The evaluation of a weighted sum of Gauss hypergeometric functions and its connection with Galton-Watson processes
aa r X i v : . [ m a t h . C A ] D ec The evaluation of a weighted sum of Gausshypergeometric functions and its connection withGalton-Watson processes
R. B. Paris ∗ Division of Computing and Mathematics,Abertay University, Dundee DD1 1HG, UK
Vladimir V. Vinogradov
Department of Mathematics, Ohio University,Athens, OH 45701, USA
Abstract
We evaluate a weighted sum of Gauss hypergeometric functions for certain ranges of theargument, weights and parameters. The domain of absolute convergence of this series isestablished by determining the growth of the hypergeometric function for large summationindex. An application to Galton-Watson branching processes arising in the theory of stochasticprocesses is presented. A new class of positive integer-valued distributions with power tails isintroduced.
MSC:
Keywords: asymptotic behaviour, characteristic function, discrete distribution with powertail, Fourier transform, Galton-Watson process, hypergeometric function with large parame-ters, law of the total progeny, probability-generating function, scaled Sibuya distribution
1. Introduction
In the probabilistic treatment of the law of the total progeny of a particular subcritical Galton-Watson branching process that we undertook in 2017, the following sum of Gauss hypergeometricfunctions was encountered: X k ≥ − k (1 − x ) k − F ( k, k + ; 2; x ) ≡ , x ∈ [0 , , (1.1)where, for complex z [10, p. 384], F ( a, b ; c ; z ) = X k ≥ ( a ) k ( b ) k ( c ) k z k k ! ( | z | < ∗ E-mail address: [email protected] ; [email protected] a ) k = Γ( a + k ) / Γ( a ) is the Pochhammer symbol. A collection of the terms of the infiniteseries in (1.1) constitutes a probability distribution on the set of positive integers. In particular,for x = 1, this collection degenerates into a point mass at 1 (compare to (2.2)). In contrast, for x = 0 this collection becomes the set of the probabilities of the geometric distribution on the set N of positive integers with equal chances of successes and failures. It turns out that for a fixed x ∈ (0 , F function that appears on the left-handside of that formula by an arbitrary fixed real constant c > . Specifically, we established that theset c − c − √ x (1 − √ x ) ℓ − F ( ℓ, ℓ + ; c ; x ) , x ∈ (0 ,
1) (1.2)constitutes a genuine probability function on N whose upper tail decays like ℓ / − c as ℓ → + ∞ .This follows with some effort from a combination of (2.1) and Theorem 1 below. However, a detailedinvestigation of properties of class (1.2) of the probability distributions of random variables X c isbeyond the scope of the present paper. The study of random variables X c − c > is deferredto our work in progress [16].To some extent, the analysis parts of this paper provide the analytical foundation for our work inprogress [16], which is devoted to some aspects of distribution theory. Thus, the need to determinea closed-form expression for the probability-generating function for the class of distributions (1.2)motivated us to consider a more general problem. The following summation theorem for theweighted sum of Gauss hypergeometric functions, denoted by S ( η, c ; x ), will be established inSection 3: Theorem 1 . Let c and η be positive parameters and x ∈ [ − , . Provided the sum in (1.4) isabsolutely convergent (see Theorem 2) we have the evaluation S ( η, c ; x ) := X k ≥ (cid:18) − x η (cid:19) k F ( k + , k + 1; c ; x )= 1 X F (cid:18) , c ; xX (cid:19) , X := x + η η (1.3) when η ≥ for x ∈ [ − , and for x ∈ ( − η, η ] when < η < . The investigation of the sum that emerges in (1.3) is the primary topic of this article.An analogous sum has been derived independently by Letac in [9, Lemma 2.2] in the form S L := X k ≥ z k F ( k, k + ; c ; x ) = z − z F (cid:18) , c ; x (1 − z ) (cid:19) (1.4)for 0 < z <
1, 0 < x < (1 − z ) with c >
0, using a combination of probabilistic and analyticalarguments. The sum S L is proportional to S ( η, c ; x ) when z = (1 − x ) / (1 + η ). An importantdifference between S ( η, c ; x ) and the sum S L is that the variable x in (1.3) can assume negativevalues, whereas in (1.4) both x and z are restricted to positive values. We stress that negative values of the variable x do emerge in the case where one considers the characteristic function (i.e.,Fourier transform) of a probability distribution on non-negative integers instead of a simpler butless general probability-generating function. See also the end of Section 5 for more detail on therelation of our work with that of Letac, as well as Remark 2 therein. sum of hypergeometric functions S ( η, c ; x ) it is necessary to investigate theconvergence of the series. This is carried out in Section 2, where the convergence conditions areestablished and stated as Theorem 2. In Section 3, we prove the summation formula (1.3) for S ( η, c ; x ). In Section 4 we present some special cases concerning the evaluation of the sum (1.3) aswell as an extension of (1.1). In the concluding section we present an application to Galton-Watsonprocesses arising in the theory of stochastic processes. For the reader’s convenience some technicalcomputations are deferred to the appendix.
