On Integral Representations and Asymptotics of Some

Home , Appell series

arXiv:1707.06275v2 [math-ph] 9 Nov 2017 keywords S 00nmes 36,3C0 22,33C20 82C23, 33C70, 33C65, numbers: 2010 MSC systems quantum many-body theorem; Tauberian functions; b a IS nentoa colfrAvne tde,vaBonomea via Studies, Advanced for School International - SISSA rued hsqeSaitqe ´preetd hsqed aMa la de Physique Département de Statistique, Physique de Groupe 2 1 nitga ersnain n smttc fsome of asymptotics and representations integral On [email protected] [email protected] ntttJa aor(NSUR79) nvri´ eLran Na Lorraine Université de 7198), UMR (CNRS Lamour Jean Institut hoe.Sm nerl fteHmetfntosaeas an also are functions Humbert t the is of asymptotics integrals the Some are and theorem. functions transformations these Laplace of inverse var representations as independent two integral the New of values large. absolute the when found, is h edn smttcbhvoro h ubr ucin Φ functions Humbert the of behaviour asymptotic leading The yegoercfntosi w aibe;Hmetfnto;as function; Humbert variables; two in functions hypergeometric : T ¨rc,SeaoFasiiPaz3 H-89 Zürich, Switze 8093 - CH 3, Zürich, Stefano-Franscini-Platz ETH c yegoercfntosi w variables two in functions hypergeometric d ehegs¨tt hskdrWrsoe ntttfür Baust Institut Werkstoffe, der Rechnergestützte Physik etod ´sc eoiaeCmuainl nvriaed Lisboa, de Universidade F´ısica Computacional, Teórica de e Centro 40 adur `sNnyCdx France Cedex, lès Nancy Vandœuvre 54506 – F acaWald Sascha –7906Lso,Portugal Lisboa, P–1749-016 a,b, 1 Abstract and at Henkel Malte ie.Teeaere-expressed are These given. e on rmaTauberian a from found hen alsbcm simultaneosly become iables alysed. a,c,d, 2 Φ , 6,I316Tise Italy Trieste, I–34136 265, 2 3 Ξ , ie ee e Matériaux, des tière et ff (IfB), offe 2 c,BP 70239, B.P. ncy, ftovariables two of mttc;special ymptotics; rland 1 Introduction

Generalised hypergeometric functions, usually denoted by pFq(z), and of which Gauss’ hypergeometric function 2F1(z) is the most important special case, have been studied very thorougly and have found numerous applications in almost all fields of science, see e.g. [3, 4, 6, 1, 9, 5, 8, 2, 7] and refs. therein. A little more than a century old, hypergeometric functions of two variables [13, 14, 15, 10, 11, 12] have also received a lot of scientific interest and recently, many new applications in many different fields of mathematics and physics are being discovered see for example [16, 18, 17]. It is often convenient to define these functions via double power series. Most of the mathematical studies of these functions are either focussed on the analysis of do- mains of convergence, or on relating special cases to other known functions or else to derive functional relationships between different hypergeometric functions of two variables, see e.g. [10, 4, 28, 29, 30, 2, 21, 22, 23, 24, 25, 26, 20, 19, 27]. Relatively little seems yet to be known on the asymptotic behaviour of such double series, in contrast to the classic study of Wright [31, 32] on the asymptotics of F (z) when z . Here, we shall present results on the p q | |→∞ leading asymptotics of some hypergeometric functions when the absolute values of both variables become large simultaneously. The main tool to derive these are Eulerian and (inverse) Laplacian integral representations, and a Tauberian theorem [34, 33]. The results are stated as theorems in section 3, see eqs. (3.1-3.4,3.6).

We shall consider the third Appell series F3

m n ∞ ∞ (α)m(β)m(α′)n(β′)n x y F (α,α′,β,β′; γ; x, y)= (1.1) 3 (γ) m! n! m=0 n=0 n+m X X where (α)m = Γ(α + m)/Γ(α) denotes the Pochhammer symbol for α N. We shall also study the conﬂuent forms (Humbert functions) − 6∈

m n ∞ ∞ (α)m(β)m(α′)n x y Ξ (α,α′, β; γ; x, y)= (1.2a) 1 (γ) m! n! m=0 n=0 m+n X X ∞ ∞ (α) (β) xm yn Ξ (α, β; γ; x, y)= m m (1.2b) 2 (γ) m! n! m=0 n=0 m+n X X m n ∞ ∞ (β)m(β′)n x y Φ (β, β′; γ; x, y)= (1.2c) 2 (γ) m! n! m=0 n=0 m+n X X ∞ ∞ (β) xm yn Φ (β; γ; x, y)= m (1.2d) 3 (γ) m! n! m=0 n=0 m+n X X Throughout, we shall implicitly assume that the parameters α,α′,β,β′, γ, . . . are such that any singularity in the coeﬃcients is avoided, without restating this explicitly. While the series F3 converges for max( x , y ) < 1, the series Ξ and Ξ converge for x < 1 and y < and | | | | 1 2 | | | | ∞ Φ2 and Φ3 converge for x < and y < [29]. For reduction formulæ to generalised hypergeometric functions| of| a single∞ variable,| | ∞ see [19, 27]. We shall also be interested in the

