A spectral decomposition for the block counting process and the ﬁxation line of the beta(3,1)-coalescent

A spectral decomposition for the generator of the block counting process of the β (3 , 1) coalescent is provided. This decomposition is strongly related to Riordan matrices and particular Fuss–Catalan numbers. The result is applied to obtain formulas for the distribution function and the moments of the absorption time of the β (3 , 1) coalescent restricted to a sample of size n . We also provide the analog spectral decomposition for the ﬁxation line of the β (3 , 1) -coalescent. The main tools in the proofs are generating functions and Siegmund duality. Generalizations to the β ( a, 1) coalescent with parameter a ∈ (0 , ∞ ) are discussed leading to fractional differential or integral equations.

For t ≥ 0 let N t denote the number of blocks of Π t . The process (N t ) t≥0 is called the block counting process of Π. In [16] a spectral decomposition for the generator of the We focus on the particular beta coalescent with Λ = β(3, 1) the beta distribution with parameters a = 3 and b = 1 having density x → 3x 2 , x ∈ (0, 1). One reason why we focus on this particular beta coalescent is the fact that its block counting process moves from state i ∈ N with i ≥ 2 to any state j ∈ N with j < i with equal probability 1/(i − 1) not depending on j. The generator Q = (q ij ) i,j∈N of the block counting process of the β(3, 1)-coalescent has entries (see, for example, [4,Eq. (2.6)] with a = 3 and b = 1) (1.1) Let q i := −q ii = 3(i − 1)/(i + 1) denote the total rates, i ∈ N. In order to state the results the following definition from [21] is useful.

Definition 1.1 (Riordan matrix).
A lower left triangular matrix R = (r ij ) i,j∈N is called a Riordan matrix if for every j ∈ N the jth vertical generating function r j (z) := ∞ i=j r ij z i has the form r j (z) = f (z)(g(z)) j for some functions f and g of the form f (z) = 1 + f 1 z + f 2 z 2 + · · · and g(z) = z + g 2 z 2 + g 3 z 3 + · · · defined in some neighborhood of 0.
Our main result (Theorem 1.2 below) provides an explicit spectral decomposition for the generator Q of the block counting process of the β(3, 1)-coalescent. The result is remarkable, since the β(3, 1)-coalescent seems to be the only beta coalescent different from the Bolthausen-Sznitman coalescent where such an explicit spectral decomposition is available. For further information on this topic we refer the reader to Section 2 where the delicate question on extensions to other beta coalescents is discussed. Moreover, Theorem 1.2 sheds some new light on particular Fuss-Catalan numbers and generalized Stirling numbers as explained in the remarks after the theorem.
In the following Γ denotes the gamma function. We furthermore use the notation N 0 := {0, 1, 2, . . .}. The proof of the following theorem and of all other results are provided in Section 3.

Theorem 1.2. (Spectral decomposition of the generator of the block counting
process) The generator Q = (q ij ) i,j∈N of the block counting process of the β(3, 1)coalescent has spectral decomposition Q = RDL, where D = (d ij ) i,j∈N is the diagonal matrix with entries d ii = −3(i − 1)/(i + 1), i ∈ N, and R = (r ij ) i,j∈N and L = (l ij ) i,j∈N are the lower left triangular Riordan matrices We do not have an intuitive explanation for the fact that R has non-negative entries.
Let us provide some applications of Theorem 1.2. For n ∈ N let Π (n) = (Π (n) t ) t≥0 denote the coalescent restricted to a sample of size n (n-coalescent) and let N (n) t denote the number of blocks of Π (n) t , t ≥ 0. We are interested in τ n := inf{t > 0 : N (n) t = 1}, the absorption time of Π (n) . In the biological context τ n is called the time back to most recent common ancestor. For the β(3, 1)-coalescent it has been recently shown [15,Proposition 3.4] that τ n has the convolution representation where E 2 , E 3 , . . . are independent and E k is exponentially distributed with parameter q k , and ξ 2 , ξ 3 , . . . are independent Bernoulli random variables and independent of E 2 , E 3 , . . . with E(ξ k ) = 1/k, k ∈ {2, 3, . . .}. Formula (1.7) is intuitively clear by interpreting E k as the sojourn time of the block counting process in state k and {ξ k = 1} as the event that the jump chain of the block counting process ever visits state k ∈ {2, . . . , n} when started from state n. The independence of (E k ) k and (ξ k ) k and the fact that ξ k does not depend on the initial state n are however particular for the β(3, b)-coalescent with parameter b ∈ (0, ∞) and related to the property that for this particular class of coalescents the block counting process has constant hitting probabilities. For more details we refer the reader to the proof of [15,Proposition 3.4]. From (1.7) one may derive formulas for the distribution function of τ n . The detail computations are however not very amusing. Instead, we proceed as follows. Theorem 1.2 implies that the transition matrix P (t) = (p ij (t)) i,j∈N = e tQ of the block counting process has spectral decomposition P (t) = Re tD L. Thus, Theorem 1.2 immediately yields Consequently, τ n has moments From (1.7) and the central limit theorem it follows that (3τ n − log n)/ √ 2 log n is asymptotically standard normal distributed, in agreement with Table 2 of [5].
The total tree length of the β(3, 1)-n-coalescent has a convolution representation (see [15,Proposition 3.5]) similar to the representation (1.7) for τ n . Theorem 1.2 seems to be not directly useful to derive formulas for the distribution function or the moments of the total tree length, since these functionals cannot be expressed easily in terms of the transition matrices P (t), t ≥ 0.
We now turn to the fixation line of the β(3, 1)-coalescent. For n ∈ N and t ≥ 0 define and set L t := L (1) t for convenience. The process (L t ) t≥0 is called the fixation line of the coalescent. It is easily seen from (1.9) and well known (see [4,Theorem 2.9] or [6, Lemma 2.4]) that the block counting process is Siegmund dual (see [23]) to the fixation line, i.e. P(N (1.10) Note that the total rates g i := −g ii = 3i/(i + 2), i ∈ N, of the fixation line are related to those of the block counting process via g i = q i+1 , i ∈ N. The latter equality holds for all exchangeable coalescents (see, for example, [4, Proposition 2.5]) and is essentially a consequence of the Siegmund duality relations q j,≤i = g i,≥j , i, j ∈ N, for the generator entries (see, for example, [4, Eq. (5.1)]). Choosing j := i + 1 in these relations yields