2. Discussion of the convergence of S ( η, c ; x )To determine the domain of convergence of the sum S ( η, c ; x ) we need to examine the asymptoticbehaviour of the Gauss hypergeometric function appearing in (1.3) as k → + ∞ . This is carriedout in the appendix. From (A.1) and (A.3), we have the leading asymptotic behaviour as k → + ∞ given by F ( k + , k + 1; c ; x ) ∼ c − Γ( c ) √ π x c − k − c (1 − √ x ) − k + c − (0 < x < c )2 c − √ π | x | c − k − c (1 + | x | ) − k + c − sin Φ( k ) ( x <
0) (2.1)where Φ( k ) = ( k − c + ) φ − π ( c − ) and φ = arctan p | x | .Before proceeding with the discussion of the convergence of S ( η, c ; x ), we note the trivial eval-uations when x = 0 and x = 1 given by S ( η, c ; 0) = 1 + ηη , S ( η, c ; 1) = F ( , c ; 1) = 2 c − c − c > ) , (2.2)the second identity following from the well-known Gauss summation theorem [10, (15.4.20)]. How-ever, it should be noted that when 0 < c ≤ the sum S ( η, c ; x ) does not converge to a finite limitas x ↑
1. This can be seen from the first expression on the right-hand side of (2.1), which showsthat the late terms ( k ≫
1) in (1.3) are controlled by2 c − Γ( c ) √ πx c/ − / (cid:18) − x η (cid:19) k k − c (1 − √ x ) − k + c − = 2 c − Γ( c ) k − c √ πx c/ − / (cid:18) √ x η (cid:19) k (1 − x ) c − ; (2.3)see also Remark 1 below. With η >
1, the sum therefore diverges as x ↑ < c < . When c = , the divergent behaviour (when η >
1) is controlled by the k = 0 term in (1.3), namely F ( , ; x ), which behaves logarithmically like log (1 − x ) − / as x ↑ < x <
1, it therefore follows from (2.3) that the late terms in S ( η, c ; x ) possess thecontrolling behaviour k − c (cid:18) √ x η (cid:19) k ( k → + ∞ ) . Consequently S ( η, c ; x ) converges absolutely when 0 < x < η > √ x when 0 < c ≤ ,and η ≥ √ x when c > .When x <
0, the second expression on the right-hand side of (2.1) shows that the late terms in S ( η, c ; x ) possess the (absolute) behaviour controlled by (cid:18) | x | η (cid:19) k k − c (1 + | x | ) − k = k − c (cid:18) p | x | η (cid:19) k ( k → + ∞ ) . It therefore follows that S ( η, c ; x ) converges absolutely when x < η > p | x | − < c ≤ ) , η ≥ p | x | − c > ) . These conditions are summarised in Theorem 2.
Theorem 2 . The series S ( η, c ; x ) is absolutely convergent provided η > √ x (0 < c ≤ ) , η ≥ √ x ( c > ) (2.4) for < x < , and η > p | x | − < c ≤ ) , η ≥ p | x | − c > ) (2.5) when x < . Remark 1 : For x >
0, the inclusion of η = √ x in (2.4) is related to the existence of the extremeconjugate distribution (1.2).
