Geodesic
Symmetric identity for polynomial sequences satisfying $A_{n+1}^\prime (x)=(n+1)A_n(x)$
Communications in Mathematics, Tome 29 (2021) no. 3, pp. 343-355 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
Classification : 05A19, 05A40, 11B68
Keywords: Appell sequence; Apostol-Bernoulli polynomial; Apostol-Euler polynomial; generalized Bernoulli polynomial; primitive Dirichlet character. 
@article{COMIM_2021_29_3_a1,
     author = {Bencherif, Farid and Boumahdi, Rachid and Garici, Tarek},
     title = {Symmetric identity for polynomial sequences satisfying $A_{n+1}^\prime (x)=(n+1)A_n(x)$},
     journal = {Communications in Mathematics},
     pages = {343--355},
     year = {2021},
     volume = {29},
     number = {3},
     mrnumber = {4355417},
     zbl = {1480.05012},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/COMIM_2021_29_3_a1/}
}
TY  - JOUR
AU  - Bencherif, Farid
AU  - Boumahdi, Rachid
AU  - Garici, Tarek
TI  - Symmetric identity for polynomial sequences satisfying $A_{n+1}^\prime (x)=(n+1)A_n(x)$
JO  - Communications in Mathematics
PY  - 2021
SP  - 343
EP  - 355
VL  - 29
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/COMIM_2021_29_3_a1/
LA  - en
ID  - COMIM_2021_29_3_a1
ER  - 
%0 Journal Article
%A Bencherif, Farid
%A Boumahdi, Rachid
%A Garici, Tarek
%T Symmetric identity for polynomial sequences satisfying $A_{n+1}^\prime (x)=(n+1)A_n(x)$
%J Communications in Mathematics
%D 2021
%P 343-355
%V 29
%N 3
%U http://geodesic.mathdoc.fr/item/COMIM_2021_29_3_a1/
%G en
%F COMIM_2021_29_3_a1
Bencherif, Farid; Boumahdi, Rachid; Garici, Tarek. Symmetric identity for polynomial sequences satisfying $A_{n+1}^\prime (x)=(n+1)A_n(x)$. Communications in Mathematics, Tome 29 (2021) no. 3, pp. 343-355. http://geodesic.mathdoc.fr/item/COMIM_2021_29_3_a1/

[1] Agoh, T.: Shortened recurrence relations for generalized Bernoulli numbers and polynomials. Journal of Number Theory, 176, 2017, 149-173, Elsevier, | DOI | MR

[2] Appell, P.: Sur une classe de polynômes. Annales Scientifiques de l'École Normale Supérieure, 9, 1880, 119-144,

[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, | DOI | MR

[4] Gel'fand, I.M., Shilov, G.E.: Generalized Functions, Vol. 1: Properties and Operations. 1964, Acad. Press, New York,

[5] Gelfand, M.B.: A note on a certain relation among Bernoulli numbers. Baškir. Gos. Univ. Ucen. Zap. Vyp, 31, 1968, 215-216,

[6] Gessel, I.M.: Applications of the classical umbral calculus. Algebra Univers., 49, 4, 2003, 397-434, | DOI | MR

[7] He, Y., Zhang, W.: Some symmetric identities involving a sequence of polynomials. The Electronic Journal of Combinatorics, 17, N7, 2010, 7pp, | DOI | MR

[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,

[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, | DOI | MR

[10] Momiyama, H.: A new recurrence formula for Bernoulli numbers. Fibonacci Quarterly, 39, 3, 2001, 285-288, THE FIBONACCI ASSOCIATION, | MR

[11] Prévost, M.: Padé approximation and Apostol-Bernoulli and Apostol-Euler polynomials. Journal of computational and applied mathematics, 233, 11, 2010, 3005-3017, Elsevier, | DOI | MR

[12] Rota, G.-C.: The number of partitions of a set. The American Mathematical Monthly, 71, 5, 1964, 498-504, Taylor & Francis, | DOI

[13] Stern, M.: Beiträge zur Theorie der Bernoullischen und Eulerschen Zahlen. Abhandlungen der Königlichen Gesellschaft der Wissenschaften in Göttingen, 26, 1880, 3-46,

[14] Sun, Z.-W.: Combinatorial identities in dual sequences. European Journal of Combinatorics, 24, 6, 2003, 709-718, Elsevier, | DOI | MR

[15] Ettingshausen, A. von: Vorlesungen über die höhere Mathematik. 1827, C. Gerold, Vienna,

[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,

[17] Wu, K.-J., Sun, Z.-W., Pan, H.: Some identities for Bernoulli and Euler polynomials. Fibonacci Quarterly, 42, 4, 2004, 295-299, | MR