1 series

m n (i) ∞ ∞ (β)m(β′)n 1 x y Φ (β, β′; γ,λ; x, y) := (1.3a) 2 (γ) m + n + λ m! n! m=0 n=0 m+n X X ∞ ∞ (β) 1 xm yn Φ(i)(β; γ,λ; x, y) := m (1.3b) 3 (γ) m + n + λ m! n! m=0 n=0 m+n X X which for λ N converge for x < and y < . Clearly, for λ = γ, one has − 6∈ | | ∞ | | ∞

(i) 1 (i) 1 Φ (β, β′; γ,γ; x, y)= Φ (β, β′; γ + 1; x, y) , Φ (β; γ,γ; x, y)= Φ (β; γ + 1; x, y) (1.4) 2 γ 2 3 γ 3 and for λ = 1 these series may be rewritten as Kamp´ede F´eriet series [29]

1 (i) 1;1;0 (1); (β); Φ (β; γ, 1; x, y)= dw Φ3(β; γ; xw,yw)= F − x, y (1.5a) 3 2;0;0 (γ, 2); ; Z0 − −

1 (i) 1;1;1 (1); (β); (β′) Φ (β, β′; γ, 1; x, y)= dw Φ (β, β′; γ; xw,yw)= F x, y (1.5b) 2 2 2;0;0 (γ, 2); ; Z0 − −

(i) (i) such that Φ2 , Φ3 might be called ‘integrated Humbert functions’. We are interested in situations where both x and y become large. We shall therefore substitute x tx and y ty and study the| | limit |t | where x, y R will be kept fixed and non-zero.7→ − The minus7→ − signs in the substitution rule→ ∞ are motivated∈ from applications to quantum physics, see section 4. In section 2 the integral and inverse Laplace representations of the Humbert functions and integrated Humbert functions are derived, and the integral representations are used to define the required analytic continuations. Furthermore, the inverse Laplace representations will be used in section 3 to derive the asymptotic forms. Section 4 briefly outlines an application to many-body quantum dynamics.

2 Integral representations

The starting point for the derivation of the asymptotics of the series (1.1, 1.2) are the following Eulerian integral representations. Throughout, the parameters α, β, γ . . . of the functions, as well as x, y, are assumed constants and such that all series and integrals considered exist.

Lemma 1. Using the shorthand notations Φ = Φ (β; γ; tx, ty), Φ = Φ (β, β′; γ; tx, ty), 3 3 − − 2 2 − − Ξ2 =Ξ2(α, β; γ; tx, ty), Ξ1 =Ξ1(α,β,β′; γ; tx, ty) and F3 = F3(α,α′,β,β′; γ; tx, ty), the functions deﬁned− − in (1.1, 1.2) have the integral− − representations − −

1 γ t Γ(γ) t − γ ε 1 ε 1 Φ = dv v − − F (β; γ ε; xv) F (ε; y(t v))(t v) − (2.1a) 3 Γ(γ ε)Γ(ε) 1 1 − − 0 1 − − − − Z0 1 γ t Γ(γ) t − γ ε 1 ε 1 Φ = dv v − − F (β; γ ε; xv) F (β′; ε; y(v t))(t v) − (2.1b) 2 Γ(γ ε)Γ(ε) 1 1 − − 1 1 − − − Z0

2 1 γ t Γ(γ) t − γ ε 1 ε 1 Ξ = dv v − − F (α, β; γ ε; xv) F (ε; y(t v))(t v) − (2.1c) 2 Γ(γ ε)Γ(ε) 2 1 − − 0 1 − − − − Z0 1 γ t Γ(γ) t − γ ε 1 ε 1 Ξ = dv v − − F (α, β; γ ε; xv) F (β′; ε; y(t v))(t v) − (2.1d) 1 Γ(γ ε)Γ(ε) 2 1 − − 1 1 − − − − Z0 1 γ t Γ(γ) t − γ ε 1 F3 = dv v − − 2F1 (α, β; γ ε; xv) Γ(γ ε)Γ(ε) 0 − − × − Z ε 1 F (α′, β′; ε; y(t v))(t v) − (2.1e) × 2 1 − − − and where ε is a ﬁxed constant, which satisﬁes 0 <ε<γ. The integral representations (2.1) were already given in [25] and (2.1e) in [29, (9.4.16)]. We begin by repeating them since they are the crucial starting point of our analysis.

Proof: We illustrate the technique for the example Φ2. The double series (1.2c) is decoupled by using the decomposition m + n + γ = (m + γ ε)+(n + ε) and the identity [1, (6.2.1)] involving the Euler Beta function −

1 n+ε 1 m+γ ε 1 1 B(n + ε, m + γ ε) du (1 u) − u − − = − = 0 − (2.2) Γ(m + n + γ) Γ(n + ε)Γ(m + γ ε) Γ(n + ε)Γ(m + γ ε) − R − Inserting this into the deﬁnition (1.2c) gives, because the series are absolutely convergent

