Identities for generalized Euler polynomials

Jiu, Lin; Moll, Victor H.; Vignat, Christophe

doi:10.1080/10652469.2014.918613

Cited by 8 publications

(8 citation statements)

References 4 publications

(2 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Remark. The coefficients of 1 T h (1/z) have also been studied in [9]. There, the case of fixed h is investigated, whereas we mostly focus on the asymptotic behavior of Remark (Iverson's notation).…”

Section: Chebyshev Polynomials and Random Walksmentioning

confidence: 99%

“…To compute the asymptotic expansions for the first moments, we need β(1) = π/4, β(2) = G ≈ 0.91597, as well as β(3) = π 3 /32, where G is the Catalan constant. These values are taken from [12, At this point, all that remains to obtain asymptotic expansions is to multiply the contributions resulting from Lemma 3.2 with the correct coefficients and contributions from (9).…”

Section: Admissible Random Walks On Nmentioning

confidence: 99%

See 1 more Smart Citation

Analysis of Bidirectional Ballot Sequences and Random Walks Ending in Their Maximum

Hackl¹,

Heuberger²,

Prodinger³

et al. 2016

Ann. Comb.

View full text Add to dashboard Cite

Consider non-negative lattice paths ending at their maximum height, which will be called admissible paths. We show that the probability for a lattice path to be admissible is related to the Chebyshev polynomials of the first or second kind, depending on whether the lattice path is defined with a reflective barrier or not. Parameters like the number of admissible paths with given length or the expected height are analyzed asymptotically. Additionally, we use a bijection between admissible random walks and special binary sequences to prove a recent conjecture by Zhao on ballot sequences.2010 Mathematics Subject Classification. 05A16; 05A15, 05A10, 60C05.

show abstract

Section: Chebyshev Polynomials and Random Walksmentioning

confidence: 99%

Section: Admissible Random Walks On Nmentioning

confidence: 99%

Analysis of Bidirectional Ballot Sequences and Random Walks Ending in Their Maximum

Hackl¹,

Heuberger²,

Prodinger³

et al. 2016

Ann. Comb.

View full text Add to dashboard Cite

show abstract

“…It is surprising that the positive coefficients p (N ) also appear as transition probabilities in the context of a random walk over a finite number of sites [4,Note 4.8], which reveals the possibility to connect random walks and E (p) n (x), as well as B (p) n (x).…”

Section: Introductionmentioning

confidence: 99%

Random Walk Models for Nontrivial Identities of Bernoulli and Euler Polynomials

Jiu¹,

Simonelli²,

Yue³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

We consider the 1-dimensional reflected Brownian motion and 3-dimensional Bessel process and the general models. By decomposing the hitting times of consecutive sites into loops, we obtain identities, called loop identities, for the generating functions of the hitting times. After proving this decomposition both combinatorially and inductively, we consider the case that sites are equally distributed. Then, from loop identities, we derive expressions of Bernoulli and Euler polynomials, in terms of Euler polynomials of higherorders.

show abstract

“…via the N -th Chebychev polynomial T N (t). These coefficients p (N ) l also appear as transition probabilities in the context of a random walk over a finite number of sites [5,Note 4.8,p. 787].…”

Section: Introductionmentioning

confidence: 99%

Connection Coefficients for Higher-order Bernoulli and Euler Polynomials: A Random Walk Approach

Jiu,

Vignat

2018

Preprint

Self Cite

View full text Add to dashboard Cite

We consider the use of random walks as an approach to obtain connection coefficients for higher-order Bernoulli and Euler polynomials. In particular, we consider the cases of a 1-dimensional linear reflected Brownian motion and of a 3-dimensional Bessel process. Considering the successive hitting times of two, three, and four fixed levels by these random walks yields non-trivial identities that involve higher-order Bernoulli and Euler polynomials.

show abstract

Identities for generalized Euler polynomials

Abstract: Abstract. For N ∈ N, let TN be the Chebyshev polynomial of the first kind. Expressions for the sequence of numbers p

Cited by 8 publications

References 4 publications

Analysis of Bidirectional Ballot Sequences and Random Walks Ending in Their Maximum

Analysis of Bidirectional Ballot Sequences and Random Walks Ending in Their Maximum

Random Walk Models for Nontrivial Identities of Bernoulli and Euler Polynomials

Connection Coefficients for Higher-order Bernoulli and Euler Polynomials: A Random Walk Approach

Contact Info

Product

Resources

About