BIRTH-DEATH PROCESSES AND q-continued FRACTIONS

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1 TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 364, Number 5, May 202, Pages S (202)05522-X Article electronically published on January 7, 202 BIRTH-DEATH PROCESSES AND -CONTINUED FRACTIONS TONY FENG, RACHEL KIRSCH, ELISE VILLELLA, AND MATT WAGE Abstract In the 997 paper of Parthasarathy, Lenin, Schoutens, and Van Assche, the authors study a birth-death process related to the Rogers-Ramanujan continued fraction r() We generalize their results to establish a correspondence between birth-death processes and a larger family of -continued fractions It turns out that many of these continued fractions, including r(), play important roles in number theory, specifically in the theory of modular forms and -series We draw upon the number-theoretic properties of modular forms to give identities between the transition probabilities of different birth-death processes Introduction A birth-death process is a continuous-time Markov chain in which the states represent the size of a population The transition from state n to state n + is called a birth and the transition from state n to state n is called a death (see 2) Birth-death processes are an important object of study in probability, stochastic processes, and ueuing theory, as they can be used to model population growth, epidemics, and ueue length [8] In [6], Parthasarathy et al establish a correspondence between a particular birth-death model and the celebrated Rogers- Ramanujan continued fraction () r() := = It is this correspondence between birth-death models and -continued fractions that we examine more generally here Additionally, we explore the effect of the deep number-theoretic and analytic properties of the Rogers-Ramanujan continued fraction on the corresponding birth-death process A birth-death process is determined by its birth and death rates More precisely, for each state n in a birth-death process there is a birth rate λ n and a death rate μ n (for a formal definition of birth-death processes, see 2) Let p 0 (t) bethe probability that the population has size 0 at time t, given that it has size 0 at time 0 Define (2) f 0 (s) := 0 e st p 0 (t)dt Received by the editors July 24, 2009 and, in revised form, October 26, Mathematics Subject Classification Primary 03B48; Secondary A c 202 American Mathematical Society Reverts to public domain 28 years from publication License or copyright restrictions may apply to redistribution; see

2 2704 TONY FENG, RACHEL KIRSCH, ELISE VILLELLA, AND MATT WAGE As we shall see in 2, it turns out that f 0 (0) has the following suggestive representation: (3) f 0 (0) = λ 0 μ λ 0 λ μ 2 λ + μ λ 2 + μ 2 λ 2μ 3 Since we are interested in finding birth-death processes corresponding to certain -continued fractions, we let {a n } := {a n ()} and {b n } := {b n ()} be seuences of rational functions of which are positive for (0, ) We now define two more seuences of rational functions of from the seuences {a n } and {b n } Let {λ n } := {λ n ()} and {μ n } := {μ n ()} be defined by λ 0 := b 0 and { λn μ (4) n := a n, n, λ n + μ n := b n, n Notice that {a n } (resp {b n }) will be the numerators (resp denominators) of the continued fraction (3) By (4) we have (5) λ n = b n a n, n λ n Call a birth-death model good if its birth rates and death rates are positive Observe that if λ n > 0, then also μ n+ > 0sinceλ n μ n+ = a n+ > 0 Therefore to show that a process is good it suffices to show that λ n > 0forn 0 In 24 we establish conditions for a seuence of recursively defined birth rates {λ n ()} to give rise to a good birth-death process Specifically, with {a n } and {b n } as above we prove the following two theorems for (0, ) which provide conditions for verifyingthataprocessisgood Theorem Suppose that the following conditions are met (a) There exists k>0 R such that for all (0, ), we have 0 <b n () <k (b) There exists C > 0 R, depending on, such that 0 d ( an ()) <C and 0 d d d (b n()) <C Then there exists [0, ] such that λ n () > 0 for all n 0 if and only if Remark The main result of [6] is the special case of Theorem where b n = and a n = n,inwhichcase 0576 Theorem 2 Let α > 0 R, and suppose that there exists k > 0 with the following properties (a) λ k α ak+ (b) For all n k, we have b n α a n α an+ Then for all n>k, we have λ n b n α a n > 0 Theorem 2 will prove especially useful later in the examples A process is recurrent if it returns to some state with probability and transient otherwise It is positive recurrent if the expected time of return is finite (For a more formal definition, see 25) It turns out that recurrence of a birth-death process is related to the convergence of (3) In 25 we prove that a birth-death License or copyright restrictions may apply to redistribution; see

3 BIRTH-DEATH PROCESSES AND -CONTINUED FRACTIONS 2705 process satisfying the hypotheses of Theorem is transient for < and positive recurrent for =, and we show that a birth-death process satisfying the hypotheses of Theorem 2 for (0, ) is transient on that range We also examine the asymptotic behavior of the birth and death rates In 23 we analyze the continued fractions from a -series perspective to compute formulas for polynomials associated with the birth-death processes We then discuss a method of recovering transition probabilities such as p 0 (t) from the corresponding continued fraction These results end up reuiring important constructions in the theory of -series For example, the following result expresses the convergents in terms of the Pochhammer symbol and the -binomial coefficient (see 23) Corollary Let A n (s)/b n (s) be the nth convergent of the continued fraction (6) Then (7) A n (s) B n (s) = s ++ n/2 /2 s ++b + ( ) j a j j2 +j n/2 a ( ) j a j j2 [ n j j a 2 s ++b 2 [ ] n j (s +)( b /2 /(s +);) n j j ( b /2 /(s +);) j ] ( b/(s +);)n j ( b/(s +);) j The above euation allows us to approximate the transition probabilities of a birth-death process from the corresponding continued fraction via (3) Remark The Rogers-Ramanujan continued fraction () is the case a =,b = s = 0 of the continued fraction (6) In 3 we give several examples of famous number-theoretic -continued fractions which correspond to good birth-death processes One of them is (8) whose power series expansion is essentially the classical Jacobi theta function (9) ϑ() = n Z n2 Given a good birth-death model B, denotebyf B () the normalized Laplace transform of p 0 (t) evaluated at s = 0 associated with B (in other words, the f 0 (s) as in euation (2) for the particular birth-death model B) In the last section we prove some surprising relations between the transition probabilities p 0 (t) fordifferent birth-death processes, including one for the birth-death process associated with r(), the Rogers-Ramanujan continued fraction () For example, let the License or copyright restrictions may apply to redistribution; see