Theorem 1.4. (Spectral decomposition of the generator of the fixation line)
The with the convention thatr ij = 0 if i − j 2 + 1 ∈ −N 0 , and In contrast to the spectral decomposition for the block counting process, here the matrix L has non-negative entries. Again we do not have an intuitive explanation for this fact.
(ii) It is readily seen thatr ij andl ij are related to the Fuss-Catalan numbers c n (α, β)

On extensions to the beta(a,1)-coalescent
In this section generalizations to the β(a, 1)-coalescent with parameter a ∈ (0, ∞) are discussed. For the β(a, 1)-coalescent with parameter a ∈ (0, ∞) the block counting process has rates (see, for example, [4, Eq. (2.6)]) q ij = a Γ(i + 1) Γ(i + a − 1) The total rate q i := i−1 j=1 q ij therefore simplifies to and total rates For a ∈ (0, ∞) and |z| < 1 define For a ∈ (0, 2) the function φ is the probability generating function of a random variable η with distribution For a ∈ [2, ∞) there is no analog probabilistic interpretation for φ, however we can still work with φ. Note that φ(z) = 0 for a = 2. The following lemma provides a partial answer towards the spectral decomposition Q = RDL of the generator Q of the block counting process of the β(a, 1)-coalescent.
In order to state the result let us briefly recall fractional integrals and derivatives.
For a function f : [0, 1) → R the Riemann-Liouville fractional integral of order α ∈ (0, ∞) provided that the integral on the right hand side exists. The Riemann-Liouville fractional where n := α + 1, provided that the expression on the right hand side exists. For α ∈ (−∞, 0) we also write (D α f )(x) := (I −α f )(x). We also use the notation D α x (f (x)) := (D α f )(x), α ∈ R, x ∈ [0, 1). For general information on fractional calculus theory we refer the reader to the books of Kilbas, Srivastava and Trujillo [8], Miller and Ross [13] and Podlubny [18]. Lemma 2.1. Let a ∈ (0, ∞) and let R = (r ij ) i,j∈N denote the matrix in the spectral decomposition Q = RDL of the generator Q of the block counting process of the β(a, 1)coalescent. Then for every j ∈ N the vertical generating function r j (z) := ∞ i=j r ij z i is a solution to the equation where φ(z) is defined via (2.5).

Remark 2.2.
For a = 1 Eq. (2.6) is of the form (1−φ(z))r j (z) = j z 0 r j (t)/t dt. Taking the derivative with respect to z yields the differential equation (1 − φ(z))r j (z) − φ (z)r j (z) = r j (z)/z, which was solved in [16, Eq. (2.7)]. For a = 3 Eq. (2.6) reduces to the differential equation (3.3) with solution (3.4). For a = 2 Eq. (2.6) leads to the uninformative equation r j (z) = r j (z). For integer a ∈ {4, 5, . . .} one may be able to solve the differential equation (2.6) of order a − 2 for r j , however the solution may turn out to have a rather complicated form. For non-integer parameter a it might be possible to solve the truly fractional equation (2.6) by applying fractional calculus theory; see for example Kilbas, Srivastava and Trujillo [8], Miller and Ross [13] or Podlubny [18]. We leave the solution of (2. where φ(z) is defined via (2.5).

