Heat kernel for higher-order differential operators and generalized exponential functions
HHeat kernel for higher-order differential operators and generalized exponentialfunctions
A. O. Barvinsky, ∗ P. I. Pronin, † and W. Wachowski ‡ Theory Department, Lebedev Physics Institute, Leninsky Prospect 53, Moscow 119991, Russia Department of Theoretical Physics, Faculty of Physics,M. V. Lomonosov Moscow State University, 119991 Moscow, Russia.
We consider the heat kernel for higher-derivative and nonlocal operators in d -dimensional Eu-clidean space-time and its asymptotic behavior. As a building block for operators of such type, weconsider the heat kernel of the minimal operator—generic power of the Laplacian—and show thatit is given by the expression essentially different from the conventional exponential WKB ansatz.Rather it is represented by the generalized exponential function (GEF) directly related to what isknown in mathematics as the Fox–Wright Ψ -functions and Fox H -functions. The structure of itsessential singularity in the proper time parameter is different from that of the usual exponentialansatz, which invalidated previous attempts to directly generalize the Schwinger–DeWitt heat ker-nel technique to higher-derivative operators. In particular, contrary to the conventional exponentialdecay of the heat kernel in space, we show the oscillatory behavior of GEF for higher-derivativeoperators. We give several integral representations for the generalized exponential function, findits asymptotics and semiclassical expansion, which turns out to be essentially different for localoperators and nonlocal operators of noninteger order. Finally, we briefly discuss further applica-tions of the GEF technique to generic higher-derivative and pseudodifferential operators in curvedspace-time, which might be critically important for applications of Hoˇrava–Lifshitz and other UVrenormalizable quantum gravity models.
1. INTRODUCTION
Physical phenomena in higher derivative and nonlocalfield theories are essentially different from conventionallocal quantum field theory (QFT) with the wave oper-ators of second order in space-time derivatives. Thereare numerous manifestations of this difference includingthe problem with unitarity which arises due to higher-derivative (Ostrogradsky) ghosts [1], violation of Lorentzinvariance and peculiar causality properties of Hoˇrava–Lifshitz gravity models (which are motivated by the at-tempts to solve this problem in renormalizable quan-tum gravity), noninteger conformal operator dimensionsin conformal field theories and so on. These peculiar-ities are deeply rooted in mathematical formalism ofhigher derivative models and one of its fundamentalingredients—the heat kernel—the building block under-lying the propagator of the theory in Feynman diagram-matic technique.Now the heat kernel method is one of the most pow-erful tools in mathematical physics, that has a widerange of applications extending from pure mathemat-ics (spectral geometry) to the analysis of financial mar-kets. Being combined with the background field methodin QFT it provides directly in the coordinate space acalculational technique for the quantum effective action,studying renormalizability of field models, their quantumanomalies, critical phenomena, etc. This makes it in-dispensable for computations in the presence of external ∗ [email protected] † [email protected] ‡ [email protected] fields or in curved space-time, which is crucially impor-tant for gauge theories and quantization of gravity [2–10].See also [11–13] and references there.Importance of the heat kernel approach was under-stood long ago both by mathematicians [14–19] andphysicists [20–22]. But the efforts of mathematician weremainly focused on general estimations and theorems re-garding the proper time expansion of the functional trace of the heat kernel with curvature invariant coefficients[17–19, 23–27], whereas the physicists would consider thetwo-point heat kernel itself with the separate point ar-guments [6, 21, 22]. This would give essentially moreflexibility and efficiency in obtaining these coefficients—with the ultimate goal of physical applications in UVrenormalization and gradient expansion. This is wherethe difference between the expression