4 2706 TONY FENG, RACHEL KIRSCH, ELISE VILLELLA, AND MATT WAGE birth-death process corresponding to (8) be B 3 ; we will show that if y>0, then we have (0) + 2e 2πy ( f B3 e 2πy ) = ( )) 2/y (+e π 2y fb3 e π 2y The above euation is a conseuence of the fact that ϑ 3 () essentially transforms to itself under the mapping τ /τ (where τ H with = e 2πiτ ) on the upper half of the complex plane, which is in turn a conseuence of the theory of modular forms (Recall that a modular form is a type of meromorphic function that satisfies a special transformation property For a formal definition, see [5]) 2 Theory of -continued fractions and birth-death processes Here we recall the formal definition of a birth-death process and important facts about orthogonal polynomials and -continued fractions We also prove Theorems and 2 and study the asymptotic behavior of the birth-death processes 2 Birth-death processes Here we closely follow the treatment of the subject as presented in [8] Given that a population has size m initially, let p m,n (t) bethe probability that the population is at size n after time t By definition, a birth-death process has the following transition probabilities for m 0: p m,m+ (t) :=λ m t + o(t), p m,m (t) :=μ m t + o(t), where lim t 0 o(t)/t =0 Wecallλ n the birth rates and μ n the death rates of this process These probabilities p m,n satisfy the Kolmogorov forward euations for birth-death processes [8]: (2) p m,0(t) = λ 0 p m,0 (t)+μ p m, (t), (22) p m,n(t) =λ n p m,n (t) (λ n + μ n )p m,n (t)+μ n+ p m,n+ (t), where the derivative is taken with respect to t We are interested in the case where the initial population is zero, so we will denote p n (t) :=p 0,n (t) Following [, p 366], let f n (s) be the normalized Laplace transform of p n (t): ( n ) f n (s) :=( ) n μ i e st p n (t)dt, f 0 (s) := 0 i= e st p 0 (t)dt Taking the normalized Laplace transforms of (2) and (22) yields the recurrence relations (23) = s + λ 0 + f (s) f 0 (s) f 0 (s), f n (s) λ n μ n (24) = f n (s) s + λ n + μ n + f n+(s) f n (s) These recurrence relations yield the continued fraction (25) f 0 (s) = s + b 0 a s + b 0 a 2 s + b 2 +, License or copyright restrictions may apply to redistribution; see

5 BIRTH-DEATH PROCESSES AND -CONTINUED FRACTIONS 2707 where b 0 := λ 0 and for n wehavea n := λ n μ n and b n := μ n +λ n (Henceforth we adopt the notation for continued fractions as in (25)) We will be examining f 0 (0) = p 0 0 (t)dt, which is well defined for < for some (0, ) to be specified in Orthogonal polynomials Central to the study of birth-death processes and continued fractions are orthogonal polynomials A seuence of polynomials is called orthogonal if there exists a distribution function ψ such that P n (x)p m (x)dψ(x) =0form n and P 2 n(x)dψ(x) 0 By Favard s Theorem [6, p 2], polynomials defined by the following recurrence relation are orthogonal: (26) P n+ (x) =(x b n )P n (x) a n P n (x), where P (x) =0,P 0 (x) =,b n R and a n > 0 In [3], Karlin and McGregor exhibit deep connections between birth-death processes and orthogonal polynomials, using the polynomials to compute approximations of the transition probabilities Orthogonal polynomials will play a key role in our methods as well, as they have many useful and interesting properties We now list some properties of orthogonal polynomials that will be used in later sections (F) The zeros of P n are real and distinct The zeros of P n and P n+ interlace ([6, p 28]) (F2) The spectrum of ψ, defined by S := {x : ψ(x + δ) ψ(x δ) > 0 for all δ>0} is countably infinite, and for each s S, every neighborhood of s contains a zero of infinitely many polynomials P n (x) ([6, p 5, p 60]) (F3) The orthonormal polynomials corresponding to P n (x) are ([6, p 23]) P n (x) p n (x) := (a a 2 a n ) /2 with a i as in (26) (F4) If s S, then ([6, p 63]) ( 0 <ψ(s+) ψ(s ) pk(s)) 2, k=0 where ψ(s+) := lim δ 0 + ψ(s + δ) andψ(s ) := lim δ 0 ψ(s + δ) 23 Explicit formulas through generating functions In this section we analyze generating functions for seuences of orthogonal polynomials associated with birth-death probability models We derive explicit formulas for the orthogonal polynomials (and conseuently the birth and death rates) and the convergents of their corresponding continued fractions, which allows us to approximate the normalized Laplace transforms of the transition probabilities License or copyright restrictions may apply to redistribution; see

