Exactly solvable birth and death processes

Title

Exactly solvable birth and death processes

Author(s)

Sasaki, Ryu

Citation

Journal of Mathematical Physics (2009), 50(10)

Issue Date

2009-10

URL

http://hdl.handle.net/2433/87411

Right

c 2009 American Institute of Physics.

Type

Journal Article

Exactly solvable birth and death processes

Ryu Sasakia兲

Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan

共Received 28 May 2009; accepted 7 August 2009; published online 1 October 2009兲 Many examples of exactly solvable birth and death processes, a typical stationary Markov chain, are presented together with the explicit expressions of the transition probabilities. They are derived by similarity transforming exactly solvable “matrix” quantum mechanics, which is recently proposed by Odake and the author关S. Odake and R. Sasaki, J. Math. Phys. 49, 053503 共2008兲兴. The 共q-兲 Askey scheme of hypergeometric orthogonal polynomials of a discrete variable and their dual poly-nomials play a central role. The most generic solvable birth/death rates are rational functions of qx _{共with x being the population兲 corresponding to the q-Racah} polynomial. © 2009 American Institute of Physics.关doi:10.1063/1.3215983兴

I. INTRODUCTION

The Brownian motion, a typical stationary Markov process with a continuous state space, is known to be described well by the Fokker–Planck equation.1,2A birth and death process, on the other hand, being a typical stationary Markov chain with a set of non-negative integers as a state space,2,3can be naturally considered as a discretization of a one-dimensional共1D兲 Fokker–Planck equation. Although birth and death processes have a wide range of applications,2,3such as demog-raphy, queueing theory, inventory models, and chemical dynamics, we will focus on their math-ematical aspect, i.e., the exact solvability. In this paper, we present 18 exactly solvable birth and death processes based on the共q-兲 Askey scheme of hypergeometric orthogonal polynomials hav-ing discrete orthogonality measures. They are also called orthogonal polynomials of a discrete variable.4–6 For example, they are the共q-兲 Racah, the 共q-兲 共dual兲Hahn, the 共q-兲 Krawtchouk, the 共q-兲 Charlier, and the 共q-兲 Meixner polynomials.4,5,7

Various expressions of the transition prob-ability are given explicitly together with the totality of the eigenvalues and the measures of the Karlin–McGregor type representation.8

It is well known that the 1D Fokker–Planck equation is related by a similarity transformation to a corresponding 1D time-independent Schrödinger equation1or the eigenvalue problem for a suitable Hamiltonian. In other words, solutions of an exactly solvable Schrödinger equation give the solutions of the corresponding Fokker–Planck equation, which is now exactly solvable. Exact solvability means that the totality of the eigenvalues 共in these cases, all are discrete兲 and the corresponding eigenfunctions are obtained exactly. Here the Hamiltonian in quantum mechanics is a Hermitian共self-adjoint兲 linear operator in a certain Hilbert space. A natural discretization of the Hamiltonians of 1D quantum mechanics is Hermitian matrices of a finite or infinite dimensions. Recently, exactly solvable “matrix” quantum mechanics was proposed by Odake and the present author9by adopting special types of tridiagonal Jacobi matrices of finite or infinite dimensions as Hamiltonians. The eigenfunctions are spanned by the abovementioned orthogonal polynomials of a discrete variable. The corresponding discretization of the Fokker–Planck equation is, as ex-pected, the birth and death process with a reflecting wall共s兲共3.20兲. Among the 18 exactly solvable birth and death processes to be explored in this paper, some are quite well known having the linear2,3,10,11 and quadratic12 birth and death rates, corresponding to the Meixner 共Sec. IV B 1兲, Charlier 共Sec. IV B 2兲, Krawtchouk 共Sec. IV A 4兲, and Hahn 共Sec. IV A 2兲 polynomials. The

a兲_{Electronic mail: [email protected].}

50, 103509-1

others have rational functions共of the population x兲 of the birth and death rates corresponding to the dual Hahn 共Sec. IV A 3兲 and Racah 共Sec. IV A 1兲 polynomials and some others have

q⫾x-linear, quadratic, and rational birth and death rates. The most generic one is the q-Racah polynomial共Sec. IV A 5兲 having qx_{rational birth and death rates共4.21兲.}

This paper is organized as follows. In Sec. II, the general properties of the Hamiltonians in 1D quantum mechanics 共and/or the Hermitian matrices兲 are reviewed in Sec. II A. The relationship between the Schrödinger equation and the corresponding Fokker–Planck equation is recapitulated in Sec. II B and the solutions of the initial value problem of the Fokker–Planck equations and the transition probabilities are expressed in terms of the orthogonal polynomials constituting the eigenfunctions of the corresponding Schrödinger equation. In Sec. III the birth and death operator is derived from the generic form of the Hamiltonian of the exactly solvable matrix quantum mechanics of Ref.9. The solutions of the initial value problem of the birth and death equations and the transition probabilities are expressed in terms of the orthogonal polynomials constituting the eigenfunctions of the corresponding Schrödinger equation of the matrix quantum mechanics. Various equivalent expressions of the transition probabilities are derived in terms of the dual polynomials. Section IV provides various data, the birth and death rates, the energy spectra and the sinusoidal coordinates, the stationary probability, the normalization constants, and the eigenpoly-nomials of the exactly solvable 18 models, which are sufficient to evaluate the transition prob-ability explicitly. These 18 models are named after the eigenpolynomials, such as the共q-兲 Racah, etc. Section V is for a brief summary and comments. The Appendix provides the collection of the definitions of basic symbols and functions for self-containedness. Throughout this paper we use the parameter q in the range of 0⬍q⬍1.

II. FOKKER–PLANCK OPERATOR FROM HAMILTONIAN

Here we recapitulate the well-known connection between the Fokker–Planck equation and the Schrödinger equation1in order to introduce appropriate notation and settings for the main purpose of the paper; connecting the birth and death process to the matrix quantum mechanics to be explored in Sec. III.