1 m n γ ε 1 Γ(γ) ∞ (β) ( txu) ∞ (β′) ( ty(1 u)) u − − Φ = du m − n − − 2 Γ(γ ε)Γ(ε) (γ ε) m! (ε) n! (1 u)1 ε 0 m=0 m n=0 n − − Z X − X − 1 Γ(γ) ε 1 γ ε 1 = du (1 u) − u − − F (β; γ ε; txu) F (β′; ε; ty(1 u)) Γ(γ ε)Γ(ε) − 1 1 − − 1 1 − − − Z0 The assertion (2.1b) follows by rescaling v = tu. The other identities eqs. (2.1) are derived similarly. q.e.d. Comment. Recall the following deﬁnitions of some further double hypergeometric series [29]

m n ∞ ∞ (a)m+n(b)m(b′)n x y F (a, b, b′; c,c′; x, y)= (2.3a) 2 (c) (c ) m! n! m=0 n=0 m ′ n X X m n ∞ ∞ (a)m+n(b)m x y Ψ (a, b; c,c′; x, y)= (2.3b) 1 (c) (c ) m! n! m=0 n=0 m ′ n X X m n ∞ ∞ (a)m+n x y Ψ (a; c,c′; x, y)= (2.3c) 2 (c) (c ) m! n! m=0 n=0 m ′ n X X

3 ∞ z 1 u By using Γ(z)= 0 duu − e− , one may derive in a way similar to Lemma 1 the identities R 1 ∞ u a 1 F (a, b, b′; c,c′; x, y)= du e− u − F (b; c; xu) F (b′; c′; yu) (2.4a) 2 Γ(a) 1 1 1 1 Z0

1 ∞ u a 1 Ψ (a, b; c,c′; x, y)= du e− u − F (b; c; xu) F (c′; yu) (2.4b) 1 Γ(a) 1 1 0 1 Z0

1 ∞ u a 1 Ψ (a; c,c′; x, y)= du e− u − F (c; xu) F (c′; yu) (2.4c) 2 Γ(a) 0 1 0 1 Z0 see also [29, (9.4.29)], [24, (27)]. However, there is no known direct way to render these as con- volutions, which will become our main tool to analyse the t asymptotics of the functions in Lemma 1. A diﬀerent route is suggested by [24, eq. (31)].→∞

The integral representations (2.1c,2.1d,2.1e) can be used to extend the deﬁnition of the functions Ξ2, Ξ1, F3 (unecessary for Φ2, Φ3). This is based on the Eulerian integral representation, for γ>β> 0 [1, eq. (15.3.1)]

1 Γ(γ) β 1 γ β 1 α F (α, β; γ; x)= duu − (1 u) − − (1 + ux)− (2.5) 2 1 − Γ(β)Γ(γ β) − − Z0 which deﬁnes the analytic continuation of the function 2F1 (α, β; γ; x) in the domain arg x < π, which has a cut for 1 >x> [4]. − | | − −∞

Definition: The integral representations (2.1) define the (principal branch of) the functions Φ = Φ (β; γ; tx, ty), Φ = Φ (β, β′; γ; tx, ty), Ξ =Ξ (α, β; γ; tx, ty), Ξ =Ξ (α, β, 3 3 − − 2 2 − − 2 2 − − 1 1 β′; γ; tx, ty) and F = F (α,α′,β,β′; γ; tx, ty) in the regions arg xt < π and arg yt < − − 3 3 − − | | | | π. They are the analytical continuations of the series (1.1,1.2), beyond their respective domain of convergence. Indeed, from (2.1), the functions Φ , Φ are defined for all x < , y < , while Ξ , Ξ 2 3 | | ∞ | | ∞ 1 2 have a cut for 1 > xt > and F3 has cuts for 1 > xt > and 1 > yt> . All results which follow− will implicitly−∞ use these analytic− continuations.−∞ − −∞

In consequence, if we use F as a generic symbol for any of the functions in (2.1), one can recast (2.1) as follows (in the sense of the analytic continuation)

t Γ(γ) 1 γ F (t)= t − dv F (v)F (t v) Γ(γ ε)Γ(ε) 1 2 − − Z0 Γ(γ) 1 γ 1 = t − L − F (p) F (p) (t) (2.6) Γ(γ ε)Γ(ε) 1 2 − L ∞ pv where f(p) = (f(v))(p) = 0 dv e− f(v) denotes the Laplace transform. The functions F (v) are readily read oﬀ from eqs. (2.1) and are listed in the following table. 1,2 R

4 F (t) F1(v) F2(v) γ ε 1 ε 1 Φ v − − F (β; γ ε; xv) v − F (ε; yv) 3 1 1 − − 0 1 − γ ε 1 ε 1 Φ v − − F (β; γ ε; xv) v − F (β′; ε; yv) 2 1 1 − − 1 1 − γ ε 1 ε 1 Ξ v − − F (α, β; γ ε; xv) v − F (ε; yv) 2 2 1 − − 0 1 − γ ε 1 ε 1 Ξ v − − F (α, β; γ ε; xv) v − F (β′; ε; yv) 1 2 1 − − 1 1 − γ ε 1 ε 1 F v − − F (α, β; γ ε; xv) v − F (α′, β′; ε; yv) 3 2 1 − − 2 1 −

Next, we require the following list of Laplace transforms, taken from [35, (3.35.1.3,3.37.1.2,3.38.1.1)], combined with [1, (13.1.10,13.1.33)]

a 1 a y/p L v − F (a; yv) (p)=Γ(a)p− e− (2.7a) 0 1 − b 1 a b a L v− F (a; b; yv) (p)=Γ(b)p − (p + y)− (2.7b) 1 1 − c 1 a c a p L v − F (a, b; c; yv) (p)=Γ(c)p − y− U a;1+ a b; (2.7c) 2 1 − − y where U denotes the Tricomi function [1]. Combining these with the integral forms (2.6) gives Lemma 2. The Laplace transforms of the analytically continued functions in Lemma 1 are given by the following table, where U denotes the Tricomi function.

