Apostol Type \((p, q)\)-Bernoulli, \((p, q)\)-Euler and \((p, q)\)-Genocchi Polynomials and Numbers
DOI:
https://doi.org/10.26713/cma.v8i1.512Keywords:
\((p, q)\)-calculus, Apostol Bernoulli polynomials, Apostol Euler polynomials, Apostol Genocchi polynomials, Generating function, Cauchy productAbstract
The main subject of this work is to introduce and investigate a new generalizations of the Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials under the theory of post quantum calculus, denoted by \((p, q)\)-calculus. We call them Apostol type \((p, q)\)-Bernoulli polynomials of order \(\alpha\), Apostol type \((p, q)\)-Euler polynomials of order \(\alpha\) and the Apostol type \((p, q)\)-Genocchi polynomials of order \(\alpha\). We derive some of their properties involving addition theorems, difference equations, derivative properties, recurrence relationships, and so on. Also, \((p, q)\)-analogues of some familiar formulae belonging to usual Apostol-Bernoulli, Euler and Genocchi polynomials are shown. Furthermore, \((p, q)\)-generalizations of Cheon's main result [G.S. Cheon, Appl. Math. Lett. 16 (2003) 365–368] and the formula of Srivastava and Pintér [H.M. Srivastava, A. Pintér, Appl. Math. Lett. 17 (2004), 375–380] are investigated.Downloads
References
T.M. Apostol, On the Lerch Zeta function, Pasific J. Math. 1 (1951), 161–167, https://projecteuclid.org/euclid.pjm/1103052188.
R. Chakrabarti and R. Jagannathan, A (p, q)-oscillator realization of two-parameter quantum algebras, J. Phys. A: Math. Gen. 24 (1991), L711, http://iopscience.iop.org/article/10.1088/0305-4470/24/13/002/pdf.
G.S. Cheon, A note on the Bernoulli and Euler polynomials, Appl. Math. Lett. 16 (2003) 365–368, doi:10.1016/S0893-9659(03)80058-7.
R.B. Corcino, On P,Q-Binomial coefficients, Electron. J. Combin. Number Theory 8 (2008), #A29, http://www.westga.edu/char126relaxintegers/cgi-bin/get.cgi.
U. Duran, M. Acikgoz and S. Araci, On (p, q)-Bernoulli, (p, q)-Euler and (p, q)-Genocchi polynomials, J. Comput. Theor. Nanosci. 13 (2016), 7833–7846.
U. Duran, M. Acikgoz and S. Araci, On some polynomials derived from (p, q)-calculus, J. Comput. Theor. Nanosci. 13 (2016), 7903–7908.
B.S. El-Desouky and R.S. Gomaa, A new unified family of generalized Apostol-Euler, Bernoulli and Genocchi polynomials, Appl. Math. Comput. 247 (2014), 695–702, http://dx.doi.org/10.1016/j.amc.2014.09.002.
V. Gupta, (p, q)-Baskakov-Kantorovich operators, Appl. Math. Inf. Sci. 10 (4) (2016), 1551–1556, doi:doi:10.18576/amis/100433.
R. Jagannathan and K.S. Rao, Two-parameter quantum algebras, twin-basic numbers, and associated generalized hypergeometric series, arXiv:math/0602613 [math.NT], https://arxiv.org/abs/math/0602613.
Y. He, S. Araci, H.M. Srivastava and M. Acikgoz, Some new identities for the Apostol-Bernoulli polynomials and the Apostol-Genocchi polynomials, Appl. Math. Comput. 262 (2015), 31–41 http://dx.doi.org/10.1016/j.amc.2015.03.132.
V. Kurt, New identities and relations derived from the generalized Bernoulli polynomials, Euler polynomials and Genocchi polynomials, Adv. Difference Equ. 2014 (2014), 5 pages, doi:doi:10.1186/1687-1847-2014-5.
B. Kurt, A note on the Apostol type q-Frobenius-Euler polynomials and generalizations of the Srivastava-Pinter addition theorems, Filomat 30 (1) (2016), 65–72, doi:doi:10.2298/FIL1601065K.
