Title: A Unified Family of Apostol-Bernoulli Based Poly-Daehee Polynomials
Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814
Article ID: MTJPAM-D-20-00009; Volume 3 / Issue 3 / Year 2021 (Special Issue), Pages 1-11
Document Type: Research Paper
Author(s): Talha Usman a , Nabiullah Khan b , Mohd Saif c , Junesang Choi d
aDepartment of Mathematics, School of Basic and Applied Sciences, Lingaya’s Vidyapeeth, Faridabad 121002, Haryana, India
bDepartment of Applied Mathematics, Faculty of Engineering and Technology, Aligarh Muslim University, Aligarh 202002, India
cDepartment of Applied Mathematics, Faculty of Engineering and Technology, Aligarh Muslim University, Aligarh 202002, India
dDepartment of Mathematics, Dongguk University, Gyeongju 38066, Republic of Korea
Received: 22 April 2019, Accepted: 3 June 2020, Published: 25 April 2021.
Corresponding Author: Junesang Choi (Email address: firstname.lastname@example.org)
Full Text: PDF
We introduce a unified family of Apostol-Bernoulli based poly-Daehee polynomials. Then we provide a number of formulas involving these unified polynomials such as differential formulas, addition formulas, summation formulas, and an implicit summation formula. The identities presented here, being general, are pointed out to yield the corresponding formulas associated with relatively simple polynomials. Further we provide several other polynomials similar to these unified polynomials.
Keywords: Apostol-Bernoulli based poly-Daehee polynomials and numbers, Apostol-Euler based poly-Daehee polynomials and numbers, Apostol-Genocchi based poly-Daehee polynomials and numbers, Differential formula, Addition formula, Summation formula, Implicit summation formulaReferences:
- S. Araci and M. Acikgoz, A note on the Frobenius-Euler numbers and polynomials associated with Bernstein polynomials, Adv. Stud. Contemp. Math. 22 (3), 399-406, 2012.
- L. Carlitz, A note on Bernoulli and Euler polynomials of the second kind, Scripta Math. 25, 323-330, 1961.
- L. Comtet and L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions (Translated from the French by J. W. Nienhuys), Reidel, Dordrecht and Boston, 1974.
- G. Dattoli, S. Lorenzutta and C. Cesarano, Finite sums and generalized forms of Bernoulli polynomials, Rend. Math. (Ser. 8) 19, 385-391, 1999.
- A. Erdélyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, Higher Transcendental Functions Vol.I, McGraw-Hill Book Company, New York, Toronto and London, 1953.
- M. Kaneko, Poly-Bernoulli numbers, J. Théor. Nombres Bordeaux 9 (1), 221-228, 1997.
- N.U. Khan and T. Usman, A new class of Laguerre poly-Bernoulli numbers and polynomials, Adv. Stud. Contemp. Math. 27 (2), 229–241, 2017.
- N.U. Khan, T. Usman and M. Aman, Generating functions for Legendre-based poly-Bernoulli numbers and polynomials, Honam Math. J. 39 (2), 217-231, 2017.
- N.U. Khan, T. Usman and J. Choi, A new class of generalized polynomials, Turk. J. Math. 42, 1366-1379, 2018.
- T. Kim, On the multiple q-Genocchi and Euler numbers, Russ. J. Math. Phys. 15 (4), 481-486, 2008.
- D.S. Kim, T. Kim, S.-H. Lee and J.-J. Seo, Higher-order Daehee numbers and polynomials, Int. J. Math. Anal. (Ruse) 8 (6), 273-283, 2014.
- T. Kim, S.-H. Lee, T. Mansour and J.-J. Seo, A note on q-Daehee polynomials and numbers, Adv. Stud. Contemp. Math. 24 (2), 155-160, 2014.
- T. Kim, H.I. Kwon, S.-H. Lee and J.-J. Seo, A note on poly-Bernoulli numbers and polynomials of the second kind, Adv. Differen. Equ. 2014, Article Id 219, 2014.
- D.S. Kim and T. Kim, Daehee numbers and polynomials, Appl. Math. Sci. 7 (120), 5969-5976, 2013.
- D.S. Kim and T. Kim, A study on the integral of the product of several Bernoulli polynomials, Rocky Mountain J. Math. 44 (4), 1251-1263, 2014.
- D.S. Kim and T. Kim, Some identities involving Genocchi polynomials and numbers, Ars Comb. 121, 403-412, 2015.
- D.S. Kim, N. Lee, J. Na and K.H. Park, Identities of symmetry for higher order Euler polynomials in three variables (I), Adv. Stud. Contemp. Math. 22, 51-74, 2012.
- T. Komatsu and F. Luca, Some relationships between poly-Cauchy numbers and poly-Bernoulli numbers, Ann. Math. Inform. 41, 99-105, 2013.
- L. Lewin, Polylogarithms and Associated Functions, Elsevier (North-Holland), New York, London and Amsterdam, 1981.
- D.S. Lim and J. Kwon, A note on poly-Daehee numbers and polynomials, Proc. Jangjeon Math. Soc. 19 (2), 219-224, 2016.
- Q.-M. Luo, Extension for the Genocchi polynomials and its Fourier expansions and integral representations, Osaka J. Math. 48, 291-309, 2011.
- Q.-M. Luo, Apostol-Euler polynomials of higher order and the Gaussian hypergeometric function, Taiwanese J. Math. 10 (4), 917-925, 2006.
- Q.-M. Luo and H. M. Srivastava, Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials, J. Math. Anal. Appl. 308 (1), 290-302, 2005.
- J.-W. Park, S.-H. Rim and J. Kwon, The twisted Daehee numbers and polynomials, Adv. Differ. Equ. 9 (1), Article Id 1, 2014.
- E.D. Rainville, Special Functions, Macmillan Company, New York, 1960; Reprinted by Chelsea Publishing Company, Bronx, New York, 1971.
- J.Sándor and B. Crstici, Handbook of Number Theory II, Kluwer Acad. Publ., Dordrecht, 2004.
- H.M. Srivastava and J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Science Publishers, Amsterdam, London and New York, 2012.