Some new identities involving Sheffer-Appell polynomial sequences via matrix approach. (English) Zbl 1439.11086
Summary: In this contribution, some new identities involving Sheffer-Appell polynomial sequences using generalized Pascal functional and Wronskian matrices are deduced. As a direct application of them, identities involving families of polynomials as Euler, Bernoulli, Miller-Lee and Apostol-Euler polynomials, among others, are given.
MSC:
| 11B83 | Special sequences and polynomials |
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
Keywords:
Sheffer-Appell polynomial sequence; generalized Pascal functional; Wronskian matrices; identities; orthogonal polynomialsReferences:
| [1] | Aceto, L., Cação, I.: A matrix approach to Sheffer polynomials. J. Math. Anal. Appl. 446, 87-100 (2017) · Zbl 1351.11020 · doi:10.1016/j.jmaa.2016.08.038 |
| [2] | Andrews, L.C.: Special Functions for Engineers and Applied Mathematicians. Macmillan Publishing Company, New York (1985) |
| [3] | Andrews, G.E., Askey, R., Roy, R.: Special Functions, Encyclopedia of Mathematics and its Applications, vol. 71. Cambridge University Press, Cambridge (1999) · Zbl 0920.33001 |
| [4] | Apostol, T.M.: On the Lerch zeta function. Pac. J. Math. 1, 161-167 (1951) · Zbl 0043.07103 · doi:10.2140/pjm.1951.1.161 |
| [5] | 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 |
| [6] | Costabile, F.A., Longo, E.: A determinantal approach to Appell polynomials. J. Comput. Appl. Math. 234(5), 1528-1542 (2010) · Zbl 1200.33020 · doi:10.1016/j.cam.2010.02.033 |
| [7] | Costabile, F.A., Longo, E.: An algebraic approach to Sheffer polynomial sequences. Integral Transforms Spec. Funct. 25(4), 295-311 (2014) · Zbl 1345.11020 · doi:10.1080/10652469.2013.842234 |
| [8] | Dattoli, G., Migliorati, M., Srivastava, H.M.: Sheffer polynomials, monomiality principle, algebraic methods and the theory of classical polynomials. Math. Comput. Model. 45, 1033-1041 (2007) · Zbl 1117.33008 · doi:10.1016/j.mcm.2006.08.010 |
| [9] | Dere, R., Simsek, Y., Srivastava, H.M.: A unified presentation of three families of generalized Apostol type polynomials based upon the theory of the umbral calculus and the umbral algebra. J. Number Theory 133, 3245-3263 (2013) · Zbl 1295.11023 · doi:10.1016/j.jnt.2013.03.004 |
| [10] | di Bucchianico, A., Loeb, D.: A selected survey of umbral calculus. Electron. J. Combin. 2, 1-34 (2000) · Zbl 0851.05012 |
| [11] | Garza, L.G., Garza, L.E., Marcellán, F., Pinzón-Cortés, N.C.: A matrix approach for the semiclassical and coherent orthogonal polynomials. Appl. Math. Comput. 256, 459-471 (2015) · Zbl 1338.42033 |
| [12] | He, Y., Araci, S., Srivastava, H.M., 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 |
| [13] | He, M.-X., Ricci, P.E.: Differential equation of Appell polynomials via the factorization method. J. Comput. Appl. Math. 139, 231-237 (2002) · Zbl 0994.33008 · doi:10.1016/S0377-0427(01)00423-X |
| [14] | Khan, S., Riyasat, M.: A determinantal approach to Sheffer-Appell polynomials via monomiality principle. J. Math. Anal. Appl. 421, 806-829 (2015) · Zbl 1310.33012 · doi:10.1016/j.jmaa.2014.07.044 |