for this kernel forsecond order and higher order operators starts explic-itly showing up. Gilkey–Seeley approach, which is basedon functorial methods [28–31], does not feel this differ-ence, while the Schwinger–DeWitt technique, which ex-plicitly generates recurrent equations for the two-pointHaMiDeW (Hadamard–Minakshisundaram–DeWitt) co-efficients and their coincidence limits, is very vulnerableto the choice of the leading order heat kernel ansatz andbreaks down when it is inappropriately chosen.Consider a generic minimal second-order operator F ( ∇ ) = − ∆ + . . . whose covariant derivatives form aLaplacian ∆ = g ab ∇ a ∇ b acting in a curved d -dimensionalspace-time with the coordinates x = x a and the Rie-mannian metric g ab . Then the ansatz for its heat kernel a r X i v : . [ h e p - t h ] N ov K F ( τ | x , y ) = e − τF ( ∇ ) δ ( x , y ) has the form K F ( τ | x , y ) = D / ( x , y )(4 πτ ) d/ exp (cid:18) − σ ( x , y )2 τ (cid:19) × ∞ (cid:88) n =0 τ n a n [ F | x , y ] , (1.1)where σ ( x , y ) = l ( x , y ) / l ( x , y ) is the geodetic distance between the points x and y . In fact this ansatz has a semiclassical nature.This is because its exponential coincides with the princi-pal Hamilton function S ( τ | x , y ) = σ ( x , y ) / τ of the par-ticle moving in the x -space and fictitious imaginary time τ with the Hamiltonian F ( ∇ ), and the preexponentialfactor is just the square root of the Van Vleck–Morettedeterminant D ( x , y ) / (2 τ ) d = det (cid:2) ∂ S ( τ | x , y ) /∂x a ∂y b (cid:3) .So the expansion in powers of the proper time corre-sponds to the conventional semiclassical expansion in (cid:126) . Two-point coefficients of this expansion then satisfysimple recurrent equations which can be systematicallysolved for their coincidence limits a n [ F | x , x ]—local in-variants of the space-time curvature and the coefficientsof the operator F ( ∇ ). Note the property of the expan-sion (1.1) that it isolates essentially singular part of theheat kernel in the exponential, which vanishes in the co-incidence limit y = x , whereas important physical infor-mation is contained in the HaMiDeW-coefficients of theregular expansion a n [ F | x , x ].It is straightforward to formally extend this semiclas-sical ansatz to higher-derivative or pseudodifferential op-erators of the minimal form F ( ∇ ) = ( − ∆) ν + . . . withsome integer or noninteger ν , but this extension fails togenerate solvable recurrent relations for the generalizedSchwinger–DeWitt coefficients. The origin of this diffi-culty is that this semiclassical approach fails to performthe resummation of all negative powers of τ in the ex-ponential, and the infinite power series in τ turns out toinclude infinitely many of its negative powers. Essentiallysingular part of the τ -expansion at τ → y = x as it happens in (1.1) forsecond-order operators.Apparently due to this difficulty the heat kernelmethod was only indirectly used in physical applicationswith higher order operators. Numerous problems likeregularization by higher order covariant derivatives orhigher derivative theories, namely, R -gravity [32, 33],nonlocal and superrenormalizable models [34, 35] andHoˇrava–Lifshitz models [36, 37] were treated by meansof the reduction to minimal second-order operators whichallow one to use the expansion (1.1). Such a reductiontechnique for a wide class of theories with the general-ized causality condition was suggested in [6] and actu-ally allowed to circumvent the use of the proper short-time expansion of the form (1.1). Discussion of the heatkernel method for higher-order operators within similarreduction, functorial or other methods can be found in [29, 38–42], see also a series of papers by Gusynin et al. [43–48].Nevertheless, the heat kernels of higher-order differen-tial operators are themselves important as explicit ob-jects, because these kernels represent the building blockof Green’s functions of these operators, which are needednot only in the UV limit of their coinciding arguments.Moreover, a consistent version of the expansion (1.1)for higher-derivative operators and nonlocal operatorsof pseudodifferential type should be a source of recur-rent relations for the generalized HaMiDeW-coefficients,and this generalization is expected to be