scholarly journals On ther-Shifted Central Coefficients of Riordan Matrices

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Sai-nan Zheng ◽  
Sheng-liang Yang
Keyword(s):  
Generating Functions ◽  
Hitting Time ◽  
Central Column ◽  
Riordan Group ◽  
The Right

By presenting Riordan matrix as a triangle, the central coefficients are entries in the central column. Starting at the central column, ther-shifted central coefficients are entries in columnrof the right part of the triangle. This paper aims to characterize ther-shifted central coefficients of Riordan matrices. Here we will concentrate on four elements of the subgroups of the Riordan group, that is, the Bell subgroup, the associated subgroup, the derivative subgroup, and the hitting time subgroup. Some examples are presented to show how we deduce the generating functions for interesting sequences by using different means of calculating theser-shifted central coefficients. Besides, we make some extensions in the Bell subgroup.

scholarly journals Standard Paths in Another Composition Poset

10.37236/1829 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Jan Snellman
Keyword(s):  
Generating Functions ◽  
Hasse Diagram ◽  
Term Orders ◽  
The Right

Bergeron, Bousquet-Mélou and Dulucq [Ann. Sci. Math. Québec 19 (1995), 139–151] enumerated paths in the Hasse diagram of the following poset: the underlying set is that of all compositions, and a composition $\mu$ covers another composition $\lambda$ if $\mu$ can be obtained from $\lambda$ by adding $1$ to one of the parts of $\lambda$, or by inserting a part of size $1$ into $\lambda$. We employ the methods they developed in order to study the same problem for the following poset, which is of interest because of its relation to non-commutative term orders : the underlying set is the same, but $\mu$ covers $\lambda$ if $\mu$ can be obtained from $\lambda$ by adding $1$ to one of the parts of $\lambda$, or by inserting a part of size $1$ at the left or at the right of $\lambda$. We calculate generating functions for standard paths of fixed width and for standard paths of height $\le 2$.

L'intégrale du mouvement brownien

1993 ◽  
Vol 30 (01) ◽  
pp. 17-27
Author(s):  
Aimé Lachal
Keyword(s):  
Fourier Transform ◽  
Brownian Motion ◽  
Hitting Time ◽  
First Hitting Time ◽  
Brownian Motion Process ◽  
Classical Technique ◽  
Motion Process ◽  
Mouvement Brownien ◽  
The Law ◽  
The Right

Let be the Brownian motion process starting at the origin, its primitive and Ut = (Xt+x + ty, Bt + y), , the associated bidimensional process starting from a point . In this paper we present an elementary procedure for re-deriving the formula of Lefebvre (1989) giving the Laplace–Fourier transform of the distribution of the couple (σ α, Uσa ), as well as Lachal's (1991) formulae giving the explicit Laplace–Fourier transform of the law of the couple (σ ab, Uσab ), where σ α and σ ab denote respectively the first hitting time of from the right and the first hitting time of the double-sided barrier by the process . This method, which unifies and considerably simplifies the proofs of these results, is in fact a ‘vectorial' extension of the classical technique of Darling and Siegert (1953). It rests on an essential observation (Lachal (1992)) of the Markovian character of the bidimensional process . Using the same procedure, we subsequently determine the Laplace–Fourier transform of the conjoint law of the quadruplet (σ α, Uσa, σb, Uσb ).

The random walk associated by the game of roulette

1981 ◽  
Vol 18 (04) ◽  
pp. 931-936
Author(s):  
James M. Hill ◽  
Chandra M. Gulati
Keyword(s):  
Random Walk ◽  
Power Series ◽  
Generating Functions ◽  
The Right ◽  
Lagrange's Theorem ◽  
Explicit Expressions

The random walk arising in the game of roulette involves an absorbing barrier at the origin; at each step either a unit displacement to the left or a fixed multiple displacement to the right can occur with probabilities q and p respectively. Using generating functions and Lagrange's theorem for the expansion of a function as a power series explicit expressions are deduced for the probabilities of the player's capital at the nth step.

L'intégrale du mouvement brownien

1993 ◽  
Vol 30 (1) ◽  
pp. 17-27 ◽  
Author(s):  
Aimé Lachal
Keyword(s):  
Fourier Transform ◽  
Brownian Motion ◽  
Hitting Time ◽  
First Hitting Time ◽  
Brownian Motion Process ◽  
Classical Technique ◽  
Motion Process ◽  
Mouvement Brownien ◽  
The Law ◽  
The Right

Let be the Brownian motion process starting at the origin, its primitive and Ut = (Xt+x + ty, Bt + y), , the associated bidimensional process starting from a point . In this paper we present an elementary procedure for re-deriving the formula of Lefebvre (1989) giving the Laplace–Fourier transform of the distribution of the couple (σ α, Uσa), as well as Lachal's (1991) formulae giving the explicit Laplace–Fourier transform of the law of the couple (σ ab, Uσab), where σ α and σ ab denote respectively the first hitting time of from the right and the first hitting time of the double-sided barrier by the process . This method, which unifies and considerably simplifies the proofs of these results, is in fact a ‘vectorial' extension of the classical technique of Darling and Siegert (1953). It rests on an essential observation (Lachal (1992)) of the Markovian character of the bidimensional process .Using the same procedure, we subsequently determine the Laplace–Fourier transform of the conjoint law of the quadruplet (σ α, Uσa, σb, Uσb).