3. The evaluation of S ( η, c ; x )To evaluate the sum in (1.3) we express the F function appearing therein by its series represen-tation to find S ( η, c ; x ) = X k ≥ (cid:18) − x η (cid:19) k X n ≥ ( k + ) n ( k + 1) n ( c ) n n ! x n ( x ∈ ( − , X k ≥ (cid:18) − x η (cid:19) k X n ≥ ( k + 1) n ( c ) n n ! (cid:18) x (cid:19) n , where we have made use of the result ( a ) n = 2 n ( a ) n ( a + ) n .Employing the fact that ( k + 1) n = (2 n )!(2 n + 1) k /k !, we obtain S ( η, c ; x ) = X n ≥ ( ) n (1) n ( c ) n n ! x n X k ≥ (2 n + 1) k k ! (cid:18) − x η (cid:19) k , provided 1 − x < η so that the interchange in the order of summation is permissible. The sumover k can be expressed in terms of a F function with the evaluation [10, (15.4.6)] F (cid:18) n + 1; ; 1 − x η (cid:19) = X k ≥ (2 n + 1) k k ! (cid:18) − x η (cid:19) k = X − n − , X := x + η η . This then yields S ( η, c ; x ) = 1 X X n ≥ ( ) n (1) n ( c ) n n ! (cid:18) xX (cid:19) n ( | x | < X ) . (3.1)Identification of the sum in (3.1) as a Gauss hypergeometric function then establishes the summa-tion result in (1.3). A similar treatment can be brought to bear on Letac’s sum S L to produce thesummation formula stated in (1.4).If we let ξ := x/X , the hypergeometric function in (1.3) supplies the analytic continuation of S ( η, c ; x ) for ξ ∈ ( −∞ ,
1] ( c > ) and ξ ∈ ( −∞ ,
1) (0 < c ≤ ) . sum of hypergeometric functions η ≥ ξ ∈ [ − ξ ∗ ,
1] for x ∈ [ − , ξ ∗ := ( η + 1) / ( η − . In the case0 < η <
1, we have ξ ∈ ( −∞ ,
1] for x ∈ ( − η, η ] and ξ ∈ ( −∞ , − ξ ∗ ] for x ∈ [ − , − η ). Finally,when x = 1, the hypergeometric function in (1.3) has unit argument, so that it is clear we require c > for its convergence to the value given in (2.2). For 0 < c ≤ , the function on the right-handside of (1.3) diverges as x ↑
4. Special cases of S ( η, c ; x ) for integer c When c = m + 1 ( m = 0 , , , . . . ) the hypergeometric function in (1.3) can be expressed in closedform. From [10, (15.8.4)] connecting hypergeometric functions of argument χ and 1 − χ and thefamiliar Euler transformation [10, (15.8.1)], we have χ c − F ( , c, χ ) = Γ( c )Γ( − c ) √ π (1 − χ ) c − / + 2 c − c − F (2 − c, − c ; − c ; 1 − χ ) . For c = 2 , . . . the hypergeometric function on the right reduces to a polynomial in 1 − χ of degree c − : F ( , c ; χ ) = (1 − χ ) − / ( c = 1 , χ ∈ ( −∞ ,
1) )2 χ { − (1 − χ ) / } ( c = 2 , χ ∈ ( −∞ ,
1] )43 χ {− χ + 2(1 − χ ) / } ( c = 3 , χ ∈ ( −∞ ,
1] )25 χ { − χ + 15 χ − − χ ) / } ( c = 4 , χ ∈ ( −∞ ,
1] ) . (4.1)It is relevant that the representations in (4.1) can also be obtained from [14, (3.1), (4.5)]. It willbe observed that when c = 1 the value of F ( , c ; χ ) diverges as χ ↑
1, whereas the other valueswith c > are finite in this limit in accordance with (2.2).When c = 1 and η ≥
1, we find from (1.3) the evaluation S ( η, x ) = 1 X (cid:18) − xX (cid:19) − / = 1 + η p (1 − x )( η − x ) . (4.2)When 0 < η <
1, we have
X > x > − η so that (4.2) still applies in this case. However, when x < − η , then X < X = | X | e πi in this case, we find with X = | X | e πi that the same result holds.Hence, we obtain S ( η, c ; x ) = 1 + η p (1 − x )( η − x ) x ∈ [ − ,
1) (4.3)provided the convergence conditions η > √ x (0 < x <
1) and η > p | x | − x <
0) hold.A similar treatment when c = 2 and c = 3 yields S ( η, x ) = 2( η + x )(1 + η ) x (cid:26) − p (1 − x )( η − x ) η + x (cid:27) , x ∈ [ − ,
1] (4.4) The evaluations in (4.1) can also be generated by
Mathematica . Closed-form evaluations in terms of arctanh √ χ are also possible when c = m + . and S ( η, x ) = 4( η + x )3(1 + η ) x (cid:26) x (1 + η ) − η + x ) + 2[(1 − x )( η − x )] / η + x (cid:27) , x ∈ [ − , . (4.5)In (4.4) and (4.5) we require the convergence conditions η ≥ √ x (0 < x <