a much morepowerful tool than the reduction technique mentionedabove. So possible applications of the standard methodto higher-order operators does not make it less interest-ing to study their heat kernels directly. This is the goalof the present paper.To understand the nature of the generalization of (1.1)for minimal higher-derivative operators it is enough toconsider the case of a flat space-time of the Euclideansignature with the world function σ ( x , y ) = ( x − y ) / F ( ∇ ) = ( − ∆) ν —the ν th power of theLaplacian ∆ = δ ab ∂ a ∂ b , so that the standard heat ker-nel takes the translationally invariant form e τ ∆ δ ( x , y ) = e τ ∆ δ ( x − y ) with e τ ∆ δ ( x ) = 1(4 πτ ) d/ exp (cid:18) − x τ (cid:19) . (1.2)Then, the basic fact for a generic and not necessarilyinteger ν can be formulated as e − τ ( − ∆) ν δ ( x ) = 1 (cid:0) πτ /ν (cid:1) d/ E ν,d/ (cid:16) − x τ /ν (cid:17) , (1.3)where E ν,d/ ( z ) is what we call generalized exponentialfunction (GEF) . It is defined as a two-parameter familyof functions represented by the Taylor series E ν,α ( z ) = 1 ν ∞ (cid:88) m =0 Γ (cid:0) α + mν (cid:1) Γ( α + m ) z m m ! . (1.4)This function obviously reduces to the usual exponentialfunction at ν = 1 E ,α ( z ) = exp( z ) (1.5)and recovers the standard Gaussian behavior of the heatkernel. On the contrary, for other values of ν it per-forms resummation of negative powers of the proper time,which is impossible with the usual semiclassical ansatz.It should be emphasized that the functions (1.3) wereoriginally utilized in higher derivative models in [38] and[45], the latter paper also including their series expansion(1.4). Later they were used in the context of anoma-lous diffusion theory [49] and in application to Hoˇrava–Lifshitz models [50]. However, thus far no systematicstudies of these functions were undertaken and their po-tential role for the extension of the HaMiDeW techniqueto modified field and gravity models was underestimated.The goal of this work is to fill up this omission.The paper is organized as follows. In Sec. 2 we derivethe heat kernel of the operator ( − ∆) ν and its associatedGEF in the form of the Taylor expansion and presentits integral representation in terms of Bessel functions.In Sec. 3 we discuss the properties of the generalizedexponential functions E ν,α ( z ) and consider their Mellin–Barnes integral representation generating their asymp-totic behavior at z → ∞ , which is responsible for theshort time, τ →
0, or large | x | → ∞ limit of the heatkernel (1.3). Interestingly, this limiting behavior turnsout to be different for fractional and integer powers ν .Contrary to the second-order case this asymptotics ispower-law for fractional ν and exponential for integer ν ,and moreover features oscillations for growing | x | . Forinteger powers ν this property is demonstrated in Sec. 4where the asymptotic behavior of GEF is compared withthe semiclassical heat kernel ansatz and the saddle-pointapproximation for the momentum space integral repre-sentation. In concluding section we briefly discuss furtherapplication of GEF to generic minimal and nonminimalhigher-derivative operators in curved space-time, whichwill allow us to build a solvable recurrent equations forthe full set of generalized HaMiDeW-coefficients.
2. THE HEAT KERNEL OF THE POWER OFLAPLACIAN
For the operator F = ( − ∆) ν its heat kernel K ν,d ( τ, x ) = e − τ ( − ∆) ν δ ( x ) (2.1)has an obvious momentum space representation K ν,d ( τ, x ) = (cid:90) d d k (2 π ) d exp (cid:0) − k ν τ + i kx (cid:1) , (2.2)where k = | k | = √ k and kx = k a x a . For ν = 1this integral defines the well-known fundamental solution(1.2).Note that the heat kernel (2.2) is invariant with respectto O ( d )-rotations and homogeneous K ν,d ( τ, x ) = K ν,d ( τ, | x | ) , (2.3) K ν,d ( c ν τ, c x ) = c − d K ν,d ( τ, x ) , (2.4)where c is an arbitrary constant. Therefore, it shouldhave the form (1.3), where E ν,d/ ( z ) is some unknownfunction of the ratio − x / τ /ν . Since it stands in placeof the exponential function in the usual expression forthe heat kernel (1.2), we will call it the generalized expo-nential function (GEF) .Let us find the expansion of the generalized exponen-tial function E ν,d/ ( z ) in its Taylor series. Using the re-lations ∂ a σ k = kσ k − x a , ∂ a (cid:0) σ k x a (cid:1) = ( d + 2 k ) σ k , (2.5) where σ = x /