| [15] | Kim, D.S., Kim, T.: A matrix approach to some identites involving Sheffer polynomial sequences. Appl. Math. Comput. 253, 83-101 (2015) · Zbl 1338.33025 |
| [16] | Lehmer, D.H.: A new approach to Bernoulli polynomials. Am. Math. Mon. 95, 905-911 (1988) · Zbl 0663.10009 · doi:10.1080/00029890.1988.11972114 |
| [17] | Luo, Q.M.: Apostol-Euler polynomials of higher order and the Gaussian hypergeometric function. Taiwan. J. Math. 10(4), 917-925 (2006) · Zbl 1189.11011 · doi:10.11650/twjm/1500403883 |
| [18] | Luo, Q.M., Srivastava, H.M.: Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials. J. Math. Anal. Appl. 308(1), 290-302 (2005) · Zbl 1076.33006 · doi:10.1016/j.jmaa.2005.01.020 |
| [19] | McBride, E.B.: Obtaining Generating Functions. Springer, Berlin (1971) · Zbl 0215.43403 · doi:10.1007/978-3-642-87682-0 |
| [20] | Pintér, Á., Srivastava, H.M.: Addition theorems for the Appell polynomials and the associated classes of polynomial expansions. Aequ. Math. 85, 483-495 (2013) · Zbl 1288.11022 · doi:10.1007/s00010-012-0148-8 |
| [21] | Rainville E.D.: Special Functions, The Macmillan Co. Inc., New York (1960). Reprinted by Chelsea Publ. Co. Bronx, New York, 1971 · Zbl 0092.06503 |
| [22] | Roman, S.: The theory of the umbral calculus. I. J. Math. Anal. Appl. 87, 58-115 (1982) · Zbl 0499.05009 · doi:10.1016/0022-247X(82)90154-8 |
| [23] | Roman, S.: The Umbral Calculus. Academic Press, New York (1984) · Zbl 0536.33001 |
| [24] | Roman, S., Rota, G.-C.: The umbral calculus. Adv. Math. 27, 95-188 (1978) · Zbl 0375.05007 · doi:10.1016/0001-8708(78)90087-7 |
| [25] | Rosen, K.: Handbook of Discrete and Combinatorial Mathematics. CRC Press, Boca Raton (2000) · Zbl 1044.00002 |
| [26] | Sheffer, I.M.: A differential equation for Appell polynomials. Bull. Am. Math. Soc. 41, 914-923 (1935) · JFM 61.0379.01 · doi:10.1090/S0002-9904-1935-06218-4 |
| [27] | Sheffer, I.M.: Some properties of polynomial sets of type zero. Duke Math. J. 5, 590-622 (1939) · JFM 65.0366.01 · doi:10.1215/S0012-7094-39-00549-1 |
| [28] | Srivastava, H.M.: Some characterizations of Appell and \(q\)-Appell polynomials. Ann. Mat. Pura Appl. 130(Ser. 4), 321-329 (1982) · Zbl 0468.33007 · doi:10.1007/BF01761501 |
| [29] | Srivastava, H.M., Nisar, K.S., Khan, M.A.: Some umbral calculus presentations of the Chan-Chyan-Srivastava polynomials and the Erkuş-Srivastava polynomials. Proyecc. J. Math. 33, 77-90 (2014) · Zbl 1301.33016 |
| [30] | Yang, Y.-Z., Micek, C.: Generalized Pascal functional matrix and its applications. Linear Algebra Appl. 423, 230-245 (2007) · Zbl 1115.15012 · doi:10.1016/j.laa.2006.12.014 |
| [31] | Yang, Y.-Z., Youn, H.-Y.: Appell polynomial sequences: a linear algebra approach. J. Algebra Number Theory Appl. 13, 65-98 (2009) · Zbl 1177.33025 |
| [32] | Youn, H.-Y., Yang, Y.-Z.: Differential equation and recursive formulas of Sheffer polynomial sequences. ISRN Discrete Math. 2011, 1-16 (2011). Article ID 476462 · Zbl 1237.33008 · doi:10.5402/2011/476462 |
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