1 γ function F (t)/ (Γ(γ)t − ) Laplace transform F (p) β γ β y/p Φ (β; γ; xt, yt) p − (p + x)− e− 3 − − β+β′ γ β β′ Φ (β, β′; γ; xt, yt) p − (p + x)− (p + y)− 2 − − α α γ y/p Ξ (α, β; γ; xt, yt) x− p − U(α;1+ α β; p/x)e− 2 − − − α α+β′ γ β′ Ξ (α,β,β′; γ; xt, yt) x− p − (p + y)− U(α;1+ α β; p/x) 1 − − − α α′ α+α′ γ p p F (α,α′,β,β′; γ; xt, yt) x− y− p − U(α;1+ α β; )U(α′;1+ α′ β′; ) 3 − − − x − y

The entries for Φ2 and Φ3 are contained in [28].

Corollary 1. Applying again eq. (2.2), the formal Kamp´ede F´eriet series

′ ; (αp); (α′ ′ ) ′ m+n m n 0;p;p − p ∞ (α1)m (αp)m (α1′ )n (αp )n ( 1) (tx) (ty) F1;q;q′ tx, ty = · · · · · · − γ; (βq); (β′ ′ ) − − ! (β1)m (βq)m (β1′ )n (βq′ ′ )n (γ)m+n m! n! q m,n=0 · · · · · · X

1 γ ε 1 Γ(γ) du u − − α1,...,αp = 1 ε pFq+1 ; txu Γ(γ ε)Γ(ε) (1 u) β1,...,βq, γ ε − × − Z0 − − − α1′ ,...,αp′ ′ p′ Fq′+1 ; ty(1 u) (2.8) × β′ ,...,β′ ′ , ε − − 1 q

1 γ 1 γ x y = Γ(γ)t − L − s− F (α ); (β ); ′ F ′ (α′ ′ ); (β′ ′ ); (t) (2.9) p q p q − s p q p q − s 5 contains all functions treated here explicitly as special cases, if (2.8) can be used as above to deﬁne an analytic continuation, in the domain arg xt < π and arg yt < π. For the derivation of (2.9), we used the identity [35, (3.38.1.1)] | | | |

µ 1 µ ω L v − F ((a );(b ),µ; ωv) (s)=Γ(µ)s− F (a );(b ); (2.10) p q+1 p q − p q p q − s (2) For q = q′ = 0 and p = p′ = 2, eq. (2.8) reduces to F3, or the Lauricella function FB in two variables. Furthermore, for p = p′ = q = q′ = 0, one has an addition theorem

∞ 1 xm yn F (γ; x + y)= 0 1 (γ) m! n! m,n=0 m+n X Γ(γ) 1 uγ ε 1 = du − − F (γ ε; xu) F (ε; y(1 u)) (2.11) Γ(γ ε)Γ(ε) (1 u)1 ε 0 1 − 0 1 − − Z0 − −

(i) (i) (i) We now turn to the variants Φ3 and Φ2 deﬁned in eq. (1.3). Since Φ2 is symmetric under the permutation (x, β) (y, β′), we can set x y without restriction of the generality. ↔ ≥ (i) (i) (i) (i) Lemma 3. With the shorthands Φ3 = Φ3 (β; γ, 1; tx, ty), Φ2 = Φ2 (β, β′; γ, 1; tx, ty) (i,s) (i) − − − − with x>y and Φ2 := Φ2 (β, β′; γ, 1; tx, tx), the following integral representations of the integrated Humbert functions (1.3) hold− true−

y/x β 1 (i) 1 γ e y − 1 1 γ y y y Φ = Γ(γ)t − L − p − Γ 1 β, Γ 1 β, + (t) (2.12a) 3 x x − x − − x p 1 γ β′ 1 (i) Γ(γ)t − x − 1 1 γ y Φ = ′ L − p − F 1 β, β′;2 β; 2 (1 β)(x y) β − 2 1 − − −(x y) − − − − β+γ 1 β (p + x)y +p (p + x) − F 1 β, β′;2 β; (t) (2.12b) 2 1 − − −p(x y) − 1 γ (i,s) Γ(γ)t − L 1 β+β′ γ 1 β β′ 1 γ Φ2 = − p − (p + x) − − p − (t) (2.12c) (1 β β′)x − − − where Γ(a, x) is the incomplete Gamma-function. Proof. Starting from (1.3b), the extra denominator is turned into an auxiliary integral

