English

Riordan arrays and generalized Euler polynomials

Number Theory 2017-09-21 v2

Abstract

Generalization of the Euler polynomials An(x)=(1x)n+1m=0mnxm{{A}_{n}}\left( x \right)={{\left( 1-x \right)}^{n+1}}\sum\nolimits_{m=0}^{\infty }{{{m}^{n}}{{x}^{m}}} are the polynomials αn(x)=(1x)n+1m=0un(m)xm{{\alpha }_{n}}\left( x \right)={{\left( 1-x \right)}^{n+1}}\sum\nolimits_{m=0}^{\infty }{{{u}_{n}}}\left( m \right){{x}^{m}}, where un(x){{u}_{n}}\left( x \right) is the polynomial of degree nn. These polynomials appear in various fields of mathematics, which causes a variety of methods for their study. In present paper we will consider generalized Euler polynomials as an attribute of the theory of Riordan arrays. From this point of view, we will consider the transformations associated with them, with a participation of such objects as binomial sequences, Stirling numbers, multinomial coefficients, shift operator, and demonstrate a constructiveness of the chosen point of view.

Keywords

Cite

@article{arxiv.1709.02229,
  title  = {Riordan arrays and generalized Euler polynomials},
  author = {E. Burlachenko},
  journal= {arXiv preprint arXiv:1709.02229},
  year   = {2017}
}

Comments

Corrected typos

R2 v1 2026-06-22T21:35:55.730Z