1) and η ≥ p | x | − x < η = 1 the above evaluations reduce to S (1 , x ) = 21 − x , x ∈ [ − , , S (1 , x ) = 2 − x, x ∈ [ − , . In the probabilistic application given in (1.1) we have12 X k ≥ (cid:18) − x (cid:19) k F ( k + , k + 1; 2; x ) = 1 (4.6)valid in the extended interval x ∈ [ − , < η <
1, the evaluation of S ( η, c ; x ) when x = η follows immediately from (1.3) bythe Gauss summation theorem. To determine the value when x = − η (corresponding to X = 0)we apply the Euler transformation [10, (15.8.1)] to obtain F (cid:18) , c ; − | x | X (cid:19) = (cid:18) | x | X (cid:19) − / F (cid:18) , c − c ; | x || x | + X (cid:19) and again apply the Gauss summation theorem. Thus, we find the evaluations S ( η, c ; − η ) = Γ( c )Γ( c − ) r πη , S ( η, c ; η ) = 2 c − c − · η ( c > ) (4.7)when 0 < η <
5. Application to Galton-Watson processes
First, we introduce a two-parameter class of scaled (or zero-modified ) Sibuya distributions W ( α ) λ . Definition 1.
Given α ∈ (0 , and λ ∈ (0 , , the probability-generating function and the proba-bility function of the scaled Sibuya random variable W ( α ) λ are as follows: P W ( α ) λ ( u ) = 1 − λ (1 − u ) α , (5.1) P {W ( α ) λ = k } = (cid:26) − λ if k = 0 − λ ( − α ) k /k ! if k ∈ N . (5.2)The case λ = 1 corresponds to the Sibuya distribution per se . The reader is referred to [5, 7, 8] formore detail on Sibuya and scaled Sibuya distributions.It is relevant that the most elaborate result of this section, which motivated the present paper,pertains to the case α = . In this case, the expression − λ ( − α ) k /k ! appearing on the right-handside of (5.2) can be rewritten as λ Γ( k − )2 √ πk ! = λ k = 1) λk ! (2 k − k ( k ≥ sum of hypergeometric functions p = therein. Letac was concerned with the casewhere the law of the total progeny is given by (5.1) with λ = 1. See [13] for the case α = and hisrecent paper [8] for the case α ∈ [ , λ ∈ (0 ,
1) and investigate the resulting law of the total progeny.Hence, one of the main goals of this section is to demonstrate that the probablity function ofthe law of the total progeny for a particular “dual” subcritical Galton-Watson process with Sibuya-type branching mechanism corresponding to α = is given by (5.15). The latter representationconstitutes the first summation theorem for Gauss hypergeometric functions that we originallyderived in 2017 by applying probabilistic arguments described in this section.We recognise that our probabilistic derivation of (5.15) can be simplified since it is not absolutelynecessary to use the classical results of queueing theory here. However, we elected to provide ourproof in the present form in order to emphasise connections between Galton-Watson processes andsome models of queueing theory. In addition, we consider that it is also instructive to discuss thegeneral case of α ∈ (0 , α < λ <
1. By (5.1), the probability Q of ultimate extinction of this process is asfollows: Q (= Q α,λ ) = 1 − λ / (1 − α ) ∈ (0 ,
1) (5.4)(compare to [5, p. 183]). A combination of (5.1), (5.4) and [1, p. 17 and Th. I.12.3] stipulates thatthe “dual” subcritical process, which is constructed starting from the supercritical Galton-Watsonprocess with probability-generating function (5.1) conditioned by ultimate extinction, is charac-terised by the branching mechanism described by the following particle-production generatingfunction: P ( α,λ ) dual ( u ) = 1 Q P W ( α ) λ ( Q u ) = 11 − λ / (1 − α ) (cid:18) − λ { − (1 − λ / (1 − α ) ) u } α (cid:19) . (5.5)It turns out to be possible to derive the closed-form expression for the probability-generatingfunction H α ( z ) of the law of total progeny