A. Properties of Hamiltonians

Throughout this paper we discuss one degree of freedom systems only. The Hamiltonians to be discussed in this paper are time independent and share the properties listed below. Most properties are common to the Hamiltonians having the continuous dynamical variable x 共to be used for the Fokker–Planck equation兲 and the discrete dynamical variable x 共to be applied to the birth and death processes兲. They are expressed by the same symbols. When they need different symbols, such as the L2 andᐉ2norms, two different expressions are shown in a curly bracket as in共2.3兲and共2.7兲. The upper共lower兲 one is for the continuous 共discrete兲 dynamical variable case. The former共the continuous variable兲 case corresponds to the ordinary quantum mechanics and the “discrete” quantum mechanics with the pure imaginary shifts,13 which gives rise to the “de-formed” Fokker–Planck equations.14

共i兲 Factorizability,

H = A†_{A, 共2.1兲}

in which † denotes the Hermitian conjugation with respect to the standard L2 共ᐉ2兲 inner product关see共2.3兲兴. This also means that the Hamiltonian H is positive semidefinite. 共ii兲 Completeness of its eigenfunctions␾n共x兲 belonging to discrete eigenvalues 共all distinct兲,

H␾n共x兲 = E共n兲␾n共x兲, E共0兲 ⬍ E共1兲 ⬍ ¯ , 共2.2兲 and all the eigenvectors are square normalizable and orthogonal with each other

共␾n,␾m兲 = def

冦

冕

␾n共x兲ⴱ␾m共x兲dx

兺

_x ␾n共x兲ⴱ␾m共x兲

冧

= 1 dn 2␦nm, 0⬍ dn⬍ ⬁. 共2.3兲

The range of the integration 共summation兲 depends on the specific Hamiltonian. Any ele-ment in the Hilbert space H is expanded by兵␾n其,

∀ f 苸 H ⇒ f =

兺

n fn␾ˆn, ␾ˆn= def dn␾n, fn= def 共␾ˆ_{n, f兲. 共2.4兲}

Here and hereafter, fˆ denotes a normalized vector fˆ =

def

f/

冑

共f , f兲. We choose all the

eigen-functions兵␾n其 to be real, which is always possible in 1D quantum mechanics. 共iii兲 The ground state wave function␾₀ is annihilated byA and is positive everywhere,

A␾0共x兲 = 0 ⇒ H␾0共x兲 = 0, E共0兲 = 0, ␾0共x兲 ⬎ 0. 共2.5兲

共iv兲 The eigenfunction␾n共x兲 is ␾0共x兲 times a polynomial,

␾n共x兲 =␾0共x兲Pn共␩共x兲兲, n = 0,1,2, ... , P0⬅ 1, 共2.6兲

in which a real function␩共x兲 is called a sinusoidal coordinate.15,9,13In other words, Pn共␩兲 is an orthogonal polynomial with the orthogonality measure␾₀共x兲2_,

冦

冕

␾0共x兲2Pn共␩共x兲兲Pm共␩共x兲兲dx

兺

_x ␾0共x兲2Pn共␩共x兲兲Pm共␩共x兲兲

冧

= 1

dn

2␦nm. 共2.7兲

共v兲 The similarity transformed HamiltonianH with respect to␾0共x兲

H˜ =def␾0−1ⴰ H ⴰ␾0 共2.8兲 provides a differential (difference) equation governing the polynomial Pn共␩共x兲兲.

B. Fokker–Planck equation

The Fokker–Planck equation in one dimension reads as

⳵

⳵tP共x;t兲 = LFPP共x;t兲, P共x;t兲 ⱖ 0,

冕

P共x;t兲dx = 1, 共2.9兲

in whichP共x;t兲 is the probability distribution over certain continuous range of the parameter x; for example,共−⬁,⬁兲, 共0,⬁兲, or 共0,␲兲. The Fokker–Planck operator LFPcorresponding to the

Hamil-tonianH共2.2兲is defined by1,14 LFP= def −␾0ⴰ H ⴰ␾0 −1 , 共2.10兲

in which␾0is defined in共2.5兲.共It should be emphasized that the inverse similarity transformation

in terms of␾0 is used here: LFP= −␾20ⴰH˜ ⴰ␾0−2.兲 This guarantees that the eigenvalues of LFPare negative semidefinite. The square normalized ground state eigenfunction ␾0共x兲 provides the sta-tionary distribution␾ˆ₀共x兲2 _{of the corresponding Fokker–Planck operator,}

⳵ ⳵t␾

ˆ

0共x兲2= LFP␾ˆ0共x兲2= 0,

冕

␾ˆ0共x兲2dx = 1. 共2.11兲

It is obvious that␾₀共x兲␾n共x兲 is the eigenvector of the Fokker–Planck operator LFP,

LFP␾0共x兲␾n共x兲 = − E共n兲␾0共x兲␾n共x兲, n = 0,1, ... . 共2.12兲 Corresponding to an arbitrary initial probability distributionP共x;0兲 关with 兰P共x;0兲dx=1兴, which can be expressed as a linear combination of兵␾ˆ0共x兲␾ˆn共x兲其, n=0,1,...,

P共x;0兲 =␾ˆ 0共x兲

兺

n=0 ⬁ cn␾ˆn共x兲, c0= 1, cn= def 共␾ˆ_n,␾ˆ 0共x兲−1P共x;0兲兲, n = 1,2, ... , 共2.13兲

we obtain the solution of the Fokker–Planck equation,

P共x;t兲 =␾ˆ 0共x兲

兺

n=0

⬁

cne−E共n兲t␾ˆn共x兲, t ⬎ 0. 共2.14兲 This is a consequence of the completeness of the eigenfunctions兵␾n共x兲其 共the polynomials兲 of the HamiltonianH. The positivity of the spectrum E共n兲⬎0, nⱖ1共2.2兲guarantees that the stationary distribution␾ˆ₀2共x兲 is achieved at future infinity,

lim

t_→⬁P共x;t兲 =␾ ˆ

0

2_{共x兲. 共2.15兲}

The transition probability from y at t = 0关i.e., P共x;0兲=␦共x−y兲兴 to x at t is given by

P共y,x;t兲 =␾ˆ

0共x兲␾ˆ0共y兲−1

兺

n=0

⬁

e−E共n兲t␾ˆn共x兲␾ˆn共y兲, t ⬎ 0. 共2.16兲 In terms of the polynomial Pn共␩共x兲兲 共2.6兲, it is expressed as

P共y,x;t兲 =␾0共x兲2

兺

n=0 ⬁ dn 2 e−E共n兲tPn共␩共x兲兲Pn共␩共y兲兲, t ⬎ 0, 共2.17兲 in which dnis the normalization constants共2.3兲and共2.7兲.

As shown in Ref. 14 in some detail, various examples of exactly solvable quantum mechanics16,17 and the discrete quantum mechanics with the pure imaginary shifts13,18,15 provide many explicit cases in which the transition probability共2.16兲and共2.17兲can be obtained exactly. The corresponding orthogonal polynomials are the Hermite, Laguerre, and Jacobi polynomials in the ordinary quantum mechanics16,17and the Meixner–Pollaczek, continuous共dual兲 Hahn, Wilson, and Askey–Wilson polynomials14,13 and their degenerate polynomials, such as the continuous

q-Hermite polynomials.