∞ m n (i) ∞ v(m+n+1) (β)m ( tx) ( ty) Φ = dv e− − − 3 (γ) m! n! 0 m,n=0 m+n Z X and the decoupling of the two series proceeds via (2.2), as in the proof of Lemma 1. A last v change of variables w = e− and using also (2.7) leads to

1 (i) 1 γ L 1 β γ β yw/p Φ3 = Γ(γ)t − − p − dw (p + xw)− e− (t) Z0 =: M | {z }

6 The integral is found as follows, reducing it to incomplete Gamma functions [1] M x+p y/x 1 β y/x+y/p 1 β y a p e px − β b = da a− exp − = db b− e− M x −x p x y Zp Zy/x 1 β ey/x px − y y y = Γ 1 β, Γ 1 β, + x y − x − − x p (i) and inserting into Φ3 gives the assertion (2.12a). (i) Turning to Φ2 , the procedure to go from (1.3a) to an integral representation follows the same lines as before. Changing variables as before and re-using (2.7), we ﬁnd

1 (i) 1 γL 1 β+β′ γ β β′ Φ2 = Γ(γ)t − − p − dw (p + xw)− (p + yw)− (t) Z0 =: N which is still symmetric under the simultaneous| exchanges{z (x, β) (y, β′}), as it should be. To evaluate this, recall the following identity on the incomplete Beta↔ function [1, (6.6.8,15.3.4)] ξ ua 1 ξa I(a, b, ξ) := du − = F (a, b;1+ a; ξ) (1 + u)b a 2 1 − Z0 Then we can evaluate the integral , now using x>y N ′ 1 β β β β′ p − p − = x− y− dw w + w + N 0 x y Z ′ 1 β β (p+x)y/(p(x y)) β β′ p(x y) − − − β β′ = x− y− − db b− (1 + b)− xy y/(x y) Z − 1 β β′ β β′ p(x y) − − (p + x)y y = x− y− − I 1 β, β′, I 1 β, β′, xy − p(x y) − − −(x y) − − ′ 1 β β β β′ 1 β p(x y) − − x− y− (p + x)y − (p + x)y = − F 1 β, β′;2 β; xy 1 β p(x y) 2 1 − − −p(x y) − " − − 1 β y − y F 1 β, β′;2 β; − x y 2 1 − − −x y − − # (i) and inserting this into the above expression for Φ2 gives the assertion (2.12b). Finally, in the symmetric case x = y we have

1 (i,s) 1 γ L 1 β+β′ γ β β′ Φ2 = Γ(γ)t − − p − dw (p + xw)− − (t) Z0 and straightforward integration gives the assertion (2.12c). q.e.d. Corollary 2. For x> 0, and λ> 0, µ> 0, one has the identity

1 λ 1 dww − Γ(λ)Γ(µ) Φ (β, β′; γ; xw, xw)= F (β + β′,λ; µ + λ,γ; x) (2.13) (1 w)1 µ 2 − − Γ(λ + µ) 2 2 − Z0 − − 7 (i,s) Proof. The lines of calculation follow the proof of Φ2 in Lemma 3. Consider

1 λ 1 µ 1 dw w − (1 w) − Φ (β, β′; γ; xtw, xtw) − 2 − − Z0 1 1 γ 1 β+β′ γ λ 1 µ 1 β β′ = Γ(γ)t − L − p − dww − (1 w) − (p + xw)− − (t) − Z0

Γ(γ)Γ(λ)Γ(µ) 1 γ 1 γ x = t − L − p− F β + β′,λ; µ + λ; (t) Γ(λ + µ) 2 1 − p Γ(λ)Γ(µ) = F (β + β′,λ; µ + λ,γ; xt) Γ(λ + µ) 2 2 − where in the third line, the integral representation [1, (15.3.1)] of 2F1 was used and in the forth line, [36, (3.35.1.10)], or else (2.10), was applied. Set t = 1. q.e.d.

Similar, but inequivalent, integral formulae involving Φ2 are stated in [23, eqs. (3.18,3.19)].

3 Asymptotic expansions

We shall use a Tauberian theorem for the asymptotic evaluations: the behaviour of a function f(t) for t is related to the one of its Laplace transform f(p) for p 0 [34], [33, ch. XIII]. Therefore,→∞ it is sufficient to analyse the behaviour of the representations→ as inverse Laplace transformations from Lemmas 2 and 3 in section 2 for p 0, before inverting. → Theorem 1. The Humbert function Φ2 = Φ2(β, β′; γ; tx, ty) has the following leading asymptotic behaviour for t , with x, y =0 being kept− fixed − →∞ 6 Γ(γ) β β′ − − Γ(γ β β′) (tx) (ty) ; for x> 0, y> 0 − −  ′ Γ(γ) yt β β+β γ  ′ e (t( y + x))− (t y ) − ; for x> 0, y< 0  Γ(β ) −  | | | |  ′ ′  Γ(γ) xt β β+β γ  e− (t(y + x ))− (t x ) − ; for x< 0, y> 0  Γ(β) | | | | Φ2  (3.1)  ′ ′ ≃  Γ(γ) x t β+β γ β  e−| | (t x ) − (t x y )− ; for x 0 and y > 0, the leading term for p 0 is found to be Φ2 1 γ 1 β+β′ γ β β′ → ≃ Γ(γ)t − L − p − x− y− (t) and direct evaluation [36, (2.1.1.1)] gives the assertion. Next, 8 for x > 0 and y < 0, one first makes the shift p = q y which produces an exponential − contribution, according to the shift theorem