2, it is easy to verify by induction that foran arbitrary function f ( cσ ), where c is a constant, thefollowing differentiation rule holds∆ m f ( cσ ) = (2 c ) m m (cid:88) k =0 C km Γ( d/ m )Γ( d/ k ) ( cσ ) k f ( m + k ) ( cσ ) , (2.6)where f ( k ) ( z ) = d k f ( z ) /dz k , C km = m ! /k !( m − k )! arethe binomial coefficients and Γ( s ) is the Euler gammafunction. Then for f ( z ) = E ν,d/ ( z ) and c = − / τ /ν we obtain at x = 0( − ∆) m K ν,d ( τ, x ) (cid:12)(cid:12)(cid:12) x =0 = τ − d/ mν (4 π ) d/ Γ( d/ m )Γ( d/ E ( m ) ν,d/ (0) . (2.7)On the other hand, these quantities can easily be calcu-lated directly( − ∆) m K ν,d ( τ, x ) (cid:12)(cid:12)(cid:12) x =0 = (cid:90) d d k (2 π ) d k m exp (cid:0) − k ν τ (cid:1) = τ − d/ mν (4 π ) d/ Γ (cid:16) d/ mν (cid:17) ν Γ( d/ . (2.8)Comparing the expressions (2.7) and (2.8), we find allthe derivatives of E ν,α ( z ) at z = 0 and thus get its Taylorexpansion (1.4). Another integral representation expresses the general-ized exponential function E ν,α ( z ) in terms of the Besselfunction J α ( z ) or the Bessel-Clifford function C α ( z ). Thelatter is determined by the series C α ( z ) = ∞ (cid:88) m =0 α + 1 + m ) z m m ! (2.9)and related to the Bessel function J α ( z ) by the equationwhich removes its branching point at z = 0 J α ( z ) = (cid:16) z (cid:17) α C α ( − z / . (2.10)Therefore, the Bessel–Clifford functions have no singular-ities and represent single-valued entire functions on thewhole complex z plane.Bessel function representation of K ν,d ( τ, x ) followsfrom integration over angles in the momentum space in-tegral (2.2), which reads as K ν,d ( τ ; x ) = S d − (2 π ) d ∞ (cid:90) dk k d − exp( − k ν τ ) × π (cid:90) dθ (sin θ ) d − e ikx cos θ , (2.11)where S d − = 2 π ( d − / / Γ( d − ) is the volume of ( d − θ is the angle between thevectors x and k in (2.2) and x = | x | . Expandingexp( ikx cos θ ) and integrating over θ on account of2 π/ (cid:90) (sin θ ) α − (cos θ ) β − dθ = Γ( α )Γ( β )Γ( α + β ) , (2.12) √ π Γ(2 z + 1) = 4 z Γ (cid:18) z + 12 (cid:19) Γ( z + 1) (2.13)one gets π (cid:90) dθ (sin θ ) d − e ikx cos θ = √ π Γ (cid:18) d − (cid:19) C d − (cid:0) − k x / (cid:1) . (2.14)As a result the heat kernel has the following integral rep-resentation with the Bessel-Clifford function K ν,d ( τ, x ) = 2(4 π ) d/ ∞ (cid:90) k d − exp( − k ν τ ) × C d − (cid:0) − k x / (cid:1) dk. (2.15)Under the change of integration variable µ = k ν τ com-parison of this expression with (1.3) gives the relevantintegral representations for the GEF E ν,α ( z ) E ν,α ( z ) = 1 ν ∞ (cid:90) dµ µ α/ν − e − µ C α − ( zµ /ν ) . (2.16)Note that substitution of the expansion (2.9) for theBessel-Clifford function into (2.16) directly leads to theexpansion (1.4), which confirms this representation.Another remark is that the functions E ν,α ( z ) and K ν,d ( τ ; x ) are directly related to Bessel-Clifford (2.9) andBessel functions (2.10) in the limit ν → ∞ . Indeed, re-placing in this limit Γ (cid:0) ( α + m ) /ν (cid:1) by ν/ ( α + m ) in theexpansion (1.4), one has E ∞ ,α ( z ) = C α ( z ) , (2.17) K ∞ ,d ( τ, x ) = 1(2 πx ) d/ J d/ ( x ) . (2.18)Interestingly, the heat kernel of ( − ∆) ∞ becomes inde-pendent of the proper time parameter τ because of theobvious limit τ /ν −−−−→ ν →∞
3. GENERALIZED EXPONENTIALFUNCTIONS AND THEIR PROPERTIES
Various properties of GEF follow from the Mellin-Barnes integral representation of this function. This rep-resentation can be obtained by converting the series (1.4) s C w wC ××××××× l k − × × × × α r k α + ν FIG. 1. The location of the poles of ε ν,α ( s ) and the contours C and C w on the complex s plane. into the contour integral in the complex plane of an aux-iliary parameter s , such that the residues at simple polesof the integrand generate various terms of this series. Itis easy to guess that this integral reads E ν,α ( − z ) = 12 πi (cid:90) C ds Γ( s )Γ (cid:0) α − sν (cid:1) ν Γ( α − s ) z − s . (3.1)Then the inverse Mellin transform obviously gives ∞ (cid:90) dz z s − E ν,α ( − z ) = Γ( s )Γ (cid:0) α − sν (cid:1) ν Γ( α − s ) ≡ ε ν,α ( s ) . (3.2)The location of the poles of ε ν,α ( s ) and the contour