V. Kurt, Some symmetry identities for the unified Apostol-type polynomials and multiple power sums, Filomat 30 (3) (2016), 583–592, doi:doi:10.2298/FIL1603583K.
Q.-M. Luo, On the Apostol-Bernoulli polynomials, Cent. Eur. J. Math. 2 (4) (2004), 509-515 https://www.degruyter.com/view/j/math.2004.2.issue-4/BF02475959/bf02475959.pdf.
Q.-M. Luo, Some results for the q-Bernoulli and q-Euler polynomials, J. Math. Anal. Appl. 363 (2010), 7–10, doi:doi:10.1016/j.jmaa.2009.07.042.
Q.-M. Luo, H.M. Srivastava, q-extension of some relationships between the Bernoulli and Euler polynomials, Taiwanese J. Math. 15 (1) (2011), 241–257, http://www.jstor.org/stable/taiwjmath.15.1.241.
Q.-M. Luo, H. M. Srivastava, Some relationships between the Apostol-Bernoulli and Apostol-Euler polynomials, Comput. Math. Appl. 51 (2006), 631–642, http://dx.doi.org/10.1016/j.camwa.2005.04.018.
Q.-M. Luo, H. M. Srivastava, Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials, J. Math. Anal. Appl. 308 (2005), 290–302, http://dx.doi.org/10.1016/j.jmaa.2005.01.020.
N.I. Mahmudov, On a class of q-Benoulli and q-Euler polynomials, Adv. Difference Equ. 2013 (2013), 108 pages, doi:10.1186/1687-1847-2013–108.
N.I. Mahmudov and M.E. Keleshteri, q-Extension for the Apostol type polynomials, J. Appl. Math. 2014, ID 868167, 7 pages, http://dx.doi.org/10.1155/2014/868167.
N.I. Mahmudov, q-Analogues of the Bernoulli and Genocchi polynomials and the Srivastava-Pintér addition theorems, Discrete Dyn. Nat. Soc. 2012, Article ID 169348 (2012), http://dx.doi.org/10.1155/2012/169348.
G.V. Milovanovic, V. Gupta and N. Malik, (p, q)-Beta functions and applications in approximation, Bol. Soc. Mat. Mex. 2016, 1–19, doi:doi:10.1007/s40590-016-0139-1.
M.A. Ozarslan, Unified Apostol-Bernoulli, Euler and Genochi polynomials, Comp. Math. Appl. 62 (2011), 2452–2462, http://dx.doi.org/10.1016/j.camwa.2011.07.031.
P.N. Sadjang, On the fundamental theorem of (p, q)-calculus and some (p, q)-Taylor formulas, arXiv:1309.3934 [math.QA], https://arxiv.org/abs/1309.3934.
H.M. Srivastava, Some generalizations and basic (or q-)extensions of the Bernoulli, Euler and Genocchi polynomials, Appl. Math. Inform. Sci. 5 (2011), 390–444, http://mobile.www.naturalspublishing.com/files/published/819u1z9mqbk1d8.pdf.
H.M. Srivastava and A. Pinter, Remarks on some relationships between the Bernoulli and Euler polynomials, Appl. Math. Lett. 17 (2004), 375–380, doi:doi:10.1016/S0893-9659(04)90077-8.
H.M. Srivastava, Some formulas for the Bernoulli and Euler polynomials at rational arguments, Math. Proc. Cambridge Philos. Soc. 129 (2000) 77–84, https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/some-formulas-for-the-bernoulli-and-euler-polynomials-at-rational-arguments/A77F3A8708924E9E99F2E9F8ECDFE983.
R. Tremblay, S. Gaboury and B.-J. Fugère, A new class of generalized Apostol–Bernoulli polynomials and some analogues of the Srivastava-Pintér addition theorem, Appl. Math. Lett. 24 (2011) 1888–1893, http://dx.doi.org/10.1016/j.aml.2011.05.012.
W. Wang, C. Jia and T. Wang, Some results on the Apostol-Bernoulli and Apostol-Euler polynomials, Comput. Math. Appl. 55 (2008), 1322–1332, http://dx.doi.org/10.1016/j.camwa.2007.06.021.
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