1 ωt L − f¯(p ω) (t) = e f(t) − Second, one takes the leading term for q 0. Then → 1 γ y t 1 β+β′ γ β β′ Φ = Γ(γ)t − e−| | L − (q + y ) − (q + y + x)− q− (t) 2 | | | | q 0 1 γ y t 1 β+β′ γ β β′ → Γ(γ)t − e−| | L − y − ( y + x)− q− (t) ≃ | | | | and direct evaluation gives the assertion. For x < 0 and y > 0 one merely has to permute (x, β) (y, β′). Finally, for x< 0 and y < 0 ↔ 1 γ 1 β+β′ γ β β′ Φ = Γ(γ)t − L − p − (p x )− (p y )− (t) 2 −| | −| | If x y , one makes the shift q = p x and the stated result follows as before. | | | | −| | If y

Theorem 2. The Humbert function Φ3 = Φ3(β; γ; tx, ty) has the following asymptotic behaviour for t , with x, y =0 being kept ﬁxed − − →∞ 6 β (1+β γ)/2 Γ(γ)(tx)− (ty) − Jγ β 1(2√yt ) ;for x> 0, y > 0 − −

 β (1+β γ)/2 Φ3  Γ(γ)(tx)− (t y ) − Iγ β 1(2 y t ) ; for x> 0, y < 0 (3.2) ≃  | | − − | |  Γ(γ) β γ y/ x x t p (t x ) − e− | |−| | ; for x< 0 Γ(β) | |   where Jν is a Bessel function and Iν the corresponding modified Bessel function and neither γ nor β are non-positive integers. The qualitative behaviour of the leading asymptotic term is only influenced by the signs of β and β γ. − Proof. Use the representation of Φ3 as an inverse Laplace transformation in Lemma 2. For x> 0, simply retain the lowest order in p 0 and use (2.7a). Expressing the hypergeometric → function 0F1 as a Bessel or a modified Bessel function, respectively, gives the assertion for y > 0 and y < 0. For x< 0, make the shift q = p x such that −| | 1 γ x t 1 β γ β y/(q+ x ) Φ = Γ(γ)t − e−| | L − (q + x ) − q− e− | | (t) 3 | | q 0 1 γ x t 1 β γ β y/ x → Γ(γ)t − e−| | L − x − q− e− | | (t) ≃ | | 1 β β 1 and re-use Γ(β)L − q− (t)= t − [36, (2.1.1.1)]. q.e.d. Theorem 3. The Humbert function Ξ = Ξ (α, β; γ; tx, ty) has the following asymptotic 2 2 − −

9 behaviour for t , with x, y =0 and x> 0 being kept ﬁxed →∞ 6 Γ(α)Γ(α β) β (γ β 1)/2 − (tx)− (ty)− − − Jγ β 1(2√yt ); y > 0, α > β Γ(α) − − ∀  Γ(α)Γ(α β) β (γ β 1)/2 − − − − −  Γ(α) (tx) (t y ) Iγ β 1(2 y t ); y < 0, α > β  | | − − | | ∀   Γ(γ) α (γ α 1)/2 π p  (tx)− (ty)− − − Yγ α 1(2√yt )  Γ(α) 2 − −  1  +Jγ α 1(2 y t ) 2 ln(tx) + ln(x/y) ψ(α) 2CE ; y > 0, α = β  − − | | − − ∀ Ξ2  i (3.3) ≃  Γ(γ) α p (γ α 1)/2  (tx)− (t y )− − − Iγ α 1(2 y t ) Γ(α) − − 1 | | | | ln(tx) + ln(x/ y ) ψ(α) 2CE ; y < 0, α = β  × 2 | | − − p ∀   Γ(α)Γ(β α) α (γ α 1)/2  − (tx)− (ty)− − − J (2√yt ); y > 0, α < β  Γ(β) γ α 1  − − ∀   Γ(α)Γ(β α) α (γ α 1)/2  − (tx)− (t y )− − − Iγ α 1(2 y t ); y < 0, α < β  Γ(β) − −  | | | | ∀  p where Jνand Yν are the Bessel and Neuman functions, respectively, Iν is a modiﬁed Bessel function, ψ(x) is the digamma function and C 0.5772 ... is Euler’s constant [1]. For x< 0, E ≃ the function Ξ2 has a cut. Proof. In order to apply the inverse Laplace representation of Lemma 2, the small-p expansion