III. BIRTH AND DEATH PROCESS FROM MATRIX QUANTUM MECHANICS

The birth and death equation is a discretization of the Fokker–Planck equation in one dimen-sion共2.9兲. It reads as

⳵

⳵tP共x;t兲 = 共LBDP兲共x;t兲, P共x;t兲 ⱖ 0,

兺

x

P共x;t兲 = 1, 共3.1兲

in whichP共x;t兲 is the probability distribution over a certain discrete set of the parameter x. Here, we simply take a set of consecutive non-negative integers, either finite or infinite,

The exactly solvable birth and death operator or a matrix LBDis derived from the generic form of

an exactly solvable HamiltonianH of a discrete quantum mechanics with real shifts,

H = def

−

冑

B共x兲e⳵

冑

D共x兲 −

冑

D共x兲e−⳵

冑

B共x兲 + B共x兲 + D共x兲, 共3.3兲

in which the two functions B共x兲 and D共x兲 are real and positive but vanish at the boundary,

B共x兲 ⬎ 0, D共x兲 ⬎ 0, D共0兲 = 0, B共N兲 = 0 for the finite case. 共3.4兲

The explicit forms of the functions B共x兲 and D共x兲 are given in each subsection of Sec. IV, which are named after the orthogonal polynomials appearing as the main part of the eigenfunctions. In the Hamiltonian共3.3兲e⫾⳵is a formal shift operator acting on a function f of x as

共e⫾⳵_{f兲共x兲 = f共x ⫾ 1兲.}

Thus the Schrödinger equationH␺共x兲=E␺共x兲 is a difference equation with real shifts, 共B共x兲 + D共x兲兲␺共x兲 −

冑

B共x兲D共x + 1兲␺共x + 1兲 −

冑

B共x − 1兲D共x兲␺共x − 1兲 = E␺共x兲,

x = 0,1, . . . ,共N兲, ... . 共3.5兲

The boundary condition D共0兲=0 is necessary for the term ␺共−1兲 not to appear, and B共N兲=0 is necessary for the term␺共N+1兲 not to appear in the finite dimensional matrix case.

Although the HamiltonianH共3.3兲is presented in a difference operator form, it is, in fact, a real symmetric tridiagonal共Jacobi兲 matrix,

H = 共Hx,y兲, Hx,y=Hy,x, 共3.6兲

Hx,y= −

冑

B共x兲D共x + 1兲␦x+1,y−

冑

B共x − 1兲D共x兲␦x−1,y+共B共x兲 + D共x兲兲␦x,y. 共3.7兲 As mentioned above, the Hamiltonian is factorizable共2.1兲,H=A†_A,

A†₌

冑

_{B共x兲 −}

冑

_D共x兲e−⳵_{, A =}

冑

_{B共x兲 − e}⳵

冑

_{D共x兲. 共3.8兲}

In the matrix form,A†_{has the diagonal and subdiagonal elements only andA has the diagonal and}

superdiagonal elements only 共A†_兲

x,y=

冑

B共x兲␦x,y−

冑

D共x兲␦x−1,y, Ax,y=

冑

B共x兲␦x,y−

冑

D共x + 1兲␦x+1,y. 共3.9兲 Equation共2.5兲determining the ground state wave function ␾₀is easy to solve sinceA␾₀= 0 is a two term recurrence relation,

␾0共x + 1兲

␾0共x兲

=

冑

B共x兲

D共x + 1兲. 共3.10兲

It can be solved elementarily with the boundary共initial兲 condition ␾₀共0兲=1,

␾0共x兲 =

冑

兿

y=0 x−1

B共y兲

D共y + 1兲, x = 1,2, . . . . 共3.11兲

With the standard convention兿_k=nn−1ⴱ =1, the expression共3.11兲is valid for x = 0, too. For the infinite matrix case, the requirement of the finiteᐉ2 _{norm of the eigenvectors}

兺

x=0 ⬁ ␾0共x兲2=

兺

x=0 ⬁

兿

y=0 x−1 B共y兲 D共y + 1兲⬍ ⬁ 共3.12兲

imposes constraints on the asymptotic behaviors of B共x兲 and D共x兲.

With the above explicit form of the ground state wave function ␾0共x兲, the similarity

trans-formed Hamiltonian共2.8兲is easily obtained

H˜ =def␾0−1ⴰ H ⴰ␾0= B共x兲共1 − e⳵兲 + D共x兲共1 − e−⳵兲. 共3.13兲

As mentioned above,H˜ provides the difference equation for the polynomial eigenfunctions, 共H˜ Pn兲共␩共x兲兲 = E共n兲Pn共␩共x兲兲, 共3.14兲 that is,

B共x兲共Pn共␩共x兲兲 − Pn共␩共x + 1兲兲兲 + D共x兲共Pn共␩共x兲兲 − Pn共␩共x − 1兲兲兲 = E共n兲Pn共␩共x兲兲. 共3.15兲 The eigenpolynomials 兵Pn其 are the orthogonal polynomials of a discrete variable. See Sec. 5 of Ref.9for various forms of B共x兲 and D共x兲 and the corresponding orthogonal polynomials. It is also

recapitulated in Sec. IV of this paper. For example, they are the共q-兲 Racah, the 共q-兲 共dual兲Hahn, the共q-兲 Krawtchouk, the 共q-兲 Charlier, and the 共q-兲 Meixner polynomials.4,5,7As a matrix, H˜ is another tridiagonal matrix

H˜ = 共H˜

x,y兲, H˜x,y= B共x兲共␦x,y−␦x+1,y兲 + D共x兲共␦x,y−␦x−1,y兲. 共3.16兲 Corresponding to共2.10兲, the inverse similarity transformation of the HamiltonianH supplies the

birth and death operator LBD, LBD=

def

−␾0ⴰ H ⴰ␾0−1=共e−⳵− 1兲B共x兲 + 共e⳵− 1兲D共x兲. 共3.17兲

Obviously, the stationary probability is given by␾ˆ₀共x兲2_{= d} 0 2_␾

0共x兲2. In the matrix form, LBDis again

tridiagonal

LBD=共LBD_x,y兲, LBD_x,y= B共x − 1兲␦x−1,y− B共x兲␦x,y+ D共x + 1兲␦x+1,y− D共x兲␦x,y. 共3.18兲 In fact, −LBDis the transposed matrix ofH˜ ,

− LBD=共H˜ 兲t, − LBD_x,y=H˜y,x. 共3.19兲 With the explicit form of the birth and death operator LBD, the birth and death equation共3.1兲in

our notation reads as

⳵

⳵tP共x;t兲 =

兺

y

LBD_x,yP共y;t兲 = − 共B共x兲 + D共x兲兲P共x;t兲 + B共x − 1兲P共x − 1;t兲 + D共x + 1兲P共x + 1;t兲.