6 2708 TONY FENG, RACHEL KIRSCH, ELISE VILLELLA, AND MATT WAGE Let {λ n } be a seuence of real numbers satisfying λ 0 =andλ n (+b n λ n )= a n for n Define R n by R 0 =andλ n = R n+ /R n Thenwehave R n+ =(+b n )R n a n R n Let {P n (x)} be a seuence of polynomials satisfying the conditions P (x) =0, P 0 (x) =, and (27) P n+ (x) =(x + b n )P n (x) a n P n (x) so that P n () = R n Observe that the P n (x) are orthogonal polynomials with respect to some distribution ψ (see (26)) Theorem 2 With notation as above, (28) n/2 x n P n(x) = ( ) j a j j2 /x 2j [ n j j ] ( b/x; )n j ( b/x; ) j Above and throughout, (α; ) n := n i=0 ( αi )isthe-pochhammer symbol, the infinite extension of the -Pochhammer symbol is (α; ) := i=0 ( αk ), and [ a =(; ) b] a / ((; ) b (; ) a b ) is the -binomial coefficient Proof We follow the approach of Ismail in [9] First we construct the generating function F (z) := n=0 zn P n (x) Multiplying the recurrence relation (27) by z n+ and summing from n =0to yields F (z, ) = ( + bz( az/b)f (z)) xz Iterating, and using F (z, 0) = 0, we have (29) F (z, ) = k=0 z k b k k(k )/2 (az/b; ) k (xz; ) k+ Now we apply Gauss s -binomial formula (see [2, p 5]) (20) (z; ) k = k and Heine s -binomial formula [5, p 8] (; ) k (; ) j (; ) k j ( z) j j(j+)/2 (2) (αz; ) (z; ) = i=0 (α; ) i (; ) i z i License or copyright restrictions may apply to redistribution; see

7 BIRTH-DEATH PROCESSES AND -CONTINUED FRACTIONS 2709 to obtain F (z, ) = z k b k k(k )/2 k=0 i=0 ( k+ ; ) i (; ) i (xz) i k (; ) k ( az/b) j (; ) j (; ) k j j(j+)/2 Collecting terms, switching the order of summation (see [20]), and substituting n = i + j + k, wehave F (z, ) = n/2 z n n=0 ( ) j j(j+)/2 a j b j x n j (; ) n j n j x k b k k(k )/2 (; ) j (; ) k j (; ) n j k Now euating coefficients of z n and rewriting (α; ) n j k as (α;) n j k(k+)/2 k(n j) ( k+j n /α;) k ( α) k yields P n (x) = n/2 ( ) j j(j+)/2 a j b j x n j n j (; ) j k=j k=j x k b k k(n j) (j n ; ) k (; ) k j Next, set m = k j and apply Heine s binomial formula (2) to obtain P n (x) = n/2 ( ) j nj j(j )/2 a j x n 2j (j n ; ) j ( j b/x; ) (; ) j ( n j b/x; ) Euation (28) then follows after straightforward manipulations Corollary 22 With notation as above, we have (22) R n = n/2 ( ) j a j j2 [ n j j Remark This affords us an explicit formula for λ n Corollary 23 If (23) f(z) =( bz; ) ] ( b; )n j ( b; ) j ( ) j a j j2 z 2j (; ) j ( bz; ) j, then (24) lim n x n P n(x) =f(/x) Furthermore, let the positive elements of the support of the distribution function ψ with respect to which the polynomials P n (x) are orthogonal be {x,x 2,}, where x >x 2 >x 3 > > 0 and let the zeros of (23) be 0 <z <z 2 <z 3 Then we have z k =/x k Proof Euation (24) follows immediately by letting n in (28) and setting z =/x Let the zeros of P n (x) bex n,k,k =, 2,,n,wherex n, >x n,2 > x n,3 x n,n It is a well-known property of orthogonal polynomials (see 22 (F2)) that lim n x n,k = x k By (24), lim n f(/x n,k )=f(/x k )=0 License or copyright restrictions may apply to redistribution; see

8 270 TONY FENG, RACHEL KIRSCH, ELISE VILLELLA, AND MATT WAGE Remark If we set b =0,a=, then the polynomials are given by P 0 =,P =0, and P n+ (x) =xp n (x) n P n (x),n and we recover (23) in [6]: F (z; ) = k=0 ( )k z 2k k2 /(; ) k Corollary 24 Let A n (s)/b n (s) be the nth convergent of the continued fraction (25) Then (26) A n (s) B n (s) = a a 2 s ++ s ++b+ s ++b 2 + n/2 /2 Moreover, as n we find ( ) j a j j2 +j n/2 a a 2 s ++ s ++b+ s ++b 2 + = ( ) j a j j2 [ n j j ( b/(s +);) /2 [ ] n j (s +)( b /2 /(s +);) n j j ( b /2 /(s +);) j (s +)( b /2 /(s +);) ] ( b/(s +);)n j ( b/(s +);) j a j j2 +j (; ) j ( b /2 /(s +);) j a j j2 (; ) j ( b/(s +);) j Remark Ramanujan mentions the case s = 0 of (25) in his first letter to Hardy [3, p 22] Proof We have the recursive relations for convergents of a continued fraction A 0 (s) =0 A (s) = A n+ (s) =(s ++b n )A n (s) a n A n (s), n and B (s) =0 B 0 (s) = B n+ (s) =(s ++b n )B n (s) a n B n (s), n 0 Observe A n (s) andb n (s) satisfy similar recurrences to those for P n (s) In fact, it is straightforward to verify that B n (s) =P n (s +)and A n (s) = (n )/2 P n ((s +)/ /2 ) The corollary follows by applying (28) As we saw in 2, the fraction in (25) is the normalized Laplace transform of p 0 (t) whenλ n +μ n = b n and λ n μ n = a n We now outline a well-known method (see [6] for details) to compute the inverse Laplace transform of the convergents of the fraction, which allows us to approximate the transition probabilities p m (t) We know that A n (s)/b n (s) isthenth convergent of (25) Since the polynomials B n (s) are orthogonal, their zeros are real and distinct Using (26), we can then License or copyright restrictions may apply to redistribution; see