Γ(α β) p β α − − Γ(α) x ; for α > β p  1 p U α;1+ α β; Γ(α) lnx + ψ(α)+2CE ; for α = β − x ≃  −  Γ(β α)  − Γ(β) ; for α < β  according to [1, (13.5.6-13.5.12)] is required, for α β N. For x> 0 and α > β, to lowest Γ(γ)Γ(α β) 1 γ β − 1 6∈ −(γ β) y/p order in p 0, this gives Ξ − t − x− L − p− − e (t) and using (2.7a) gives → 2 ≃ Γ(α) the assertion. For x > 0 and α < β the result follows from the symmetry in α and β. For α = β and y > 0, expansion to lowest order in p 0 gives → 1 γ Γ(γ) t − 1 (γ α) y/p Ξ L − (ψ(α)+2C ln x) p− − e− (t) 2 ≃ −Γ(α) xα E − 1 (γ α) y/p + L − p− − ln p e− (t) 1 1 γ 2 (γ α 1) Γ(γ) t − 1 t t − − = ψ(α)+2CE ln x ln Jγ α 1(2√yt ) −Γ(α) xα − − 2 y y − − " 1 2 (γ α 1) t − − ∂Jν 1(2√yt ) − − y ∂ν ν=γ α# − re-using (2.7a) and [36, (2.5.7.3)]. For z , one has asymptotically ∂Jν (z ) π Y (z) [1, → ∞ ∂ν ≃ 2 ν (9.25,9.26)]. Collecting terms leads to the stated result. For y < 0 and α = β one has

10 analogously

1 γ Γ(γ) t − 1 (γ α) y /p Ξ L − (ψ(α)+2C ln x) p− − e| | (t) 2 ≃ −Γ(α) xα E − 1 (γ α) y /p + L − p− − ln p e| | (t) 1 1 γ 2 (γ α 1) Γ(γ) t − 1 t t − − = ψ(α)+2CE ln x ln Iγ α 1(2√yt ) −Γ(α) xα − − 2 y y − − " 1 2 (γ α 1) t − − ∂Iν 1(2√yt ) − − y ∂ν ν=γ α# − ∂I ν (z) ν and from the asymptotic form [1, (9.7.1)] for Iν(z) for z , we see that ∂ν z Iν(z) merely gives a sub-leading correction. Collecting terms we→ ∞complete the list of assertions≃ − if x> 0. For x< 0, the inverse Laplace representation in Lemma 2 has a cut. q.e.d. (i) (i) Theorem 4. The integrated Humbert function Φ3 = Φ3 (β; γ, 1; tx, ty) has the following leading asymptotic behaviour for t , with x, y =0 and x> 0 being− kept− ﬁxed →∞ 6 γ 1 β 1 1 3 − xt− Γ(1 β)(y/x)) 1 β 1F1(1;2 β; y/x) ; y > 0, β + γ > 2 − − − − ∀  h 1 1 i (i) Γ(γ) (β+γ+ ) y β π 1 3 Φ  (yt)− 2 2 cos 2√yt + β γ ; y > 0, β + γ < (3.4) 3 ≃  √π x 2 − − 2 ∀ 2   1 1 Γ(γ) (β γ ) β − 2 − − 2 − 2√π ( y t) (xt) exp 2 y t ; y < 0  | | | | ∀  where neither γ nor 1 β are non-positive integers.p  − Proof. Begin with the integral representation (2.12a) of Lemma 3. The leading term for p 0 → is found from the asymptotic identity [1, (6.5.30)] for x →∞ x a 1 y Γ(a, x + y) Γ(a, x) e− x − 1 e− − ≃ − − In order to invert L , we also need the identities (2.7a) and [36, (3.10.2.2)]

ν µ+ν 1 1 µ a Γ(ν) µ 1 a t − L − p− Γ ν, (t)= t − F (ν; ν +1,µ + ν; at) p Γ(µ) − νΓ(µ + ν) 1 2 − Then, for p 0 (here, x> 0 is assumed) → y/x β 1 Γ(1 β, y ) (i) 1 γ e y − L 1 x Φ3 Γ(γ)t − − −γ 1 ≃ x x p − y Γ(1 β, ) y/p β p e− p y/x − + 1 e− (t) − pγ 1 pγ 1 y − − − y/x β 1 1 γ e y − Γ(1 β,y/x) Γ(1 β) = Γ(γ)t − − − − x x Γ(γ 1) β 1 − y/x (yt) − F (1 β;2 β,γ β; yt) (1 e− ) F (γ β 2; yt) + 1 2 − − − − + − 0 1 − − − (1 β)Γ(γ β) Γ(γ β 1)(yt)β − − − −

11 Further evaluation is simpliﬁed by the identity, taken from [1, (6.5.3,6.5.12)]

y (1 β) y 1 y − − Γ(1 β) Γ 1 β, = F 1 β;2 β; x − − − x 1 β 1 1 − − −x h i − and this gives

ey/x F (1 β;2 β; y/x) Φ(i) Γ(γ) 1 1 − − − 3 ≃ xt (β 1)Γ(γ 1) − − y/x F (1 β;2 β,γ β; yt) 1 x 1 e− + 1 2 − − − − + − F (γ β 2; yt) (3.5) (1 β)Γ(γ β)(xt)β 1 (xt)β y Γ(γ β 1) 0 1 − − − − − − − − We can now distinguish the two cases y > 0 and y < 0. For y > 0, recall the asymptotic identity [37, (07.22.06.0011.01)]

y Γ(b )Γ(b ) π 1 →∞ 1 2 η/2 2 1F2(a1; b1, b2; y) y cos η +2√y 1+O(y− ) − ≃ √π Γ(a1) 2 Γ(b1)Γ(b2) a1 1 + y− 1+O(y− ) Γ(b a )Γ(b a ) 1 − 1 2 − 1 with η =(a b b + 1 ). Also, the function F can be expressed in terms of Bessel functions 1 − 1 − 2 2 0 1 Jν [1]. Inserting into (3.5) the above expansion and using the asymptotics of Jν [1] leads to