共3.20兲 The standard interpretation is that x is the population of a group,P共x;t兲 is the probability for the group to have the population x at the time t, B共x兲 is the birth rate, and D共x兲 is the death rate when the population is x. It is quite easy to remember. This is to be compared to the standard notation, for example8共Sec. XVII.5 of Ref.2 and Sec. 5.2 of Ref.5兲,

⳵

in which␭n is the birth rate and␮nis the death rate. The above translation table of the notation will be helpful共see TableI兲.

The boundary condition for the finite case,␭N= 0共B共N兲=0兲共3.4兲is said that the system has a reflecting wall at the population N.

The transition probability from y at t = 0 关i.e., P共x;0兲=␦_x,y兴 to x at t has exactly the same expression as that in the Fokker–Planck equation共2.16兲,

P共y,x;t兲 =␾ˆ

0共x兲␾ˆ0共y兲−1

兺

n=0 e−E共n兲t_␾ˆ

n共x兲␾ˆn共y兲, t ⬎ 0. 共3.22兲 In terms of the polynomial Pn共␩共x兲兲 共2.6兲, it is expressed as

P共y,x;t兲 =␾0共x兲2

兺

n=0 dn

2

e−E共n兲tPn共␩共x兲兲Pn共␩共y兲兲, t ⬎ 0. 共3.23兲 It should be emphasized that in these formulas共3.22兲and共3.23兲everything is known including the measure in contradistinction to the general formula by Karlin–McGregor.8

Let us mention several equivalent expressions of the transition probability共3.23兲in terms of the dual polynomials.19–21,9 It is well known that with proper normalization

␩共0兲 = 0 = E共0兲, P0⬅ 1 ⬅ Q0, Pn共0兲 = Qx共0兲 = 1, 共3.24兲 the two polynomials, 兵Pn共␩兲其, and its dual polynomial, 兵Qx共E兲其, coincide at the integer lattice points9

Pn共␩共x兲兲 = Qx共E共n兲兲, n = 0,1, ... ,共N兲, ... , x = 0,1, ... ,共N兲, ... . 共3.25兲 The dual polynomial 兵Qx共E共n兲兲其, x=0,1,..., is a right eigenvector of the similarity transformed HamiltonianH˜ matrix with the eigenvalue E共n兲,

兺

_y H˜

x,yQy共E共n兲兲 = E共n兲Qx共E共n兲兲. 共3.26兲

The above equation is the three term recurrence relation for the dual polynomials兵Qx共E兲其, 共B共x兲 + D共x兲兲Qx共E共n兲兲 − B共x兲Qx+1共E共n兲兲 − D共x兲Qx−1共E共n兲兲 = E共n兲Qx共E共n兲兲, 共3.27兲 Q0= 1, Q1共E兲 = 共B共0兲 − E兲/B共0兲, Q2共E兲 = 共B共0兲 − E兲共B共1兲 + D共1兲 − E兲/共B共0兲B共1兲兲, ... .

共3.28兲 For historical reasons, this polynomial Qx共E兲 is called the birth and death polynomial or the Karlin–McGregor polynomial.8

In terms of the dual polynomials or the Karlin–McGregor polynomial, the transition probabil-ity is

TABLE I. Translation table.

Standarda This paper

Population n = 0 , 1 , . . . ,共N兲,... x = 0 , 1 , . . . ,共N兲,...

Probability pn共t兲 P共x;t兲

Birth rate ␭n共␭N= 0兲 B共x兲 共B共N兲=0兲

Death rate ␮n共␮0= 0兲 D共x兲 共D共0兲=0兲 a_{References2and5.}

P共y,x;t兲 =␾0共x兲2

兺

n=0 dn

2

e−E共n兲tQx共E共n兲兲Qy共E共n兲兲, t ⬎ 0. 共3.29兲 Following Ref.5, let us introduce

Fx共E共n兲兲 =

def

␾0共x兲2Qx共E共n兲兲. 共3.30兲 Since LBDandH˜ are related by

L_BD= −␾₀2ⴰ H˜ ⴰ␾₀−2, 共3.31兲 it is easy to see that Fx共E共n兲兲 is a left eigenvector of H˜ and thus a right eigenvector of the birth and death operator LBD,

兺

_y L_BD x,yFy共E共n兲兲 = −␾0共x兲 2

兺

y H˜

x,yQy共E共n兲兲 = − E共n兲␾0共x兲2Qx共E共n兲兲 = − E共n兲Fx共E共n兲兲. 共3.32兲 In terms of the right eigenvectors of LBD, we obtain another expression of the transition

probability5

P共y,x;t兲 = 1

␾0共y兲2n=0

兺

dn

2

e−E共n兲tFx共E共n兲兲Fy共E共n兲兲, t ⬎ 0. 共3.33兲 The explicit forms of the transition probability共3.22兲,共3.23兲,共3.29兲, and共3.33兲can be evalu-ated straightforwardly if the HamiltonianH of an exactly solvable discrete quantum mechanics is given. Thus we may call the functions B共x兲 and D共x兲 in the Hamiltonian H of an exactly solvable discrete quantum mechanics共3.3兲the birth and death rates of an exactly solvable birth and death

process. As mentioned above, the association of the birth and death rates and the orthogonal

polynomial in this paper and in literatures8,5,12are dual to each other. Therefore the names of the polynomials in Sec. IV are the dual of the corresponding Karlin–McGregor polynomial except for the self-dual cases of the Krawtchouk 共Sec. IV A 4兲, Meixner 共Sec. IV B 1兲, and Charlier 共Sec. IV B 2兲.

In Sec. IV we will present 18 examples of exactly solvable birth and death processes. IV. 18 EXAMPLES

Now let us proceed to give the 18 explicit examples of exactly solvable birth and death processes. The input is simply the function forms of the birth and death rates B共x兲 and D共x兲. The rest is calculable. However, here we also provide other data, taken from Ref.9, such as the energy eigenvalueE共n兲, the sinusoidal coordinate ␩共x兲, the unnormalized stationary probability ␾₀共x兲2_,

the normalization constants d_n2, and the polynomials Pn共␩兲. Following the order of our previous work on the exactly solvable discrete quantum mechanics,9 we handle the most generic one first and then followed by the simpler ones. There is a logical reason for this order. The simpler ones are usually obtained by specializing or restricting the parameters of the generic ones. Each ex-ample is called by the name of the corresponding orthogonal polynomial Pn共␩兲 with the number, e.g., 关KS3.2兴 attached to it indicating the subsection in the standard review of Koekoek and Swarttouw.7The finite共N兲 cases are discussed first and then the infinite ones. In each group the Askey scheme of hypergeometric orthogonal polynomials共non-q polynomials兲 will be discussed first and followed by the q-scheme polynomials.