for the “dual” subcritical Galton-Watson process.Specifically, one ascertains that H α ( z ) = 11 − λ / (1 − α ) (cid:26) − (cid:18) λzt s (cid:18) − z ( λz ) / (1 − α ) (cid:19)(cid:19) / (1 − α ) (cid:27) . (5.6)This follows from a combination of (5.4) and (5.5) with [6, p. 39], since it must satisfy the followingfunctional equation: QH α ( z ) ≡ z P ( α,λ ) dual ( QH α ( z )) , where z ∈ [0 , . (5.7)Let y := QH α ( z ). In view of (5.1), equation (5.7) is reduced to solving the following equation for y : y = z (1 − λ (1 − y ) α ) . (5.8)Next, by [11, Eq. (4.4)], [15, Eq. (11)], there exists a unique solution t s (= t s ( v )) > t α/ (1 − α ) s ( t s −
1) = v ( v > , (5.9)which is expressed in terms of the ‘reduced’ Wright function ϕ by t s ( v ) = (cid:26) v − log (cid:18)Z ∞ e − vy y (1 + y ) ϕ (cid:18) − α, − yy α (cid:19) dy (cid:19)(cid:27) − α , (5.10)where, for real x and arbitrary complex δ , ϕ ( ρ, δ ; x ) := ∞ X n =0 x n n !Γ( ρn + δ ) ( ρ ∈ ( − , ∪ (0 , ∞ )) . The consideration of (5.9) and (5.10) has its origin in [2, Prop. 1]. It is a routine exercise todemonstrate that (5.8) can be reduced to (5.9). A subsequent application of (5.10) implies thevalidity of (5.6).For α = , (5.6) simplifies to yield the probability-generating function H / ( z ) given by H / ( z ) = z − λ (cid:18) − λ z − λ p λ z − z + 4 (cid:19) (5.11)on { z ≤ z − } ∪ { z ≥ z + } , where z − := 2 / (1 + p − λ ) ∈ (1 , , z + := 2 / (1 − p − λ ) > . (5.12)We stress that only the lower value z − is meaningful from the probabilistic point of view. Thus,the exponential tilting transformation of this distribution corresponding to the value log z − resultsin the extreme conjugate distribution that belongs to the class (1.2) when c = 2. It is also relevantthat it was the existence of such extreme conjugate distributions and the necessity of finding itspossible extensions that motivated us to consider more general problems of analysis leading to theresults of Theorems 1 and 2. Moreover, the condition z ≤ z − is equivalent to the condition η ≥ √ x ,which emerges in (2.4) of Theorem 2 in the special case c = 2.Identification of the discrete probability distribution with probability-generating function givenby (5.11) is not a trivial exercise. We were able to solve this problem by combining severalprobability and analytical methods and results. Thus, it is of separate interest that the probabilitylaw { p ℓ , ℓ ∈ N } with the probability-generating function (5.11) is a shifted Poisson mixture withthe mixing measure given by the distribution of a busy period of the server in the M/M/ arrival rate /λ − > departure rate /λ > traffic intensity − λ . Evidently,in the case of α = this coincides with the probability of extinction Q given by (5.4).It then follows from [12, p. 530] that for ℓ ≥ p ℓ = 1 √ − λ ( ℓ − Z ∞ u ℓ − e − u/λ I (cid:18) uλ p − λ (cid:19) du, where I ( u ) denotes the modified Bessel function of the first kind. Then, upon use of [17, p. 385(2)],we find that for ℓ ≥ p ℓ = 2 − ℓ (1 − Q ) ℓ − F ( ℓ, ℓ + ; 2; Q ) . (5.13)It is straightforward to verify that a combination of (1.3), (5.11) and (5.13) with the second formulain (4.1) implies the following theorem: Theorem 3 . The probability-generating function of the distribution (5.13) is given by ∞ X ℓ =1 p ℓ z ℓ = z/ − λ z/ F (cid:18) ,
1; 2; 1 − λ (1 − λ z/ (cid:19) ≡ H / ( z ) . (5.14)It is relatively easy to see that the domain of this function, which is given above (5.12), isconsistent with that of the second special case in (4.1). In turn, a combination of (5.13) and (5.14)yields the following summation theorem, which is closely related to (1.1) and the representation(4.6) of the previous section. Corollary 1.