γ 1 y β 1 1 y Φ(i) − Γ(1 β) − F 1;2 β; 3 ≃ xt − x − 1 β 1 1 − x − β 1 1 Γ(γ) y (β+γ+ ) π 1 + (yt)− 2 2 cos 2√yt + β γ √π x 2 − − 2 3 3 Herein, the first line dominates for β + γ > 2 and the second line for β + γ < 2 . This is the first part of the assertion. For y = y < 0, recall the asymptotic form [37, (07.22.06.0005.01)] −| | y Γ(b )Γ(b ) 1 1 →∞ 1 2 2 (a1 b1 b2+ 2 ) 2√y 1F2(a1; b1, b2; y) y − − e ≃ 2√π Γ(a1) and now express 0F1 in terms of a modified Bessel function Iν [1]. Insertion into (3.5) and using the known asymptotic behaviour leads to

1 1 2 (β γ 2 ) (i) Γ(γ) ( y t)− − − 2√ y t γ 1 1 y Φ | | e | | + − F 1;2 β; | | 3 ≃ 2π1/2 (xt)β β 1 xt 1 1 − − x − Clearly, the second term is always sub-dominant. This completes the proof. q.e.d. (i) (i) Theorem 5. The integrated Humbert function Φ2 = Φ2 (β, β′; γ, 1; tx, ty) has the following leading asymptotic behaviour for t , with x>y> 0 being kept− ﬁxed − →∞ β′ β (i) Γ(γ) (xt) − xy Φ2 2F2 1 β, β′;2 β,γ β; t ≃ (1 β)Γ(γ β) ((x y)t)β − − − −x y − − − ′ − γ 1 1 x β y + − F 1 β, β′;2 β; (3.6) β 1 xt x y 2 1 − − −x y − − −

12 Herein, none of γ, γ β or 1 β is a non-positive integer. For y>x> 0, one permutes (x, β) − − and (y, β′). Proof. Begin with the integral representation (2.12b) of Lemma 3. For p 0, this simpliﬁes → into 1 γ 1 β p 0 Γ(γ) t − x − xy 1 (i) → 1 Φ ′ ′ L − F 1 β, β′;2 β; 2 ≃ 1 β (x y)β x1 β pγ β 2 1 − − −x y p − − − − − γ 1 y p − F 1 β, β′;2 β; (t) − 2 1 − − −x y − We need the identity [36, (3.35.1.10)]

ν 1 1 ν ω t − L − p F a, b; c; (t)= F (a, b; c, ν; ωt) 2 1 − p Γ(ν) 2 2 − Then straightforward algebra leads to the assertion. For y>x> 0, it is enough to exchange β β′ and x y. q.e.d. ↔ ↔ (i,s) The symmetric case x = y > 0, hence Φ2 , is a special case of the corrollary 2, eq. (2.13). More explicit asymptotics of 2F2 can be found in [32, 37, 38]. These expressions derived in this section can be checked numerically. However, the convergence towards the given asymptotics is in general quite slow. Finally, it is now straightforward to obtain the asymptotics of the special Kamp´ede F´eriet series (2.9), by using the known asymptotics of the generalised hypergeometric functions pFq(z) [32].

4 An example from physics

The quantum spherical model [41, 40, 39] is a simple exactly solvable model of quantum phase transitions, in d spatial dimensions, with a non-trivial quantum critical behaviour at zero temperature (that is, the model cannot be described by a simple mean-ﬁeld approximation, at least for 1

1 d 3 g 1 d 3 g eγZ (4πγt)d/2 = Φ ; ; gZt, t + Cg2t2 dw Φ ; ; tw, gZtw (4.3) 2 3 2 2 − −γ 3 2 2 −γ − Z0 For the physically interesting long-time behaviour of Z = Z(t) for t , the asymptotics →(i) ∞ of the Humbert function Φ3 and of the integrated Humbert function Φ3 , as studied in this work, are required. In contrast to the original formulation in eqs. (4.1,4.2), the reformulation in eq. (4.3) contains the spatial dimension d merely as a parameter. This allows to discuss also the model’s behaviour at non-integer dimensions d R, which often provides useful physical insight. ∈ The ﬁnal long-time behaviour obtained from (4.3) turns out to depend subtly on the dimension d. For d 2, there is a single solution with Z(t) = Z(t) < 0. Then, for t ≥ 1 2 1 −| | → ∞ it follows that Z(t) t− ln t for d > 2 and Z(t) t− for d = 2. In both cases, this is quite diﬀerent from| the| ∼ form Z(t) ln t obtained| for| a ∼ classical, non-coherent dynamics (limit g 0) [45]. | | ∼ →

Acknowledgements

We are grateful to the Group ‘Rechnergestützte Physik der Werkstoffe’ at ETH Zürich, Switzer- land, where this work was done, for their warm hospitality. SW thanks UFA-DFH for financial support through grant CT-42-14-II.

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