Please note that the set of parameters is slightly different from the conventional ones4,5,7for some polynomials, the reason explained in Ref.9. For some polynomials, for example, the共q-兲 Racah, 共dual, q-兲 Hahn, etc., there are many nonequivalent parametrizations of B共x兲 and D共x兲, which could lead to nonequivalent birth and death processes. Here we give only one of them as a representative since the purpose of the paper is to show exactly solvable structure not to provide

an exhaustive list of all solvable models. See Ref. 9 for more general parametrizations and the allowed ranges of the parameters. In the same spirit, we did not include some of the polynomials listed in Ref.9.

A. Finite dimensional cases

1. Racah_†KS1.2‡

The Racah polynomial is the most generic hypergeometric orthogonal polynomial of a dis-crete variable. All the other共non-q兲 polynomials are obtained by restriction or limiting procedure. The function B共x兲 and D共x兲 depend on four real parameters a, b, c, and d, with one of them, say

c, being related to N, c⬅−N,

B共x兲 = −共x + a兲共x + b兲共x + c兲共x + d兲

共2x + d兲共2x + 1 + d兲 , D共x兲 = −

共x + d − a兲共x + d − b兲共x + d − c兲x

共2x − 1 + d兲共2x + d兲 . 共4.1兲 The other data are

E共n兲 = n共n + d˜兲, ␩共x兲 = x共x + d兲, d˜ =defa + b + c − d − 1, 共4.2兲 aⱖ b, d ⬎ 0, a ⬎ N + d, 0 ⬍ b ⬍ 1 + d, 共4.3兲 ␾0共x兲2= 共a,b,c,d兲x 共1 + d − a,1 + d − b,1 + d − c,1兲x 2x + d d , 共4.4兲 dn 2 = 共a,b,c,d˜兲n 共1 + d˜ − a,1 + d˜ − b,1 + d˜ − c,1兲n 2n + d˜ d ˜ ⫻ 共− 1兲N_{共1 + d − a,1 + d − b,1 + d − c兲} N 共d˜ + 1兲N共d + 1兲2N . 共4.5兲 Here共a兲nis the Pochhammer symbol共A1兲. Throughout this section, the format for dn

2

consists of two parts separated by a ⫻ symbol: dn2=共dn2/d02兲⫻d02. The second part d02 satisfies the relation

兺x␾0共x兲2= 1/d02. The polynomial is

Pn共␩共x兲兲 =4F3

冉

冏

− n,n + d˜,− x,x + d

a,b,c

冏

1

冊

, 共4.6兲

in which₄F3is the standard hypergeometric series共A3兲. The dual polynomial is again the Racah

polynomial with the parameter correspondence 共a,b,c,d兲↔共a,b,c,d˜兲. The rational 共a quartic polynomial divided by a quadratic polynomial兲 birth and death rates 共4.1兲 have not yet been discussed but the Racah polynomial appears in Ref.12.

2. Hahn_†KS1.5‡

This is a well-known example of quadratic共in x兲 birth and death rates with two real positive parameters a and b,

B共x兲 = 共x + a兲共N − x兲, D共x兲 = x共b + N − x兲. 共4.7兲

It has a quadratic energy spectrum

E共n兲 = n共n + a + b − 1兲, ␩共x兲 = x, ␾0共x兲2= N! x !共N − x兲! 共a兲x共b兲N−x 共b兲N , 共4.8兲

dn2= N! n !共N − n兲! 共a兲n共2n + a + b − 1兲共a + b兲N 共b兲n共n + a + b − 1兲N+1 ⫻ 共b兲N 共a + b兲N , 共4.9兲 Pn共␩共x兲兲 =3F2

冉冏

− n,n + a + b − 1,− x a,− N

冏

1

冊

. 共4.10兲

The dual polynomial is the dual Hahn polynomial of Sec. IV A 3 The quadratic birth and death rates are discussed in Ref.12associated with the dual Hahn polynomial.

3. Dual Hahn_†KS1.6‡

The set of parameters is the same as the Hahn polynomial case. The birth and death rates are rational functions of x,

B共x兲 =共x + a兲共x + a + b − 1兲共N − x兲

共2x − 1 + a + b兲共2x + a + b兲, D共x兲 =

x共x + b − 1兲共x + a + b + N − 1兲

共2x − 2 + a + b兲共2x − 1 + a + b兲, 共4.11兲 giving rise to a linear energy spectrum

E共n兲 = n, ␩共x兲 = x共x + a + b − 1兲, ␾0共x兲2= N! x !共N − x兲! 共a兲x共2x + a + b − 1兲共a + b兲N 共b兲x共x + a + b − 1兲N+1 , 共4.12兲 dn 2 = N! n !共N − n兲! 共a兲n共b兲N−n 共b兲N ⫻ 共b兲N 共a + b兲N , 共4.13兲 Pn共␩共x兲兲 =3F2

冉冏

− n,x + a + b − 1,− x a,− N

冏

1

冊

. 共4.14兲 4. Krawtchouk_{†KS1.10‡ „self-dual…}

The case of linear birth and death rates are a very well-known example 共the Ehrenfest model兲11

of an exactly solvable birth and death processes2,3

B共x兲 = p共N − x兲, D共x兲 = 共1 − p兲x, 0 ⬍ p ⬍ 1, 共4.15兲 E共n兲 = n, ␩共x兲 = x, 共4.16兲 ␾0共x兲2= N! x !共N − x兲!

冉

p 1 − p

冊

x , dn 2 = N! n !共N − n兲!

冉

p 1 − p冊⫻n_{共1 − p兲}N , 共4.17兲 Pn共␩共x兲兲 =2F1

冉冏

− n,− x − N

冏

p −1

冊

_{. 共4.18兲}

This is a simplest example of self-dual polynomials. The stationary probability ␾₀共x兲2_d 0 2 _{is the}

binomial distribution.