Given real
Q ∈ (0 , ∞ X ℓ =1 − ℓ (1 − Q ) ℓ − F ( ℓ, ℓ + ; 2; Q ) ≡ . (5.15) Remark 2.
The characteristic function of the distribution (5.13) equals H / ( e iu ), where thelatter function is defined by (5.11). It is clear that for u = A + iB , with A = (2 n + 1) π , where n sum of hypergeometric functions B ≥
0, one ascertains that e iu ∈ [ − , x to probability theory.To conclude, let us mention that in March 2018, we brought preliminary results of this work tothe attention of G´erard Letac. In turn, Letac advised us that in 1999 he had written a concludingchapter entitled “Les processus de population” in the monograph [13], although his name neverappeared in this book. In this chapter, he concentrated on finding the law of the total progeny fora critical or subcritical Galton-Watson process, which starts from one particle in the case wherethe branching mechanism is governed by a certain zero-modified geometric distribution. (We referto [4] for more detail on this class of distributions and its frequent use in the theory of branchingprocesses, which goes back to the Lotka model on the extinction probability for American malelines of descent based on the USA Census of 1920.) In [13, Th. 1.13], Letac identified the law of thetotal progeny in the case where the number of descendants of a particle is “backshifted” geometrici.e., its range is the set Z + of non-negative integers. Specifically, [13, Th. 1.13.3] demonstratesthat in this special case, the law of the total progeny belongs to the natural exponential family generated by the Sibuya distribution with index α = ; see Def. 1 and, in particular, (5.3) formore detail. In the case of the general zero-modified distribution with mean less than or equalto one, Letac contented himself with presenting just the probability-generating function of theresulting law along with some comments on the technical difficulties that emerge when attemptingto determine a closed-form expression for the probability function of the law of the total progeny(see the penultimate paragraph of Section 6 therein). Appendix: The large- k behaviour of the hypergeometric function in (1.3) From [10, (15.8.1)], the hypergeometric function appearing in S ( η, c ; x ) can be written as F ( k + , k + 1; c ; x ) = (1 − x ) − k/ − F (cid:18) k + 1 , c − − k ; c ; xx − (cid:19) . The asymptotic behaviour of the function on the right-hand side can be obtained from the resultstated in [10, (15.12.5)] in the form F ( α + λ, β − λ ; c ; − z ) ∼ ( α + β − / Γ( c ) √ π ( z + 1) ( c − α − β ) / − ( z − c/ − λ − c { z + p z − } λ +( α − β ) / as λ → + ∞ in | arg ( z − | < π . Identification of α = 1, β = c − and z = (1 + x ) / (1 − x ) thenleads, after some routine algebra, to the result F ( k + , k + 1; c ; x ) ∼ c − Γ( c ) √ π x c − k − c (1 − √ x ) − k + c − (A.1)as k → + ∞ when 0 < x < x <
0, we employ the integral representation [10, (15.6.2)] to obtain that F ( k + , k + 1; c ; − w ) = Γ( c )Γ( k + 2 − c )2 πi Γ( k + 1) Z (1+)0 h ( τ ) e kψ ( τ ) dτ ( w = | x | ) , (A.2) This book was written by a group of faculty members of the University of Paul Sabatier, Toulouse under thepseudonym P.S. Toulouse. h ( τ ) = ( τ − c − (1 + wτ ) / , ψ ( τ ) = log τ − log(( τ − wτ )) . The integration path is a closed loop that starts at τ = 0, encircles the point τ = 1 in the positivesense (excluding the singularity τ = − /w on the negative axis) and returns to τ = 0. Theexponential factor has saddle points at τ = ± i/ √ w , where ψ ′ ( τ ) = 0. Paths of steepest descentthrough these saddles emanate from the origin and pass to infinity in ℜ ( τ ) >