9 BIRTH-DEATH PROCESSES AND -CONTINUED FRACTIONS 27 write A n (s)/b n (s) as a sum of partial fractions and easily find the inverse Laplace transform using the formula ( ) (27) L An (s) = e st A n(s) B n (s) 2π B n (s) dω, where s = c + iω, c>0 ([, p 369]) 24 Conditions for good birth-death processes In this section, we prove Theorems and 2 Recall that we establish these theorems to give conditions under which a -continued fraction gives rise to a good birth-death process Proof of Theorem Since b n is bounded above by k, we may write b n = k β n, where 0 β n <k Define R n by R 0 =and R n+ = λ n, n R n By (5), we know R n+ =(k β n )R n a n R n Next, define the polynomials P n (x) =P n (x, ) byp (x) =0,P 0 (x) =, and (28) P n+ (x) =(x β n )P n (x) a n P n (x), n 0 Notice that P n (k) =R n We will study the zeros of the polynomials P n (x) inorder to determine the signs of P n (k) and hence the signs of λ n By Favard s Theorem (26), the polynomials P i (x) are orthogonal with respect to some distribution ψ, and therefore all the roots of P n (x) are distinct and real (see 22 (F)) Define x n = x n () to be the largest zero of P n (x) Also, define X() := lim x n() n By orthogonality, all roots of the polynomials P i (x) are bounded and the zeros of P i (x) andp i+ (x) interlace (see 22 (F)) Therefore, X existsandisinfactan element of the spectrum of ψ as it is a limit point of x n (see 22 (F2)) Next, notice that the zeros of P n are the eigenvalues of the tridiagonal matrix M n () := β 0 a 0 0 a β a2 0 a2 β 3 0 an 0 0 an β n Let v n =(u,u 2,,u n ) be an eigenvector corresponding to the largest eigenvalue x n By the Hellmann-Feynman Theorem [0], x n() = M n()v n,v n, v n,v n where, is the usual inner product, the derivatives are with respect to, and M n() is formed by taking the derivative of each individual entry By the Perron- Frobenius Theorem for matrices with non-negative entries [4], u i 0for i n This implies that x n() 0, so X() is a non-decreasing function License or copyright restrictions may apply to redistribution; see

10 272 TONY FENG, RACHEL KIRSCH, ELISE VILLELLA, AND MATT WAGE Now, we will show that X() is a continuous function for (0, ) Fix a 0 (0, ) By assumption, there exists a C = C such that d d an ( 0 ) <Cand d d β n( 0 )= d d b n( 0 ) <Cfor all n Hence, x n( 0 )= M n( 0 )v n,v n v n,v n C(u2 +2u u 2 + u u 2 u u n u n + u 2 n) u 2 + u u2 n 3C Using that x n( 0 ) 3C, it is not difficult to show that X is continuous at 0 Next we will prove that X() k if and only if λ n > 0 for all n First, if X() k, then P n (k) > 0 for all n, soλ n > 0 for all n Next, assume that X() >k If P n (k) =0forsomen, thenλ n does not exist for all n Otherwise P n (k) 0, so either x () <kor x () >k If x >k,thenp (k) < 0, so λ 0 < 0 If x <k, then there exists an n such that x n () <kand x n+ () >k,so P n+ (k) < 0 and therefore λ n < 0 In the cases in which X() <kfor all (0, ) or X() >kfor all (0, ), define := or := 0, respectively Otherwise, define to be the largest such that X() =k By the previous paragraph and the properties of X(), this satisfies the theorem Proof of Theorem 2 We will induct on n By assumption, we know that λ k α ak+ Therefore λ k+ = b k+ a k+ b k+ α a k+ λ k This establishes the base case For the inductive hypothesis, suppose that λ n b n a n for n >kthenwehave λ n = b n a n a n b n λ n b n α b n α a n = b n α a n a n an By assumption, b n α a n a n+ /α, which is greater than zero since α, a n > 0 25 Asymptotic behavior of birth-death processes We will now examine the limiting behavior of the birth-death processes to determine whether they will be recurrent or transient Let us first define these terms and others used in the statements of the theorems which follow Definition 25 ([2], p 55, p 58) A process is recurrent if 0 p n (t)dt =, where p n (t) is a transition probability as in 2 A process is transient if it is not recurrent Moreover, a recurrent process is positive recurrent if for each state n, lim t p n (t) > 0andnull recurrent if for each state n, lim t p n (t) =0 License or copyright restrictions may apply to redistribution; see