5. q-Racah_†KS3.2‡

This is the first example of the q-scheme of the orthogonal polynomials. Among them the

q-Racah polynomial is the most generic. The set of parameters is four real numbers 共a,b,c,d兲,

which is different from the standard one in the same manner as for the Racah polynomial. We restrict them

c = q−N, aⱕ b, 0 ⬍ d ⬍ 1, 0 ⬍ a ⬍ qNd, qd⬍ b ⬍ 1, d˜ ⬍ q−1, ˜ =d def

abcd−1q−1. 共4.19兲 The functions B共x兲 and D共x兲 are

B共x兲 = −共1 − aq x_{兲共1 − bq}x_{兲共1 − cq}x_{兲共1 − dq}x_兲 共1 − dq2x_{兲共1 − dq}2x+1_兲 , 共4.20兲 D共x兲 = − d˜共1 − a −1_dqx_{兲共1 − b}−1_dqx_{兲共1 − c}−1_dqx_{兲共1 − q}x_兲 共1 − dq2x−1_{兲共1 − dq}2x_兲 . 共4.21兲

The other data are

E共n兲 = 共q−n_{− 1兲共1 − d˜q}n_{兲, ␩共x兲 = 共q}−x_{− 1兲共1 − dq}x_{兲, 共4.22兲} ␾0共x兲2= 共a,b,c,d;q兲x 共a−1_dq,b−1_dq,c−1_dq,q;q兲 x˜dx 1 − dq2x 1 − d , 共4.23兲 dn 2 = 共a,b,c,d˜;q兲n 共a−1_˜q,bd −1_d˜q,c−1_d˜q,q;q兲 ndn 1 − d˜q2n 1 − d˜ ⫻ 共− 1兲N_共a−1_dq,b−1_dq,c−1_dq;q兲 N˜dNq1/2N共N+1兲 共d˜q;q兲N共dq;q兲2N , 共4.24兲 Pn共␩共x兲兲 =4␾3

冉

冏

q−n,d˜qn,q−x,dqx a,b,c

冏

q;q

冊

, 共4.25兲

in which ₄␾₃ is the basic hypergeometric series共A4兲 and 共a;q兲n is the q-Pochhammer symbol 共A2兲. The dual q-Racah polynomial is again the q-Racah polynomial with the parameter corre-spondence共a,b,c,d兲↔共a,b,c,d˜兲.

6. q-Hahn_†KS3.6‡

The q-Hahn polynomial has two positive parameters a and b and the birth and death rates are quadratic polynomials in qx_,

B共x兲 = 共1 − aqx兲共qx−N− 1兲, D共x兲 = aq−1共1 − qx兲共qx−N− b兲, 0 ⬍ a, b ⬍ 1. 共4.26兲

The other data are

E共n兲 = 共q−n_{− 1兲共1 − abq}n−1_{兲, ␩共x兲 = q}−x_{− 1, 共4.27兲} ␾0共x兲2= 共q;q兲N 共q;q兲x共q;q兲N−x 共a;q兲x共b;q兲N−x 共b;q兲Nax , 共4.28兲 dn 2 = 共q;q兲N 共q;q兲n共q;q兲N−n 共a,abq−1_;q兲 n 共abqN_,b;q兲 nan 1 − abq2n−1 1 − abq−1 ⫻ 共b;q兲NaN 共ab;q兲N , 共4.29兲 Pn共␩共x兲兲 =3␾2

冉冏

q−n,abqn−1,q−x a,q−N

冏

q;q

冊

. 共4.30兲

7. Dual q-Hahn_†KS3.7‡

For obvious reasons, we adopt the same parameters 共a,b兲 for the q-Hahn and dual q-Hahn polynomials. The birth and death rates are rational functions of qx,

B共x兲 =共q x−N_{− 1兲共1 − aq}x_{兲共1 − abq}x−1_兲 共1 − abq2x−1_{兲共1 − abq}2x_兲 , 0⬍ a, b ⬍ 1, 共4.31兲 D共x兲 = aqx−N−1共1 − q x_{兲共1 − abq}x+N−1_{兲共1 − bq}x−1_兲 共1 − abq2x−2_{兲共1 − abq}2x−1_兲 , 共4.32兲 E共n兲 = q−n_{− 1, ␩共x兲 = 共q}−x_{− 1兲共1 − abq}x−1_{兲, 共4.33兲} ␾0共x兲2= 共q;q兲N 共q;q兲x共q;q兲N−x 共a,abq−1_;q兲 x 共abqN_,b;q兲 xax 1 − abq2x−1 1 − abq−1 , 共4.34兲 dn 2 = 共q;q兲N 共q;q兲n共q;q兲N−n 共a;q兲n共b;q兲N−n 共b;q兲Nan ⫻共b;q兲NaN 共ab;q兲N , 共4.35兲 Pn共␩共x兲兲 =3␾2

冉冏

q−n,abqx−1,q−x a,q−N

冏

q;q

冊

. 共4.36兲 8. Quantum q-Krawtchouk_†KS3.14‡

This has one positive parameter p⬎q−N. The birth and death rates are quadratic polynomials in qx, B共x兲 = p−1qx共qx−N− 1兲, D共x兲 = 共1 − qx兲共1 − p−1qx−N−1兲, 共4.37兲 E共n兲 = 1 − qn , ␩共x兲 = q−x− 1, 共4.38兲 ␾0共x兲2= 共q;q兲N 共q;q兲x共q;q兲N−x p−xqx共x−1−N兲 共p−1_q−N_;q兲 x , 共4.39兲 dn 2 = 共q;q兲N 共q;q兲n共q;q兲N−n p−nq−Nn 共p−1_q−n_;q兲 n ⫻ 共p−1_q−N_;q兲 N, 共4.40兲 Pn共␩共x兲兲 =2␾1

冉冏

q−n,q−x q−N

冏

q; pq n+1

冊

_{. 共4.41兲} 9. q-Krawtchouk_†KS3.15‡

This has one positive parameter p⬎0 and the birth and death rates are linear in qx_,

B共x兲 = qx−N_{− 1, D共x兲 = p共1 − q}x_{兲, 共4.42兲} E共n兲 = 共q−n_{− 1兲共1 + pq}n_{兲, ␩共x兲 = q}−x_{− 1, 共4.43兲}

␾0共x兲2= 共q;q兲N 共q;q兲x共q;q兲N−x p−xq1/2x共x−1兲−xN, 共4.44兲 dn2= 共q;q兲N 共q;q兲n共q;q兲N−n 共− p;q兲n 共− pqN+1_;q兲 npnq1/2n共n+1兲 1 + pq2n 1 + p ⫻ pNq1/2N共N+1兲 共− pq;q兲N , 共4.45兲 Pn共␩共x兲兲 =3␾2

冉冏

q−n,q−x,− pqn q−N,0

冏

q;q

冊

. 共4.46兲

10. Affine q-Krawtchouk_{†KS3.16‡ „self-dual…}

This has one positive parameter p and the birth and death rates are quadratic polynomials in

qx, B共x兲 = 共qx−N− 1兲共1 − pqx+1兲, D共x兲 = pqx−N共1 − qx兲, 0 ⬍ p ⬍ q−1, 共4.47兲 E共n兲 = q−n_{− 1, ␩共x兲 = q}−x_{− 1, 共4.48兲} ␾0共x兲2= 共q;q兲N 共q;q兲x共q;q兲N−x 共pq;q兲x 共pq兲x , dn 2 = 共q;q兲N 共q;q兲n共q;q兲N−n 共pq;q兲n 共pq兲n ⫻ 共pq兲 N , 共4.49兲 Pn共␩共x兲兲 =3␾2

冉冏

q−n,q−x,0 pq,q−N

冏

q;q

冊

. 共4.50兲

B. Infinite dimensional cases

In contrast to the finite dimensional case, the structure of the polynomials is severely con-strained by the asymptotic forms of the functions B共x兲 and D共x兲 共3.12兲.

