symmetric identity
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2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Farid Bencherif ◽  
Rachid Boumahdi ◽  
Tarek Garici

Abstract Using umbral calculus, we establish a symmetric identity for any sequence of polynomials satisfying A′ n +1(x) = (n + 1)An (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-Euler polynomials and generalized Bernoulli polynomials attached to a primitive Dirichlet character.


Mathematics ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 238
Author(s):  
Kwang-Wu Chen

A preferential arrangement on [ [ n ] ] = { 1 , 2 , … , n } is a ranking of the elements of [ [ n ] ] where ties are allowed. The number of preferential arrangements on [ [ n ] ] is denoted by r n . The Delannoy number D ( m , n ) is the number of lattice paths from ( 0 , 0 ) to ( m , n ) in which only east ( 1 , 0 ) , north ( 0 , 1 ) , and northeast ( 1 , 1 ) steps are allowed. We establish a symmetric identity among the numbers r n and D ( p , q ) by means of algebraic and combinatorial methods.


2018 ◽  
Vol 27 (1) ◽  
pp. 98-118
Author(s):  
Emanuele Munarini

Abstract In 1997, Richard André-Jeannin obtained a symmetric identity involving the reciprocal of the Horadam numbers Wn, defined by a three-term recurrence Wn+2 = P Wn+1 − QWn with constant coefficients. In this paper, we extend this identity to sequences {an}n∈ℕ satisfying a three-term recurrence an+2 = pn+1an+1 + qn+1an with arbitrary coefficients. Then, we specialize such an identity to several q-polynomials of combinatorial interest, such as the q-Fibonacci, q-Lucas, q-Pell, q-Jacobsthal, q-Chebyshev and q-Morgan-Voyce polynomials.


2017 ◽  
Vol 11 ◽  
pp. 277-281
Author(s):  
Karina Kfouri Sartori ◽  
Sergio Mota Alves

2009 ◽  
Vol 161 (1) ◽  
pp. 11-30 ◽  
Author(s):  
A. S. Dzhumadil’daev
Keyword(s):  

1999 ◽  
Vol 35 (23) ◽  
pp. 2022
Author(s):  
DaeHun Nyang ◽  
JooSeok Song

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