ON THE 2-ORTHOGONAL POLYNOMIALS AND THE GENERALIZED BIRTH AND DEATH PROCESSES

The birth and death processes are closely related to the orthogonal polynomials. The latter allows determining the stochastic matrix associated with these processes. Let us also note that these processes are stationary Markov processes whose state space is the nonnegative integers. Many authors treated the question of the existing relationship between the birth and death processes and the orthogonal polynomials, in particular, in the works of Karlin and McGregor [6] and Ismail et al. [5]. The properties of these processes and of the orthogonal polynomials were the subject of other works, we can quote by the way of example Ismail et al. [3, 4], Maki [8], and Letessier and Valent [7]. In this paper, we will consider not only the orthogonal polynomials, but the 2-orthogonal polynomials and we will try to establish a bond between the latter and certain birth and death processes that will be called “generalized.” These processes will have like transition probabilities


Introduction
The birth and death processes are closely related to the orthogonal polynomials.The latter allows determining the stochastic matrix associated with these processes.
Let us also note that these processes are stationary Markov processes whose state space is the nonnegative integers.
Many authors treated the question of the existing relationship between the birth and death processes and the orthogonal polynomials, in particular, in the works of Karlin and McGregor [6] and Ismail et al. [5].
The properties of these processes and of the orthogonal polynomials were the subject of other works, we can quote by the way of example Ismail et al. [3,4], Maki [8], and Letessier and Valent [7].
In this paper, we will consider not only the orthogonal polynomials, but the 2-orthogonal polynomials and we will try to establish a bond between the latter and certain birth and death processes that will be called "generalized."These processes will have like transition probabilities which satisfy, when h → 0, 2 2-orthogonal polynomials and birth and death processes where λ i are the birth rates and μ i and μ i are the death rates.It is also assumed that These processes being stationary, P i j (t) does not depend on the way taken by the system to reach the state j, but depends only on the states i and j and of the laps of time t taken while going from state i towards state j.This is equivalent to In this work, we start initially by giving some properties of the "generalized" birth and death processes by determining the sequence of 2-orthogonal polynomials associated with this type of processes.Then, the sufficient conditions are given, which allow giving an integral representation of the transition probabilities from these processes.
We can quote as an example that this type of processes can be the modelling of problems met while studying the kinetics of enzymes, in particular, those which catalyze reactions to a substrate in the presence of noncompetitive inhibitors.
We will treat this type of model in future, when we study the generalized linear processes.

The generalized Chapman-Kolmogorov equations
Proposition 2.1.Let be a "generalized" birth and death process, where the transition probabilities are given by (1.2), then these probabilities satisfy two systems of differential recurrence relations called "Chapman-Kolmogorov (or C-K) equations."The forward C-K equations are d dt P i j (t) = μ j+2 P i, j+2 (t) + μ j+1 P i, j+1 (t) + λ j−1 P i, j−1 (t), − μ j + μ j + λ j P i j (t). (2.1) The backward C-K equations are (2.2) Proof.Since the process is stationary, on one hand we can write On the other hand, by applying the semigroup property, we have also then from (1.2), we have ( So, we get (2.2).
Definition 2.2.The matrix Ꮽ = (a i j ) i, j∈N defined by is called the infinitesimal generator of the process.
Lemma 2.3.A generalized birth and death process has the following properties: , where ᏼ(t) = (P i j (t)) i, j∈N is the transition matrix.Now, we will seek a solution of the Chapman-Kolmogorov equations, by using the method of separation of variables.So if we put we will get the following lemma.
Remark 2.5.(a) We have written F i (x) and Q j (x) to exhibit the dependence of F i and Q j on the constant x.
(b) It is easy to see that the functions F i (x) and Q j (x) defined, respectively, by the recurrence formulas (2.10) and (2.11) are polynomials.Moreover, let us note that degF k = k for all k ∈ N.
To characterize the sequence of polynomials {F i (x)} i∈N , we will introduce d-orthogonality notion.

d-orthogonality
Definition 3.1.Let {B n (x)} n≥0 be a monic sequence of polynomials and {ᏸ n } n≥0 a sequence of linear forms.{ᏸ n } n≥0 is called the dual sequence of {B n (x)} n≥0 if and only if Proposition 3.3 [2,9].Let {B n (x)} n≥0 be a monic sequence of polynomials, then the following statements are equivalent.
Remark 3.4.From this proposition, we deduce that the sequence of polynomials {F n } n∈N is 2-orthogonal with respect to ᏸ = (ᏸ 0 ,ᏸ 1 ) T .

Integral representation
In this section, we try to give an integral representation of the P i j (t), which are the solutions of the Chapman-Kolmogov equations.First, we give the following lemma.
Lemma 4.1.Let there be a generalized birth and death process, where the transition probabilities are given by (1.2), then the sequence {F n } n≥0 given by (2.10) satisfies the following properties.
(A) Putting F n (x) = k≥0 a n,k x k for all n ≥ 0, then 6 2-orthogonal polynomials and birth and death processes The moments of the two orthogonality forms c σ n = ᏸ σ (x n ), n ≥ 0, and σ = 0,1 are given by The following equation holds: (C) The sequence {F n (0)} n≥0 is a nondecreasing sequence and F n (0) ≥ 1 (n ≥ 0).
Proof.(A) If we put F n (x) = n k=0 a n,k x k , n ≥ 0, then from (2.10) we will deduce that where So, we get (4.1).From the definition of the dual sequence we find (4.2). (B) Let { f n } n∈N be a sequence of functions defined by or f (x,t) = ᏼ(t)Ᏺ(x), where f (x,t) = ( f n (x,t)) n∈N and Ᏺ(x) = (F n (x)) n∈N , then From (2.10), we have −xᏲ(x) = ᏭᏲ(x).So, Then, we get (4.3).
Proof.From (4.3), we deduce that {P ni (t)F i (x)} i≥0 is a bounded sequence for all n ∈ N and x, t ≥ 0. In particular, there exists a sequence of functions M n (t) such that then we have  (4.17)Corollary 4.5.Let ᏼ(t) = (P i j (t)) i, j∈N be a solution of (1.3), (2.1), and (2.2).Then the P i j (t) given by relation (4.14) can be written in the following form: Proof.We have for t ≥ 0 and taking into account that Z. Ebtissem and B. Ammar 9 for all n and m ∈ N, which is a system with lower triangular matrix of the form where the determinant of Ꮾ m is So, this system admits a unique solution given by (4.18) for all n ≥ 0 and σ = 0,1.Now, we give the sufficient conditions so that the P i j (t) given by the integral representation (4.14) is indeed a transition probability.
First, we give the following lemma.
Lemma 4.6.Let there be a system with a lower triangular matrix of the form M n X n = C n , where (4.25) Then the following properties are equivalent.
(A) n i=0 x i ≤ 1. (B) detD n and det M n have the same sign, where D n is the matrix defined by

.26)
Proof.Using Cramer's rule, on one hand we can get and on the other hand, we have  (ii) detᏰ nm ≥ 0 for all n,m ∈ N, where Ᏸ nm is the following matrix: b 00 0 0 b 10 b 11 0 where q n j and b i j are given by ( 4
.28) So, detD n = detM n (1 − n i=0 x i ) with detM n = 0, consequently we deduce that (A) and (B) are equivalent.From Corollary 4.5 and Lemma 4.1, we have the following result.