11 BIRTH-DEATH PROCESSES AND -CONTINUED FRACTIONS 273 The potential coefficients are defined following the notation in [6]: (29) π 0 :=, (220) π n := λ 0λ λ n μ μ 2 μ n = λ2 0 λ2 λ2 n a a 2 a n = P 2 n(k) a a 2 a n, with a i as in (26) and where k = if the continued fraction is of the second type As in [6], we will use Anderson s notation [2] to study the limiting behavior of the birth-death processes Define the uantities A := n=0 λ n π n, B := π n The process corresponding to these potential coefficients is recurrent if A =, and is positive recurrent if A = and B< The following theorems will prove that for < the birth-death processes are transient, and for = (0, ) the processes are positive recurrent Notice that this means that f 0 (0) is well defined because the integral f 0 (0) = p 0 0 (t)dt is finite Theorem 26 Let a n, β n, and k be defined as in Theorem and its proof Suppose for (0, ) that 0 <a n Moreover, suppose there exists some N N such that sup{a n } n=n = c, where0 <c< and c 3 k X() Then the birth-death process is transient If we also have lim n a n =0and lim n β n = β for some β R, then lim λ n = k β and lim μ n =0 n n Proof Notice P n (k) =(k x n, )(k x n,2 ) (k x n,n ), where the x n,j are the zeros of P n (x) such that x n,i <x n,j for all i<j Since the polynomials P n (x) are orthogonal polynomials, the zeros interlace (see 22 (F)) and (k x n,j ) < (k x n+,j ) Thus we have (22) P n+ (k) > (k x n+,n+ )P n (k) > (k X())P n (k) c 3 Pn (k) Recall X() = lim n x n (), where x n is the largest zero of P n Notice X() is also the largest element in the spectrum of ψ, the distribution with respect to which the polynomials P n are orthogonal (see 22 (F2)) Euation (22) implies (222) P n (k) (c 3 ) n because P 0 (x) := We will now show that B diverges With the above ineuality and the fact that a n c for n N, wehave B =+ n= P 2 n(k) a a 2 a n + n= n=0 (c 3 ) 2n N + a a 2 a n n= c 2n 3 + c N (c 3 ) n, which diverges because c 3 > Thus B = and the process is not positive recurrent Next we will show that the process is not null-recurrent because A := /(λ n π n ) < n=0 n=n License or copyright restrictions may apply to redistribution; see

12 274 TONY FENG, RACHEL KIRSCH, ELISE VILLELLA, AND MATT WAGE From the same definition of π n (220), and again using the fact that a n <cfor all n N and P n (k) (c 3 ) n, we arrive at the following ineuality: A k β 0 n= N a a 2 a n P n (k)p n+ (k) (c 3 ) + 2n+ n= n=n c n N+ (c 3 ) 2n+ = c 2 2N 3 ( c 3 ) c 2N 3 (c 3 c) + c 2 3 ( N) c 3 Since this is finite, A is finite and the process therefore is not null-recurrent Thus theprocessistransient We will now show that if a n 0andβ n β, thenμ n 0andλ n k β Because the zeros of P n (x) andp n (x) interlace, we find P n (k)/(k x n,n ) P n (k) Recall that for <,wehavek>x() x n,n,wherex n,n is the largest zero of the nth polynomial P n (x), so the denominator (k x n,n ) > 0 From the interlacing of the zeros of P n (x) andp n+ (x) (see 22 (F)), the largest zeros of the polynomials P n are increasing for a fixed, thefactors(k x n,n ) are decreasing as n increases, and bounded below by (k X) > 0 From the recursive definition of the polynomials P n, and the fact that λ n = P n+ (k)/p n (k), we have P n (k) λ n = β n a n P n (k) We will now show that lim n λ n is bounded above and below by k β Combining the facts deduced in the discussion above about λ n,wehave ( ) lim λ P n (k) n = lim β n a n n n P n (k) = k β lim a P n (k) n n P n (k) But a n P n (k)/p n (k) a n /(k x n,n ) a n /(k X), so lim n λ n k β (lim n a n )/(k X) Since λ n β n by definition for all n N, wehave lim n λ n = k b This implies lim n μ n = 0 because λ n + μ n = β n for all n Thus, when the numerators a n converge to 0, the death rates converge to zero and the birth rates converge to k β Theorem 27 Suppose {a n } and {b n } are defined as in Theorem 2, andalso suppose that lim n a n =0and lim n b n = b Then lim λ n = b, n and the process is transient lim μ n =0, n Proof By the definition of the λ n,wehaveb n λ n for all n, andfromtheorem 2 we have λ n b n α a n for sufficiently large n and some constant α>0as in Theorem 2 Combining these two facts with the assumption that a n and b n converge, we have b = lim sup n b n lim sup λ n lim inf λ n lim inf (b n α a n )=b, n n n and thus lim n λ n = b Since b n := λ n + μ n,wehavelim n μ n = lim n (b n λ n ) = 0 Thus lim n μ n =0 Considering again the uantity A, we will now prove A<, so the process described by these μ n and λ n is transient License or copyright restrictions may apply to redistribution; see