0. The integrationpath can therefore be expanded to infinity to pass over these saddles, which will both contributeequally to the integral.Consider the saddle τ s = − i/ √ w . Since h ( τ s ) = (1 − i √ w ) c − / ( i √ w ) c − , ψ ′′ ( τ s ) = 2 w / e πi (1 − i √ w ) , the direction of the steepest descent path through τ s is π − arg ψ ′′ ( τ s ) = − π − iφ, φ := arctan √ w. Application of the saddle-point method (see, for example, [10, p. 47]), where we approximate thegamma function factor multiplying the integral in (A.2) by Γ( c )( k ) − c , shows that the contribu-tion to the integral from the saddle τ s is (to leading order)Γ( c )( k ) − c πi ih ( τ s )(1 − i √ w ) k s π kψ ′′ ( τ s ) = Γ( c )2 c − i √ π w c − k − c (1 − i √ w ) − k + c − e − πi ( c − ) as k → + ∞ . The contribution from the saddle at τ = i/ √ w is given by the conjugate expressionto finally yield the result F ( k + , k + 1; c ; − w ) ∼ Γ( c )2 c − √ π w c − k − c (1 + w ) − k + c − sin (cid:20) ( k − c + ) φ − π ( c − ) (cid:21) (A.3)as k → + ∞ for w > Acknowledgement:
We are grateful to G´erard Letac for his timely feedback and for providingus with copies of his work. We also acknowledge very helpful discussions with Opher Baron. Oneof the authors (VVV) is grateful to Huaxiong Huang, Neil Madras and Tom Salisbury for theirwarm hospitality at the Fields Institute and York University.We also thank the anonymous referee for constructive suggestions, which led in particular tothe first unnumbered formula of Section 4 and an alternative derivation of (4.1), and for bringingthe article [14] to our attention.
References [1] K.B. Athreya and P.E. Ney,
Branching Processes , Springer, New York 1972.[2] J. Burridge, A. Kuznetsov, M. Kwasnicki and A.E. Kyprianou, New families of subordinators with explicittransition probability semigroup, Stoch. Proc. Appl. (2014) 3480–3495.[3] E.T. Copson,
An Introduction to the Theory of Functions of a Complex Variable , Oxford University Press,London 1972. sum of hypergeometric functions [4] K.J. Hochberg and V.V. Vinogradov, Structural continuity and asymptotic properties of a branching particlesystem, Lithuanian Math. J. (2009) 241–270.[5] T.E. Huillet, On Mittag-Leffler distributions and related stochastic processes, J. Comp. Appl. Math. (2016) 181–211.[6] P. Jagers, Branching Processes with Biological Applications , Wiley, London 1972.[7] T.J. Kozubowski and K. Podg´oski, A generalized Sibuya distribution, Ann. Inst. Stat. Math. (2018) 855–887.[8] G. Letac, Is the Sibuya distribution a progeny?, J. Appl. Probab. (2019) 52–56.[9] G. Letac, Progenies for multitype linear fractional branching processes, (2019) [submitted to Econometrics].[10] F.W.J. Olver, D.W. Lozier, R.F. Boisvert and C.W. Clark (eds.), NIST Handbook of Mathematical Functions ,Cambridge University Press, Cambridge 2010.[11] R.B. Paris and V.V. Vinogradov, Asymptotic and structural properties of special cases of the Wright functionarising in probability theory, Lith. Math. J. (2016) 377–409.[12] W.J. Stewart, Probability, Markov Chains, Queues and Simulation: the Mathematical Basis of PerformanceModeling , Princeton University Press, Princeton 2009.[13] P.S. Toulouse, Les processus de population, in
Th`emes de probabilit´es et statistique , Dunod, Paris 1999.[14] R. Vidunas, Dihedral Gauss hypergeometric functions, Kyushu J. Math. (2011) 141–167.[15] V.V. Vinogradov and R.B. Paris, On Poisson-Tweedie mixtures, J. Stat. Distrib. Appl. 4:14 (2017).[16] V.V. Vinogradov and R.B. Paris, A new class of discrete infinitely divisible distributions with power tails.(2020) [to appear in J. Stat. Distrib. Appl.].[17] G.N. Watson, A Treatise on the Theory of Bessel Functions , 2 ndnd