Some new formulas for the products of the Apostol type polynomials. (English) Zbl 1419.11040
Summary: In [Adv. Stud. Contemp. Math., Kyungshang 24, No. 4, 535–547 (2014; Zbl 1366.11049)], D. S. Kim et al. computed a kind of new sums of the products of an arbitrary number of the classical Bernoulli and Euler polynomials by using the Euler basis for the vector space of polynomials of bounded degree. Inspired by their work, in this paper, we establish some new formulas for such a kind of sums of the products of an arbitrary number of the Apostol-Bernoulli, Euler, and Genocchi polynomials by making use of the generating function methods and summation transform techniques. The results derived here are generalizations of the corresponding known formulas involving the classical Bernoulli, Euler, and Genocchi polynomials.
MSC:
| 11B68 | Bernoulli and Euler numbers and polynomials |
| 05A19 | Combinatorial identities, bijective combinatorics |
Keywords:
Apostol-Bernoulli polynomials; Apostol-Euler polynomials; Apostol-Genocchi polynomials; summation formulas; recurrence relationsCitations:
Zbl 1366.11049References:
| [1] | Agoh, T, Dilcher, K: Convolution identities and lacunary recurrences for Bernoulli numbers. J. Number Theory 124, 105-122 (2007) · Zbl 1137.11014 · doi:10.1016/j.jnt.2006.08.009 |
| [2] | Araci, S: Novel identities involving Genocchi numbers and polynomials arising from applications of umbral calculus. Appl. Math. Comput. 233, 599-607 (2014) · Zbl 1334.05013 |
| [3] | Cohen, H: Number Theory - Volume II: Analytic and Modern Tools. Graduate Texts in Mathematics, vol. 240. Springer, Berlin (2007) · Zbl 1119.11002 |
| [4] | Nielsen, N: Traité élémentaire des nombres de Bernoulli. Gauthier-Villars, Paris (1923) · JFM 50.0170.04 |
| [5] | Srivastava, HM, Choi, J: Zeta and q-Zeta Functions and Associated Series and Integrals. Elsevier, Amsterdam (2012) · Zbl 1239.33002 |
| [6] | Srivastava, HM, Pintér, Á: Remarks on some relationships between the Bernoulli and Euler polynomials. Appl. Math. Lett. 17, 375-380 (2004) · Zbl 1070.33012 · doi:10.1016/S0893-9659(04)90077-8 |
| [7] | Luo, Q-M: Extension for the Genocchi polynomials and its Fourier expansions and integral representations. Osaka J. Math. 48, 291-309 (2011) · Zbl 1268.33002 |
| [8] | Luo, Q-M, Srivastava, HM: Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials. J. Math. Anal. Appl. 308, 290-302 (2005) · Zbl 1076.33006 · doi:10.1016/j.jmaa.2005.01.020 |
| [9] | Luo, Q-M, Srivastava, HM: Some relationships between the Apostol-Bernoulli and Apostol-Euler polynomials. Comput. Math. Appl. 51, 631-642 (2006) · Zbl 1099.33011 · doi:10.1016/j.camwa.2005.04.018 |
| [10] | Apostol, TM: On the Lerch zeta function. Pac. J. Math. 1, 161-167 (1951) · Zbl 0043.07103 · doi:10.2140/pjm.1951.1.161 |
| [11] | Srivastava, HM: Some formulas for the Bernoulli and Euler polynomials at rational arguments. Math. Proc. Camb. Philos. Soc. 129, 77-84 (2000) · Zbl 0978.11004 · doi:10.1017/S0305004100004412 |
| [12] | Lu, D-Q, Srivastava, HM: Some series identities involving the generalized Apostol type and related polynomials. Comput. Math. Appl. 62, 3591-3602 (2011) · Zbl 1236.33020 · doi:10.1016/j.camwa.2011.09.010 |
| [13] | Luo, Q-M: Fourier expansions and integral representations for the Apostol-Bernoulli and Apostol-Euler polynomials. Math. Comput. 78, 2193-2208 (2009) · Zbl 1214.11032 · doi:10.1090/S0025-5718-09-02230-3 |
| [14] | Luo, Q-M, Srivastava, HM: Some generalizations of the Apostol-Genocchi polynomials and the Stirling numbers of the second kind. Appl. Math. Comput. 217, 5702-5728 (2011) · Zbl 1218.11026 |
| [15] | Navas, LM, Ruiz, FJ, Varona, JL: Asymptotic estimates for Apostol-Bernoulli and Apostol-Euler polynomials. Math. Comput. 81, 1707-1722 (2011) · Zbl 1268.11033 · doi:10.1090/S0025-5718-2012-02568-3 |
| [16] | Prévost, M: Padé approximation and Apostol-Bernoulli and Apostol-Euler polynomials. J. Comput. Appl. Math. 233, 3005-3017 (2010) · Zbl 1191.41005 · doi:10.1016/j.cam.2009.11.050 |
| [17] | Srivastava, HM: Some generalizations and basic (or q-) extensions of the Bernoulli, Euler and Genocchi polynomials. Appl. Math. Inf. Sci. 5, 390-444 (2011) |