13 BIRTH-DEATH PROCESSES AND -CONTINUED FRACTIONS 275 First, fix some ɛ>0 Since λ n and μ n converge, there exists some N N such that for n N we have μ n <ɛand λ n <b+ ɛ Then N μ μ 2 μ n A = μ μ N λ 0 λ λ n λ 0 λ N n=0 n=n μ N μ n λ N λ n μ μ N λ 0 λ N n=n ɛ n N+ (b + ɛ) n N+ and so we have N μ μ 2 μ n A μ μ N ɛ λ n=0 0 λ λ n λ 0 λ N b Thus, since A is finite, the process described by these μ n and λ n is transient Theorem 28 Let a n,b n,β n,andk be defined as in Theorem and its proof For = (0, ), iflim n a n =0and lim n β n = β for a n and b n = k β n defined as in Theorem, the birth-death process is positive recurrent, and, if lim n λ n exists, lim λ n =0and lim μ n = k β n n Proof Consider the corresponding orthonormal polynomials p n (x) := P n (x)/ a a 2 a n mentioned in 22 (F3) These polynomials satisfy the recurrence relation an+ p n+ (x) =(x β n )p n (x) a n p n (x) with initial conditions p =0andp 0 = Thep n are orthonormal with respect to the distribution ψ, andx( )=k is in the spectrum of ψ, soby 22 (F4) we have P 2 B = π n =+ n(k) = p 2 a a 2 a n(k) n ψ(k+) ψ(k ) < n=0 n= Since n=0 π n converges, lim n π n =0,so/π n diverges, and thus n=0 /π n = Weknowλ n b n k, so A = = λ n=0 n π n k π n=0 n Thus A = and B<, so the process is positive recurrent We now use the fact that B< to determine the asymptotic behavior of the birth and death rates when = We know that B =+ n= λ2 0 λ2 λ2 n / (a a 2 a n ) converges, so, by the ratio test for series convergence, for all N N, there exists n N N such that λ 2 n N /a nn + Thus, 0 <λ 2 n N a nn + By our assumption that lim n a n = 0, lim N λ nn = 0 An infinite subseuence of {λ n } converges to 0, so if {λ n } converges, then lim n λ n =0 Sinceλ n +μ n = b n, this implies lim n μ n = k β 3 Modular forms and special birth-death processes In this section we first classify modular forms (see [5] for more on modular forms) satisfying relations which will turn out to have very interesting conseuences for the transition probabilities of birth-death processes We then examine modular continued fractions which correspond to special birth-death processes Finally, we examine interesting relations between some special birth-death processes n=0 License or copyright restrictions may apply to redistribution; see

14 276 TONY FENG, RACHEL KIRSCH, ELISE VILLELLA, AND MATT WAGE 3 Modular forms Here we establish definitions of modularly symmetric and modularly reflexive and illustrate the definition with two examples Let f be a weight k modular form on a congruence subgroup Γ SL 2 (Z), and let γ = [a, b; c, d] GL 2 (R) Then we define γ(f) :=(af + b)/(cf + d) Definition 3 Suppose that f and f 2 are weight k modular forms on a congruence subgroup Γ SL 2 (Z) Furthermore, suppose that there exists ρ R, α and l>0inz, γ GL 2 (R), and χ : GL 2 (R) C with χ(γ) =suchthat f ( /(ατ)) l = ρχ(γ)τ k γ(f 2 (τ) l ) Then we say that f (τ) ismodularly symmetric to f 2 (τ) Moreover, if f (τ) is modularly symmetric to f 2 (τ) =f (τ), then we say that f (τ)ismodularly reflexive Examples () If f(τ) is a modular ( ) form on SL 2 (Z), then f(τ) is modularly reflexive with 0 α =,l =,γ =, ρ =,andχ(γ) = 0 (2) Slightly abusing notation, let r(τ) = /5 r(), where = e 2πiτ Itisknown(see 33) that ( ) r = φr(τ)+ τ r(τ)+φ, where φ =(+ 5)/2 is the golden ratio ( Therefore, ) r() is a weight 0 modularly φ reflexive form with α =,l =,γ =, ρ =,andχ(γ) = φ As we shall see, birth-death processes corresponding to forms that are modularly symmetric satisfy very unexpected symmetry relations themselves (more precisely, the Laplace transforms of the transition probabilities evaluated at 0 satisfy interesting symmetry relations) The transformation τ /τ on the positive imaginary axis maps = e 2πiτ from the interval (0, ) onto itself, and this is precisely the interval on which our birth-death models are good In 33, we exhibit identities between the Laplace transforms of transition probabilities of different birth-death processes Fundamentally, these identities exist because the modular forms associated to the birth-death processes are modularly symmetric 32 Interesting -continued fractions corresponding to birth-death processes Here we examine examples of modular continued fractions due to Ramanujan, Eisenstein, and Jacobi which correspond to birth-death probability models with positivity of the birth rates guaranteed by Theorems and 2 Throughout, let λ n and μ n be the birth and death rates respectively as in euation (4) In the examples in Table, define μ 0 := 0, and b 0 = λ 0 = Note the relation between f B3 () and the classical Jacobi theta function, a modular form (see [8, p 290]): ϑ() = n= n2 = =+2f B 3 () License or copyright restrictions may apply to redistribution; see

15 BIRTH-DEATH PROCESSES AND -CONTINUED FRACTIONS 277 Table Example a n b n f Bi () Reference B 2n + n B 2 2n + 2n B 3 2n+ + 2n+ B 4 2n + 2n = (2 ; 3 ) (; 3 ) [3, p 20] + 2 = (3 ; 4 ) (; 4 ) [3, p 22] Also note the relationship between f B4 () and the second Jacobi theta function, a modular form [8, p 290]: (3) ϑ 2 () = n= (n+/2)2 = 2/ =2/4 f B4 () The proofs that these examples constitute good birth-death processes are simple applications of Theorem 2 For example, for the first one, we observe that λ 0 =, λ = + = b a Furthermore,a n b n =+ n 2n 2 > n > 2n+ 2 By Theorem 2, we have 0 < a n <b n a n <λ n <b n Therefore, this birthdeathprocessisgood Before continuing with the next 5 examples, we introduce the notion of an even part of a continued fraction (see [, p 4]) Let (32) K(a n /b n )= a 0 b 0 + b + b 2 + be a continued fraction Denote its nth convergent by C n Let Cn = C 2n Then the even part of K(a n /b n ) is the continued fraction (33) K(a n/b n)= a 0 a a 2 b 0 + b + b 2 + with convergents Cn Clearly if K(a n /b n )converges,thenk(a n/b n)convergesto the same value The following examples all converge for < Furthermore, we have the euations (34) a 0 = a 0b, b 0 = a + b 0 b a k = a 2ka 2k b 2k+ /b 2k,k b k = a 2kb 2k+ /b 2k + a 2k+ + b 2k b 2k+,k a a 2 33 Symmetry between birth-death processes Here we give examples of modularly symmetric forms (see 3) corresponding to birth-death processes exhibited in 32 Theorem 32 Let = e 2πiτ,whereIm(τ) > 0 The following are true: () ϑ() is a weight /2 modularly reflexive form with α =4, l =, γ =( 0 ρ = 2,andχ(γ) = i 0 ), License or copyright restrictions may apply to redistribution; see

