Handbook of Number Theory, Vol.II. Kluwer Academic Publishers, Dordrecht Boston and London, 2004. | ||

[41] | Srivastava, H.M and Choi, J: Series associated with the Zeta and related functions, Kluwer Academic Publishers, Dordrecht, Boston and London, 2001. | ||

[42] | Srivastava, H.M and Pinter, A: Remarks on some relationships between the Bernoulli and Euler polynomials, Appl.Math.Lett. 17(2004), 375-380. | ||

[43] | Yang, S: An identity of symmetry for the Bernoulli polynomials, Discrete Math. Vol. 308 (2008), 550-554. | ||

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