1. Meixner_{†KS1.9‡ „self-dual…}

This is the best known example of exactly solvable birth and death processes10 and the birth and death rates are both linear in x with simple linear energy spectraE共n兲=n and ␩共x兲=x. It has two positive parameters␤ and c:

B共x兲 = c 1 − c共x +␤兲, D共x兲 = 1 1 − cx, ␤⬎ 0, 0 ⬍ c ⬍ 1, 共4.51兲 E共n兲 = n, ␩共x兲 = x, 共4.52兲 ␾0共x兲2= 共␤兲xcx x! , dn 2₌共␤兲ncn n! ⫻ 共1 − c兲 ␤_{, 共4.53兲} Pn共␩共x兲兲 =2F1

冉冏

− n,− x ␤

冏

1 − c−1

冊

. 共4.54兲 2. Charlier_{†KS1.12‡ „self-dual…}

This is another best known example of exactly solvable birth and death processes with con-stant birth rates a⬎0 and linear death rates,

B共x兲 = a, D共x兲 = x, 共4.55兲 E共n兲 = n, ␩共x兲 = x, 共4.56兲 ␾0共x兲2= ax x!, dn 2 =a n n!⫻ e −a_{, 共4.57兲} Pn共␩共x兲兲=2F0

冉冏

− n,− x −

冏

− a −1

冊

_{. 共4.58兲}

The stationary probability␾0共x兲2d02 共4.57兲is the Poisson distribution.

3. little q-Jacobi_†KS3.12‡

This has two parameters a and b. The birth and death rates grow exponentially as x tends to infinity, B共x兲 = a共q−x_{− bq兲, D共x兲 = q}−x_{− 1, 0⬍ a ⬍ q}−1_{, b⬍ q}−1_{, 共4.59兲} E共n兲 = 共q−n_{− 1兲共1 − abq}n+1_{兲, ␩共x兲 = 1 − q}x_{, 共4.60兲} ␾0共x兲2= 共bq;q兲x 共q;q兲x 共aq兲x , 共4.61兲 dn2= 共bq,abq;q兲nanqn 2 共q,aq;q兲n 1 − abq2n+1 1 − abq ⫻ 共aq;q兲⬁ 共abq2_;q兲 ⬁ , 共4.62兲 Pn共␩共x兲兲 = 共− a兲−nq−1/2n共n+1兲 共aq;q兲n 共bq;q兲n2 ␾1

冉冏

q−n,abqn+1 aq

冏

q;q x+1

冊

_{. 共4.63兲} The normalization of the polynomial is different from the conventional one.

4. q-Meixner_†KS3.13‡

This has two positive parameters b and c. The birth and death rates are quadratic in qxand as

x goes to infinity, the birth rates tend to zero and the death rates tend to unity,

B共x兲 = cqx共1 − bqx+1兲, D共x兲 = 共1 − qx兲共1 + bcqx兲, 0 ⬍ b ⬍ q−1, c⬎ 0, 共4.64兲 E共n兲 = 1 − qn , ␩共x兲 = q−x− 1, 共4.65兲 ␾0共x兲2= 共bq;q兲x 共q,− bcq;q兲x cxq1/2x共x−1兲, dn 2 = 共bq;q兲n 共q,− c−1_q;q兲 n ⫻共− bcq;q兲⬁ 共− c;q兲⬁ , 共4.66兲 Pn共␩共x兲兲 =2␾1

冉冏

q−n,q−x bq

冏

q;− c −1_qn+1

冊

_{. 共4.67兲} 5. Little q-Laguerre/Wall_†KS3.20‡

This has one positive parameter a and both the birth and death rates grow exponentially as x tends to infinity,

B共x兲 = aq−x, D共x兲 = q−x− 1, 0⬍ a ⬍ q−1, 共4.68兲 E共n兲 = q−n_{− 1, ␩共x兲 = 1 − q}x_{, 共4.69兲} ␾0共x兲2= 共aq兲x 共q;q兲x , dn 2 = a n qn2 共q,aq;q兲n ⫻ 共aq;q兲_⬁, 共4.70兲 Pn共␩共x兲兲 =2␾0

冉冏

q−n_,q−x −

冏

q;a −1_qx

冊

. 共4.71兲

The normalization of the polynomial is different from the conventional one.

6. Al-Salam–Carlitz II_†KS3.25‡

This has one positive parameter a and the birth and death rates are quadratic in qx_{. As x goes} to infinity the birth rates tend to zero and death rates tend to unity,

B共x兲 = aq2x+1, D共x兲 = 共1 − qx兲共1 − aqx兲, 0 ⬍ a ⬍ q−1, 共4.72兲 E共n兲 = 1 − qn , ␩共x兲 = q−x− 1, 共4.73兲 ␾0共x兲2= axqx2 共q,aq;q兲x , dn2= 共aq兲n 共q;q兲n ⫻ 共aq;q兲_⬁, 共4.74兲 Pn共␩共x兲兲 =2␾0

冉冏

q−n_,q−x −

冏

q;a −1_qn

冊

. 共4.75兲

The normalization of the polynomial is different from the conventional one.

7. Alternative q-Charlier_†KS3.22‡

This has one positive parameter a. The birth rates are constant a, whereas the death rates grow exponentially as x goes to infinity,

B共x兲 = a, D共x兲 = q−x− 1, a⬎ 0, 共4.76兲 E共n兲 = 共q−n_{− 1兲共1 + aq}n_{兲, ␩共x兲 = 1 − q}x_{, 共4.77兲} ␾0共x兲2= ax_q1/2x共x+1兲 共q;q兲x ,dn2= an_q1/2n共3n−1兲 共q;q兲n 共− a;q兲_⬁ 共− aqn_;q兲 ⬁ 1 + aq2n 1 + a ⫻ 1 共− aq;q兲_⬁, 共4.78兲 Pn共␩共x兲兲 = qnx2␾1

冉冏

q−n,q−x 0

冏

q;− a −1_q−n+1

冊

_{. 共4.79兲}

The normalization of the polynomial is different from the conventional one.