16 278 TONY FENG, RACHEL KIRSCH, ELISE VILLELLA, AND MATT WAGE Table 2 Example b 0 (= λ 0 ) a n b n B 5 + 4n + 2n + 2n+ B ( 2n + 4n )( 2n + 4n 2 ) +( 2n + 4n ) +( 2n+ + 4n+ ) B 7 + 2n ( n + 2n ) +( n + 2n )+ 2n+ B n ( 2n + 4n 2 ) + 2n+ + 4n + 4n+2 B 9 8n 4 ( 4n+ ) ( 4n ) + n Table 3 Example Even part of f Bi () Reference B /5 5 + B 6 + B 7 + B 8 + B 9 ϑ 3 () = (; 5 ) (; 4 ) ( 2 ; 5 ) ( 3 ; 5 ) [7] = + + (; 2 ) ( 3 ; 6 ) [3, p 59] n= + + ( ; 2 ) 2( ; 2 ) [3, p 23] 8 + (; 8 ) (; 7 ) ( 3 ; 8 ) ( 5 ; 8 ) [3, p 23] ( ) n n ( ϑ 3 ())/(2) [8, p 282] (2) ϑ 2 () is a weight /2 modularly symmetric form to ϑ 3 () with α =4, l =, γ =( 0 0 ), ρ = 2,andχ(γ) = i ( ) (3) r() is a weight 0 modularly reflexive form with α =, l =, γ = φ φ, where φ = is the golden ratio, ρ =,andχ(γ) = (4) Let u(τ) = 2 /8 (+ 2n ) n= (+ 2n+ Thenu(τ) is a weight 0 modularly reflexive form with α =/4, l =2, γ = ( ), ρ =,andχ(γ) = (5) Let v(τ) = /2 n=0 ( 8n+ )( 8n+6 ) ( 8n+3 )( 8n+5 ) ( reflexive form with α =8, l =2, γ = χ(γ) = σ 2 σ 2 Then v(τ) is a weight 0 modularly ),whereσ = + 2, ρ =,and Proof () This is well known See, for example, Chapter 3 of Koblitz [4] Alternatively, since (35) ϑ() = η(2τ)5 η(τ) 2 η(4τ) 2, we can substitute τ = /τ and, using the transformation law for the Dedekind η-function [5, p 7], obtain (36) η( /(2τ)) 5 2iτη( /τ)2 η( /(4τ)) = η(2τ)5 2 η(τ) 2 η(4τ) 2 License or copyright restrictions may apply to redistribution; see

17 BIRTH-DEATH PROCESSES AND -CONTINUED FRACTIONS 279 (2) We have the following η-uotient representations [5, p 7]: (37) η(τ) 2 η(2τ) = n= ( ) n n2 and η(4τ)2 η(2τ) = (n+/2)2 Now make the transformation τ /τ in η(τ) 2 /η(2τ) (3) This identity dates back to Ramanujan (for a modern proof, see Proposition 2 in [7]): ( ) (38) r = φr(τ)+ τ r(τ)+φ, where φ =(+ 5)/2 is the golden ratio (4) This follows from the identity [7, p 2] (39) u 2 ( 4/τ) = u2 (τ) +u 2 (τ) (5) This follows from the identity [7, p 22] (30) v 2 ( /(8τ)) = v2 (τ) σ 2 σ 2 v 2 (τ), where σ = + 2 Corollary 33 For y>0, the following identities are true: () +2e π(2y) f B3 (e π(2y)) = ( 2/y (+e π/(2y) f B3 e π/(2y))) (2) ( 2e πy/2 f B4 e π(2y)) = ( e π/(2y) f B9 e π/(2y)) 2y 2 2y (3) ( ( e 2π/(5y) f B5 e 2π/y) = φe2πy f ) B5 e 2πy + e 2πy f B5 (e 2πy )+φ (4) e πy/2 ( 2 f B 2 7 e 8π/y) = e πy/2 ( 2 f ) B 2 7 e 2πy + eπy/2 2 fb 2 5 (e 2πy ) (5) ( ( e π/(4y) fb 2 8 e π/(4y)) = e2πy f ) B 2 8 e 2πy σ 2 σ 2 e 2πy fb 2 8 (e 2πy ) Proof () Recall that for the birth-death process B 3 we have the euation (3) ϑ()=+2f B3 () Set τ = iy in (36), and substitute the identity (3) in the resulting euation (2) Make the transformation τ /τ in the B 9 identity from Table 3, substitute this identity and (3) into (37), and then set τ = iy (3) Substitute the B 5 identity from Table 3 into (38) and then set τ = iy (4) Substitute the B 7 identity from Table 3 into (39) and set τ = iy (5) Substitute the B 8 identity from Table 3 into (30) and set τ = iy n=0 License or copyright restrictions may apply to redistribution; see

