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

@article{Hackl2015AnalysisOB,
  title={Analysis of Bidirectional Ballot Sequences and Random Walks Ending in Their Maximum},
  author={Benjamin Hackl and Clemens Heuberger and H. Prodinger and Stephan G. Wagner},
  journal={Annals of Combinatorics},
  year={2015},
  volume={20},
  pages={775-797}
}
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… 
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References

SHOWING 1-10 OF 19 REFERENCES
Constructing MSTD Sets Using Bidirectional Ballot Sequences
Identities for generalized Euler polynomials
For N∈ℕ, let TN be the Chebyshev polynomial of the first kind. Expressions for the sequence of numbers , defined as the coefficients in the expansion of 1/TN(1/z), are provided. These coefficients
The On-Line Encyclopedia of Integer Sequences
  • N. Sloane
  • Computer Science
    Electron. J. Comb.
  • 1994
TLDR
The On-Line Encyclopedia of Integer Sequences (or OEIS) is a database of some 130000 number sequences which serves as a dictionary, to tell the user what is known about a particular sequence and is widely used.
A Course in Enumeration
Basics.- Fundamental Coefficients.- Formal Series and Infinite Matrices.- Methods.- Generating Functions.- Hypergeometric Summation.- Sieve Methods.- Enumeration of Patterns.- Topics.- The Catalan
On the Coefficients of the Asymptotic Expansion of n
Applying a theorem of Howard for a formula recently proved by Brassesco and M\'endez, we derive new simple explicit formulas for the coefficients of the asymptotic expansion to the sequence of
NIST Handbook of Mathematical Functions
TLDR
This handbook results from a 10-year project conducted by the National Institute of Standards and Technology with an international group of expert authors and validators and is destined to replace its predecessor, the classic but long-outdated Handbook of Mathematical Functions, edited by Abramowitz and Stegun.
Concrete mathematics - a foundation for computer science
From the Publisher: This book introduces the mathematics that supports advanced computer programming and the analysis of algorithms. The primary aim of its well-known authors is to provide a solid
...
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