| [18] | Dilcher, K: Sums of products of Bernoulli numbers. J. Number Theory 60, 23-41 (1996) · Zbl 0863.11011 · doi:10.1006/jnth.1996.0110 |
| [19] | Chen, K-W: Sums of products of generalized Bernoulli polynomials. Pac. J. Math. 208, 39-52 (2003) · Zbl 1057.11012 · doi:10.2140/pjm.2003.208.39 |
| [20] | He, Y, Araci, S: Sums of products of Apostol-Bernoulli and Apostol-Euler polynomials. Adv. Differ. Equ. 2014, Article ID 155 (2014) · Zbl 1343.11030 · doi:10.1186/1687-1847-2014-155 |
| [21] | Huang, I-C, Huang, S-Y: Bernoulli numbers and polynomials via residues. J. Number Theory 76, 178-193 (1999) · Zbl 0940.11009 · doi:10.1006/jnth.1998.2364 |
| [22] | Kim, M-S, Hu, S: Sums of products of Apostol-Bernoulli numbers. Ramanujan J. 28, 113-123 (2012) · Zbl 1250.11026 · doi:10.1007/s11139-011-9340-z |
| [23] | Kim, T: Sums of products of q-Bernoulli numbers. Arch. Math. 76, 190-195 (2001) · Zbl 0986.11010 · doi:10.1007/s000130050559 |
| [24] | Kim, T, Adiga, C: Sums of products of generalized Bernoulli numbers. Int. Math. J. 5, 1-7 (2004) · Zbl 1208.11030 |
| [25] | Simsek, Y, Kurt, V, Kim, D: New approach to the complete sum of products of the twisted (h,q)\((h,q)\)-Bernoulli numbers and polynomials. J. Nonlinear Math. Phys. 14, 44-56 (2007) · Zbl 1163.11015 · doi:10.2991/jnmp.2007.14.1.5 |
| [26] | Simsek, Y: Complete sum of products of (h,q)\((h,q)\)-extension of Euler polynomials and numbers. J. Differ. Equ. Appl. 16, 1331-1348 (2010) · Zbl 1223.11027 · doi:10.1080/10236190902813967 |
| [27] | Kim, DS, Kim, T, Mansour, T: Euler basis and the product of several Bernoulli and Euler polynomials. Adv. Stud. Contemp. Math. 24, 535-547 (2014) · Zbl 1366.11049 |
| [28] | Abramowitz, M, Stegun, IA (eds.): Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series, vol. 55. National Bureau of Standards, Washington (1964). Reprinted by Dover, New York, 1965 · Zbl 0171.38503 |
| [29] | Pan, H, Sun, Z-W: New identities involving Bernoulli and Euler polynomials. J. Comb. Theory, Ser. A 113, 156-175 (2006) · Zbl 1085.05017 · doi:10.1016/j.jcta.2005.07.008 |
| [30] | Agoh, T: Convolution identities for Bernoulli and Genocchi polynomials. Electron. J. Comb. 21, Article ID P1.65 (2014) · Zbl 1331.11013 |
| [31] | Dunne, GV, Schubert, C: Bernoulli number identities from quantum field theory and topological string theory. Commun. Number Theory Phys. 7, 225-249 (2013) · Zbl 1297.11009 · doi:10.4310/CNTP.2013.v7.n2.a1 |
| [32] | Faber, C, Pandharipande, R: Hodge integrals and Gromov-Witten theory. Invent. Math. 139, 173-199 (2000) · Zbl 0960.14031 · doi:10.1007/s002229900028 |
| [33] | He, Y, Zhang, WP: Some sum relations involving Bernoulli and Euler polynomials. Integral Transforms Spec. Funct. 22, 207-215 (2011) · Zbl 1217.11021 · doi:10.1080/10652469.2010.511209 |
| [34] | He, Y, Araci, S, Srivastava, HM, Acikgoz, M: Some new identities for the Apostol-Bernoulli polynomials and the Apostol-Genocchi polynomials. Appl. Math. Comput. 262, 31-41 (2015) · Zbl 1410.11016 |
| [35] | Kim, DS, Kim, T, Lee, S-H, Kim, Y-H: Some identities for the product of two Bernoulli and Euler polynomials. Adv. Differ. Equ. 2012, Article ID 95 (2012) · Zbl 1346.33008 · doi:10.1186/1687-1847-2012-95 |
| [36] | Kim, DS, Kim, T: Identities arising from higher-order Daehee polynomial bases. Open Math. 13, 196-208 (2015) · Zbl 1307.05019 |
| [37] | Carlson, BC: Special Functions of Applied Mathematics. Academic Press, New York (1977) · Zbl 0394.33001 |
| [38] | Srivastava, HM, Niukkanen, AW: Some Clebsch-Gordan type linearization relations and associated families of Dirichlet integrals. Math. Comput. Model. 37, 245-250 (2003) · Zbl 1076.33008 · doi:10.1016/S0895-7177(03)00003-7 |
| [39] | Andrews, GE: Euler’s pentagonal number theorem. Math. Mag. 56, 279-284 (1983) · Zbl 0523.01011 · doi:10.2307/2690367 |
| [40] | Bell, J: A summary of Euler’s work on the pentagonal number theorem. Arch. Hist. Exact Sci. 64, 301-373 (2010) · Zbl 1208.01013 · doi:10.1007/s00407-010-0057-y |
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