18 2720 TONY FENG, RACHEL KIRSCH, ELISE VILLELLA, AND MATT WAGE Remark The identities in Corollary 33 stemmed from the distinctive behavior of modular forms under the transformation τ /τ There are further results which depend on algebraic euations satisfied by modular forms For example, let f() =f B5 () For = e 2πiτ with Im (τ) > 0, one can show the following (see Theorems 75, 753, and 754 in [5] and Entry 24 and Entry 25 in [3]): () f()f(2 ) 2 = f(2 ) f() 2 f( 2 )+f() 2 (2) (3) 3 f()2 f( 3 ) 2 = ( f( 3 ) f() 3) (+ f()f(3 ) 2 2 ) ( 5 f()f( 2 f()2 f( 4 ) 2 4 ) 2 ) f()f(4 ) ( f() 5 = + f(4 ) 5 )( f()f( 4 ) ) 4 + f()5 f( 4 ) 5 5 (4) (5) f() 5 f( 5 ) = 2 f( 5 ) +4 f(5 ) 2 3 f(5 ) f(5 ) f(5 ) +4 f(5 ) 2 +2 f(5 ) f(5 ) f()5 f( 2 ) 5 ( f() 0 f( 2 ) 5 + f( 2 ) 0) + f() 0 f( 2 ) ( f()f( f()5 f( 2 ) 5 2 ) 5 3 f()5 + f(2 ) 5 ) 2 + =0 Acknowledgments The authors wish to thank Professors Ken Ono and Amanda Folsom for their guidance and suggestions They also thank Mourad Ismail, Frank Thorne, and Kathrin Bringmann for their helpful comments Finally, they thank the National Science Foundation, the Manasse family, and the Hilldale Foundation for supporting this work References [] W A Al-Salam and M E H Ismail, Orthogonal Polynomials Associated With the Rogers- Ramanujan Continued Fraction, Pacific J of Math, 04 (983), MR (85g:33007) [2] W J Anderson, Continuous-Time Markov Chains: An Applications-Oriented Approach, Springer-Verlag, New York, 99 MR8840 (92k:6070) [3] G E Andrews, B C Berndt, L Jacobsen, and R L Lamphere, The Continued Fractions Found in the Unorganized Portions of Ramanujan s Notebooks, MemAmerMathSoc,99 (992), No 477, Providence, RI MR2409 (93f:008) [4]BTAnhandDDXThanh,A Perron-Frobenius Theorem for Positive Quasipolynomial Matrices Associated with Homogeneous Difference Euations, J Appl Math vol 2007, Article ID 26075, 6 pages, 2007 MR License or copyright restrictions may apply to redistribution; see

19 BIRTH-DEATH PROCESSES AND -CONTINUED FRACTIONS 272 [5] B Berndt, Number Theory in the Spirit of Ramanujan, Student Mathematical Library, vol 34, Amer Math Soc, Providence, 2006 MR (2007f:00) [6] T S Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 978 MR (58:979) [7] W Duke, Continued Fractions and Modular Functions, Bull Amer Math Soc 42 (2005), MR (2006c:042) [8] A Folsom, Modular forms and Eisenstein s continued fractions, J Number Theory 7 (2006), MR (2007a:056) [9] M E H Ismail, The Zeros of Basic Bessel Functions, the Functions J v+ax (x), and Associated Orthogonal Polynomials, J Math Anal Appl 86 (982), 9 MR (83c:3300) [0] M E H Ismail and R Zhang, On the Hellmann-Feynman Theorem and the Variation of Zeros of Certain Special Functions, Adv in Appl Math 9, (988), MR (89i:850) [] W B Jones and W J Thron, Continued Fractions: Analytic Theory and Applications, Addison-Wesley, Reading, 980 MR (82c:3000) [2] V Kac and P Cheung, Quantum Calculus, Springer, New York, 2002 MR (2003i:3900) [3] S Karlin and J L McGregor, The Differential Euations of Birth-and-Death Processes, and the Stieltjes Moment Problem, Trans Amer Math Soc, 85 (957), MR (9:989d) [4] N Koblitz, Introduction to Elliptic Curves and Modular Forms, Graduate Texts in Math, vol 97, Springer (2nd ed), 993 MR2636 (94a:078) [5] K Ono, The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and - series, Amer Math Soc, Providence, 2003 MR (2005c:053) [6] P R Parthasarathy, R B Lenin, W Schoutens, and W Van Assche, A Birth and Death Process Related to the Rogers-Ramanujan Continued Fraction, J Math Anal Appl, 224 (998), MR (99g:6056) [7] K G Ramanathan, Ramanujan s Continued Fraction, Indian J Pure Appl Math, 6 (985), MR8080 (87e:05) [8] S M Ross, Stochastic Processes, John Wiley and Sons, Inc, New York, 996 MR (97a:60002) [9] W Van Assche, The Ratio of -like Orthogonal Polynomials, J Math Anal and Appl, 28 (987), MR97386 (89a:33007) [20] G N Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, 944 MR (6:64a) 58 Plympton Street, 479 Quincy Mail Center, Cambridge, Massachusetts address: tfeng@collegeharvardedu 722 Longwood Drive, Bethesda, Maryland address: rkirsch@mathunledu 46 Harrison Drive, Edinboro, Pennsylvania 642 address: elisemccall@gmailcom 4 N Briarcliff Drive, Appleton, Wisconsin address: mwage@princetonedu License or copyright restrictions may apply to redistribution; see