| Title:
|
Symmetric identity for polynomial sequences satisfying $A_{n+1}^\prime (x)=(n+1)A_n(x)$ (English) |
| Author:
|
Bencherif, Farid |
| Author:
|
Boumahdi, Rachid |
| Author:
|
Garici, Tarek |
| Language:
|
English |
| Journal:
|
Communications in Mathematics |
| ISSN:
|
1804-1388 (print) |
| ISSN:
|
2336-1298 (online) |
| Volume:
|
29 |
| Issue:
|
3 |
| Year:
|
2021 |
| Pages:
|
343-355 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Using umbral calculus, we establish a symmetric identity for any sequence of polynomials satisfying $A_{n+1}^\prime (x) =(n+1)A_{n}(x)$ with $A_0(x)$ a constant polynomial. This identity allows us to obtain in a simple way some known relations involving Apostol-Bernoulli polynomials, Apostol\HH Euler polynomials and generalized Bernoulli polynomials attached to a primitive Dirichlet character. (English) |
| Keyword:
|
Appell sequence |
| Keyword:
|
Apostol-Bernoulli polynomial |
| Keyword:
|
Apostol-Euler polynomial |
| Keyword:
|
generalized Bernoulli polynomial |
| Keyword:
|
primitive Dirichlet character. |
| MSC:
|
05A19 |
| MSC:
|
05A40 |
| MSC:
|
11B68 |
| idZBL:
|
Zbl 1480.05012 |
| idMR:
|
MR4355417 |
| . |
| Date available:
|
2022-01-10T09:58:44Z |
| Last updated:
|
2022-04-28 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/149321 |
| . |
| Reference:
|
[1] Agoh, T.: Shortened recurrence relations for generalized Bernoulli numbers and polynomials.Journal of Number Theory, 176, 2017, 149-173, Elsevier, MR 3622124, 10.1016/j.jnt.2016.12.014 |
| Reference:
|
[2] Appell, P.: Sur une classe de polynômes.Annales Scientifiques de l'École Normale Supérieure, 9, 1880, 119-144, |
| Reference:
|
[3] Sun, W.Y.C. Chen,L.H.: Extended Zeilberger's algorithm for identities on Bernoulli and Euler polynomials.Journal of Number Theory, 129, 9, 2009, 2111-2132, Elsevier, MR 2528056, 10.1016/j.jnt.2009.01.026 |
| Reference:
|
[4] Gel'fand, I.M., Shilov, G.E.: Generalized Functions, Vol. 1: Properties and Operations.1964, Acad. Press, New York, |
| Reference:
|
[5] Gelfand, M.B.: A note on a certain relation among Bernoulli numbers.Baškir. Gos. Univ. Ucen. Zap. Vyp, 31, 1968, 215-216, |
| Reference:
|
[6] Gessel, I.M.: Applications of the classical umbral calculus.Algebra Univers., 49, 4, 2003, 397-434, MR 2022347, 10.1007/s00012-003-1813-5 |
| Reference:
|
[7] He, Y., Zhang, W.: Some symmetric identities involving a sequence of polynomials.The Electronic Journal of Combinatorics, 17, N7, 2010, 7pp, MR 2587757, 10.37236/456 |
| Reference:
|
[8] Kaneko, M.: A recurrence formula for the Bernoulli numbers.Proceedings of the Japan Academy, Series A, Mathematical Sciences, 71, 8, 1995, 192-193, The Japan Academy, |
| Reference:
|
[9] Luo, Q.-M., Srivastava, H.M.: Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials.Journal of Mathematical Analysis and Applications, 308, 1, 2005, 290-302, Elsevier, MR 2142419, 10.1016/j.jmaa.2005.01.020 |
| Reference:
|
[10] Momiyama, H.: A new recurrence formula for Bernoulli numbers.Fibonacci Quarterly, 39, 3, 2001, 285-288, THE FIBONACCI ASSOCIATION, MR 1840040 |
| Reference:
|
[11] Prévost, M.: Padé approximation and Apostol-Bernoulli and Apostol-Euler polynomials.Journal of computational and applied mathematics, 233, 11, 2010, 3005-3017, Elsevier, MR 2592274, 10.1016/j.cam.2009.11.050 |
| Reference:
|
[12] Rota, G.-C.: The number of partitions of a set.The American Mathematical Monthly, 71, 5, 1964, 498-504, Taylor & Francis, 10.1080/00029890.1964.11992270 |
| Reference:
|
[13] Stern, M.: Beiträge zur Theorie der Bernoullischen und Eulerschen Zahlen.Abhandlungen der Königlichen Gesellschaft der Wissenschaften in Göttingen, 26, 1880, 3-46, |
| Reference:
|
[14] Sun, Z.-W.: Combinatorial identities in dual sequences.European Journal of Combinatorics, 24, 6, 2003, 709-718, Elsevier, MR 1995582, 10.1016/S0195-6698(03)00062-3 |
| Reference:
|
[15] Ettingshausen, A. von: Vorlesungen über die höhere Mathematik.1827, C. Gerold, Vienna, |
| Reference:
|
[16] Seidel, P.L. von: Uber eine einfache Entstehungsweise der Bernoullischen Zahlen und einiger verwandten Reihen, Sitzungsberichte der Münch.Akad. Math. Phys. Classe, 7, 1877, 157-187, |
| Reference:
|
[17] Wu, K.-J., Sun, Z.-W., Pan, H.: Some identities for Bernoulli and Euler polynomials.Fibonacci Quarterly, 42, 4, 2004, 295-299, MR 2110081 |
| . |