8. q-Charlier_†KS3.23‡

This has one positive parameter a, and as x goes to infinity, the birth rates tend to zero and the death rates tend to unity,

E共n兲 = 1 − qn , ␩共x兲 = q−x− 1, 共4.81兲 ␾0共x兲2= axq1/2x共x−1兲 共q;q兲x , dn 2 = q n 共− a−1_q,q;q兲 n ⫻ 1 共− a;q兲⬁, 共4.82兲 Pn共␩共x兲兲 =2␾1

冉冏

q−n,q−x 0

冏

q;− a −1_qn+1

冊

_{. 共4.83兲} V. SUMMARY AND COMMENTS

Following the simple line of arguments summarized in the following diagram, we presented 18 models of exactly solvable birth and death processes and their solutions, the transition prob-abilities. In the diagram “ES” stands for exactly solvable.

ES 1d Quantum

Mechanical systems −−−−−−−−−−−−−−−−−−→give solutions

ES 1d Fokker-Planck equations ⏐ ⏐ ⏐ ⏐ discretisation ⏐ ⏐ ⏐ ⏐ discretisation ES ‘Matrix’ Quantum

Mechanical systems −−−−−−−−−−−−−−−−−−→give solutions

ES Birth and Death processes

The exactly solvable matrix quantum mechanics or the 1D discrete quantum mechanics with real shifts was explored in detail in Ref. 9 to cover most of the hypergeometric orthogonal polynomials of a discrete variable in the共q-兲 Askey scheme.4,5,7For the “explanation” of the exact solvability, see a recent work.22 By comparing the present simple results with those in literatures8,3,5,12 one would realize the essential role played by the energy spectrumE共n兲 and the sinusoidal coordinate␩共x兲. They are the eigenvalues of the two operators, called the Leonard pair, which characterize the orthogonal polynomials completely.19–21

In this paper we did not discuss the generalization of the birth and death processes which has

␮0⬎0 共D共0兲⬎0兲, the nonvanishing death rate at zero population, although this has led to a new

type of orthogonal polynomials in the cases when the birth and death rates B共x兲 and D共x兲 are linear and quadratic in x.23,24It would be interesting to try further generalization in this direction for which B共x兲 and D共x兲 are rational, e.g., the Racah case 共Sec. IV A 1兲 or q-linear, e.g., the

q-Krawtchouk共Sec. IV A 9兲, or q-quadratic, e.g., the affine q-Krawtchouk 共Sec. IV A 10兲, or even

the q-rational, e.g., the q-Racah共Sec. IV A 5兲 cases.

It is a big challenge to try and find a closed form expression for

兺

n=0 dn

2

e−E共n兲tPn共␩共x兲兲Pn共␩共y兲兲, 共5.1兲 appearing as a part of the transition probability共2.17兲and共3.23兲for various examples in section four. To the best of our knowledge, such expressions are known only for the linear energy spec-trumE共n兲⬀n. For example, for the Fokker–Planck equation corresponding to the harmonic oscil-lator Hamiltonian, or the Ornshtein–Uhlenbeck process,1,14 we have

H = def − d 2 dx2+ x 2_{− 1, L} FP= d2 dx2+ 2 d dxx, E共n兲 = 2n, ␩共x兲 = x, 共5.2兲 P共y,x;t兲 =e−x 2

冑

␲

兺

n=0 ⬁ Hn共x兲Hn共y兲 2nn! e −2nt₌ 1

冑

␲

冑

1 − e−4texp

冋

− 共x − ye−2t_兲2 1 − e−4t

册

. 共5.3兲 The last equality was derived based on共6.1.13兲 of Ref.4. Another example is

H =def− d 2 dx2+ x 2₊g共g − 1兲 x2 −共1 + 2g兲, LFP= d2 dx2+ 2 d dx

冉

x − g x

冊

, 共5.4兲 En= 4n, ␩共x兲 = x2, ␤= def g − 1/2, 0 ⬍ x ⬍ ⬁, 共5.5兲 P共y,x;t兲 = 2e−x2_x2g

兺

n=0 ⬁ n ! Ln共␤兲共x2兲Ln共␤兲共y2兲 ⌫共n +␤+ 1兲 e−4nt 共5.6兲 = 2x 2g 共1 − e−4t_兲exp

冋

− 共x2_{+ y}2_e−4t_兲 共1 − e−4t_兲

册

共xye −2t_兲−␤_I ␤

冉

2xye−2t 1 − e−4t

冊

, 共5.7兲

in which I_␤is the modified Bessel function of the order of␤. The last equality was derived based on共6.2.25兲 of Ref.4. We would like to ask experts in special functions and orthogonal polyno-mials to derive such bilinear generating functions for various energy spectra,

E共n兲 = n共n + d兲, q−n_{− 1, 1 − q}n

, 共q−n− 1兲共1 − dqn兲. 共5.8兲

ACKNOWLEDGMENTS

We thank Mourad Ismail and Choon-Lin Ho who induced us to the present research. This work was supported in part by Grants-in-Aid for Scientific Research from the Ministry of Edu-cation, Culture, Sports, Science and Technology of Japan, Grant Nos. 18340061 and 19540179.

APPENDIX: SOME DEFINITIONS RELATED TO THE HYPERGEOMETRIC AND

q-HYPERGEOMETRIC FUNCTIONS

For self-containedness we collect several definitions related to the 共q-兲 hypergeometric functions.7

• Pochhammer symbol共a兲n,

共a兲n=

def

兿

k=1 n

共a + k − 1兲 = a共a + 1兲 ¯ 共a + n − 1兲 =⌫共a + n兲_⌫共a兲 . 共A1兲 • q-Pochhammer symbol共a;q兲n,

共a;q兲n=

def

兿

k=1 n

共1 − aqk−1_{兲 = 共1 − a兲共1 − aq兲 ¯ 共1 − aq}n−1_{兲. 共A2兲} • Hypergeometric seriesrFs, rFs

冉冏

a1, . . . ,ar b1, . . . ,bs

冏

z

冊

= def

兺

n=0 ⬁ 共a1, . . . ,ar兲n 共b1, . . . ,bs兲n zn n!, 共A3兲 where共a1, . . . , ar兲n= def 兿j=1 r _共a j兲n=共a1兲n¯共ar兲n.

r␾s

冉冏

a₁,¯ ,ar b1,¯ ,bs

冏

q;z

冊

= def

兺

n=0 ⬁ _共a 1,¯ ,ar;q兲n 共b1,¯ ,bs;q兲n 共− 1兲共1+s−r兲n_q共1+s−r兲n共n−1兲/2 zn 共q;q兲n , 共A4兲 where共a1, . . . , ar; q兲n= def 兿j=1 r _共a j; q兲n=共a1; q兲n¯共ar; q兲n.

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