The random walk associated by the game of roulette

1981 ◽  
Vol 18 (4) ◽  
pp. 931-936 ◽  
Author(s):  
James M. Hill ◽  
Chandra M. Gulati
Keyword(s):  
Random Walk ◽  
Power Series ◽  
Generating Functions ◽  
The Right ◽  
Lagrange's Theorem ◽  
Explicit Expressions

The random walk arising in the game of roulette involves an absorbing barrier at the origin; at each step either a unit displacement to the left or a fixed multiple displacement to the right can occur with probabilities q and p respectively. Using generating functions and Lagrange's theorem for the expansion of a function as a power series explicit expressions are deduced for the probabilities of the player's capital at the nth step.

A modified random walk in the presence of partially reflecting barriers

1976 ◽  
Vol 13 (1) ◽  
pp. 169-175 ◽  
Author(s):  
Saroj Dua ◽  
Shobha Khadilkar ◽  
Kanwar Sen
Keyword(s):  
Random Walk ◽  
Generating Functions ◽  
One Dimensional ◽  
Unit Step ◽  
The One ◽  
The Right ◽  
Reflecting Barriers

The paper deals with the one-dimensional modified random walk in the presence of partially reflecting barriers at a and –b (a, b > 0). The simple one-dimensional random walk on a line is the motion-record of a particle which may extend over (–∞, + ∞) or be restricted to a portion of it by absorbing and/or reflecting barriers. Here we introduce the possibility of a particle staying put along with its moving a unit step to the right or to the left and find the bivariate generating functions of the probabilities of a particle reaching m (0 <m <a) under different conditions.

scholarly journals The Double Riordan Group

10.37236/2034 ◽  
2012 ◽  
Vol 18 (2) ◽  
Author(s):  
Dennis E. Davenport ◽  
Louis W. Shapiro ◽  
Leon C. Woodson
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Triangular Matrices ◽  
Ordered Trees ◽  
Dyck Paths ◽  
Motzkin Numbers ◽  
The Matrix ◽  
Riordan Group ◽  
Lower Triangular ◽  
Lower Triangular Matrices

The Riordan group is a group of infinite lower triangular matrices that are defined by two generating functions, $g$ and $f$. The kth column of the matrix has the generating function $gf^k$. In the Double Riordan group there are two generating function $f_1$ and $f_2$ such that the columns, starting at the left, have generating functions using $f_1$ and $f_2$ alternately. Examples include Dyck paths with level steps of length 2  allowed at even height and also ordered trees with differing degree possibilities at even and odd height(perhaps representing summer and winter). The Double Riordan group is a generalization not of the Riordan group itself but of the checkerboard subgroup. In this context both familiar and far less familiar sequences occur such as the Motzkin numbers and the number of spoiled child trees. The latter is a slightly enhanced cousin of ordered trees which are counted by the Catalan numbers.

A modified random walk in the presence of partially reflecting barriers

1976 ◽  
Vol 13 (01) ◽  
pp. 169-175
Author(s):  
Saroj Dua ◽  
Shobha Khadilkar ◽  
Kanwar Sen
Keyword(s):  
Random Walk ◽  
Generating Functions ◽  
One Dimensional ◽  
Unit Step ◽  
The One ◽  
The Right ◽  
Reflecting Barriers

The paper deals with the one-dimensional modified random walk in the presence of partially reflecting barriers at a and –b (a, b &gt; 0). The simple one-dimensional random walk on a line is the motion-record of a particle which may extend over (–∞, + ∞) or be restricted to a portion of it by absorbing and/or reflecting barriers. Here we introduce the possibility of a particle staying put along with its moving a unit step to the right or to the left and find the bivariate generating functions of the probabilities of a particle reaching m (0 &lt;m &lt;a) under different conditions.

scholarly journals Solution and Analysis of a One-Dimensional First-Passage Problem with a Nonzero Halting Probability

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Ken Yamamoto
Keyword(s):  
Moment Generating Function ◽  
Hitting Time ◽  
Gauss Hypergeometric Function ◽  
Time Behavior ◽  
First Passage ◽  
One Dimensional ◽  
Random Walker ◽  
Calculation Technique ◽  
The Right ◽  
First Passage Problem

This paper treats a kind of a one-dimensional first-passage problem, which seeks the probability that a random walker first hits the origin at a specified time. In addition to a usual random walk which hops either rightwards or leftwards, the present paper introduces the “halt” that the walker does not hop with a nonzero probability. The solution to the problem is expressed using a Gauss hypergeometric function. The moment generating function of the hitting time is also calculated, and a calculation technique of the moments is developed. The author derives the long-time behavior of the hitting-time distribution, which exhibits power-law behavior if the walker hops to the right and left with equal probability.

scholarly journals A note on the random walk model arising in double diffusion

1982 ◽  
Vol 24 (2) ◽  
pp. 121-129 ◽  
Author(s):  
L. H. Liyanage ◽  
J. M. Hill ◽  
C. M. Gulati
Keyword(s):  
Random Walk ◽  
Probability Distribution ◽  
Diffusion Model ◽  
General Result ◽  
Generating Functions ◽  
Double Diffusion ◽  
Horizontal Position ◽  
Special Cases ◽  
One Step ◽  
The Right

AbstractThe discrete random walk problem for the unrestricted particle formulated in the double diffusion model given in Hill [2] is solved explicitly. In this model it is assumed that a particle moves along two distinct horizontal paths, say the upper path I and lower path 2. For i = 1, 2, when the particle is in path i, it can move at each jump in one of four possible ways, one step to the right with probability pi, one step to the left with probability qi, remains in the same position with probability ri, or exchanges paths but remains in the same horizontal position with probability si (pi + qi + ri + si = 1). Using generating functions, the probability distribution of the position of an unrestricted particle is derived. Finally some special cases are discussed to illustrate the general result.

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