Remark 2.4.
Again, to the best of the authors knowledge, explicit solutions of (2.7) are only known for a = 1 (see [10]) and a = 3 (see the proof of Theorem 1.4). For a = 2 Eq. (2.7) degenerates to the uninformative equationl i (z) + zl i (z) = d dz (zl i (z)).

Remark 2.5.
In this final remark we provide some further information explaining why the parameter values a = 1 and a = 3 are rather particular. For a ∈ {1, 3} the matrix R = (r ij ) i,j∈N of the spectral decomposition Q = RDL of the generator Q of the block counting process of the β(a, 1)-coalescent has entries where S(i, j; α, β, r) are the generalized Stirling numbers as defined in Hsu and Shiue [7] and [x] 0 := 1 and [x] n := x(x + 1) · · · (x + n − 1), n ∈ N, denote the ascending factorials. Note that (2.8) even holds for the limiting case a → 0 (Kingman coalescent), which follows from the formula for r ij for the Kingman coalescent provided in the appendix of [16] and from S(i, j; −1, 1, 0) = i! j! i−1 j−1 (Lah numbers).

Proofs
We first provide the proofs of Theorems 1.2 and 1.4. Both proofs are based on generating functions. We additionally present an alternative short proof of Theorem 1.4 based on Siegmund duality. We start with the proof of Theorem 1.2, since this proof turns out to be slightly less technical than the (first) proof of Theorem 1.4.
Proof. (of Theorem 1.2) As in [16] it follows that the entries r ij of R satisfy for each j ∈ N the recursion r jj = 1 and Plugging in q ik = 3/(i + 1), 1 ≤ k < i, and q i − q j = 3(i − 1) In order to solve this recursion we proceed similar as in the proof of Theorem 1.1 of [16].
For j ∈ N define the generating function r j (z) := ∞ i=j r ij z i , |z| < 1, and consider the modified generating function f j (z) : On the other hand f j (z) = ∞ i=j ir ij z i − j ∞ i=j r ij z i = zr j (z) − jr j (z). Thus, zr j (z) − jr j (z) = ((j + 1)/2)(z/(1 − z))r j (z) or, equivalently, r j (z) = j + 1 2 Proof. (of Theorem 1.4) Two proofs of Theorem 1.4 are provided. The first proof is self-contained and again based on generating functions. The second proof is rather short and exploits Theorem 1.2 and the fact that the block counting process is Siegmund dual to the fixation line.

Proof 1.
We follow the first proof of Theorem 3.1 of [10]. LetD = (d ij ) i,j∈N be the diagonal matrix with entriesd ii := −g i = g ii , i ∈ N, and letR = (r ij ) i,j∈N be the upper right triangular matrix with entries defined for each j ∈ N recursively viar jj := 1 and r ij := (g i − g j ) −1 j k=i+1 g ikrkj for i ∈ {j − 1, j − 2, . . . , 1}. Since g ii = −g i , i ∈ N, we conclude thatr ij g jj = j k=i g ikrkj , Thus, the entries ofR are defined such thatRD = GR. DefineL :=R −1 . Then, the spectral decomposition G =RDL holds. Moreover,DL =LG and, hence, g iilij = j k=il ik g kj , i, j ∈ N. Since g ii = −g i , i ∈ N, we obtain for each i ∈ N the recursionl ii = 1 and On the other hand, by the recursion (3.6), we obtain or, after some straightforward manipulation, The solution of this homogeneous second order differential equation with initial condi- The functionl i has Taylor expansionl i The coefficientl ij in front of z j in the Taylor expansion ofl i is hence given by (1.12).
For i ≤ j the coefficientr ij in front of u j in the Taylor expansion ofr i (u) is hence given by (choose r = 2i + 2l − j above) .

Remark 3.2.
For the Bolthausen-Sznitman coalescent the corresponding generating functionsr i andl i , i ∈ N, are given by (see [10, p. 954 showing that the transposed matricesR andL of the spectral decomposition G =RDL of the generator G of the fixation line of the Bolthausen-Sznitman coalescent are Riordan matrices of the formR = (1, 1 − e −z ) and L = (1, − log(1 − z)).
We finish this section with the proofs of Lemma 2.1 and Lemma 2.3.