C on the complex s plane is schematically shown in Fig. 1.The function Γ( s ) has a sequence of poles at l k = − k , k = 0 , , , . . . running to the left with residues ( − k /k !.And the function Γ(( α − s ) /ν ) has poles at r k = α + kν ,running to the right. The contour C begins at −∞ − i(cid:15) ,runs under the real axis, bends around 0 and returns to −∞ + i(cid:15) . The integral (3.1) is equal to the sum of residuesat the poles l k , which exactly gives the series (1.4).It is possible, however, that not all the points r k = α + kν are poles of the function ε ν,α ( s ), since they canbe canceled by the poles of the function Γ( α − s ) in thedenominator. In particular, the point r = α is nevera pole of ε ν,α ( s ). Three cases are possible: if ν is irra-tional, then all other points r k are poles; if ν = p/q isan irreducible fraction, then the poles r qk are also can-celed; and, finally, if ν is integer, then all the poles r k arecanceled.To determine the asymptotic behavior of the function E ν,α ( − z ), we deform the integration contour C into thecontour C w , coming from w − i ∞ vertically to w + i ∞ ,where w (cid:54) = Re r k , w > E ν,α ( − z ) = − (cid:88) Re r k
0. Finally, for thecritical value ν = 1 / | z | < / ν = 1 / . Indeed, using theLegendre duplication formula (2.13) and the well-knownexpansion (1 − z ) − γ = 1Γ( γ ) ∞ (cid:88) m =0 Γ( γ + m ) z m m ! , (3.6) We are grateful to the anonymous referee for pointing out thisfact. we can easily find that E ,α ( z ) = 4 α √ π ∞ (cid:88) m =0 Γ (cid:18) α + 12 + m (cid:19) (4 z ) m m != 4 α Γ (cid:0) α + (cid:1) √ π (1 − z ) − α − . (3.7)Thus, for ν = 1 / | z | > / z = 1 /
4. It is not difficult to ver-ify that in this case even terms of the series (3.4) vanishand odd terms converge to the function (3.7) in the cir-cle | z | > /
4. As a result, the operator √− ∆ in a flat d -dimensional space has the following heat kernel K ,d ( τ, x ) = Γ (cid:0) d +12 (cid:1) π d +12 τ ( τ + x ) d +12 . (3.8)This expression is related to holographic [58, 59] andbrane world [60–62] applications of effective action be-cause it represents the massless limit of the simplestbrane-to-bulk propagator e − τ √ M − (cid:3) δ ( x ), M →
0, [61,62] and may be interesting in the context of the discus-sion of fractional powers of generalized Laplacians in [25].The terms of the series (1.4) for E ν,α ( z ) are well definedfor complex α (cid:54) = − n − kν, where n, k = 0 , , , . . . (3.9)We note, however, that the singularities at the points α = − n (and hence at all the points for integers ν ) areremovable, since the poles of the gamma functions in thenumerator and denominator cancel each other. Expand-ing Γ( − n + (cid:15) ) ∼ ( − n /n ! (cid:15) we can redefine the coefficientsin (1.4)Γ( − n/ν ) ν Γ( − n ) = (cid:40) ( − − n + n/ν n !( n/ν )! , if n/ν = 0 , , . . . , , otherwise . (3.10)Thus, for integer ν the function E ν,α ( z ) is an entirefunction of z for any values of α , and for noninteger ν > / α except α = − n − kν (cid:54) = − m with positive integer k, n and m .Consequently, the functions K ν,d ( τ, x ) are well definednot only for all integer ν and d , but also for fractional ν and d satisfying these conditions.The graphs of the functions E ν,α ( z ) and K ν,d ( τ, x ) forvarious values of the parameters, obtained by numericalsummation of the series (1.4) in MATLAB, are shownin Figs. 2–4. Important distinction from the case of amonotonic exponential falloff for ν = 1 is that the heatkernel for ν (cid:54) = 1 oscillates as a function of x /τ /ν .Other interesting properties of E ν,α ( z ) include the fol-lowing simple differentiation rule d β dz β E ν,α ( z ) = E ν,α + β ( z ) . (3.11) − . − . . . . . . z E , α ( − z ) ν = 10 α / / /
20 5 10 15 20 25 − . − . . . . . x K , d ( , x ) τ = 1, ν = 10 d FIG. 3. Graphs of the functions E ,α ( − z ) and K ,d (1 , x )for various values of the parameters α and d . For integer β , it can be verified directly by differenti-ating the definition (3.5). However, this relation makessense also for all β such that E ν,α + β ( z ) is well definedfor noninteger ν (and for any complex β if ν is integer).For negative integer β it will give the principal primitiveof the function E ν,α ( z ). For noninteger β , the symbol d β /dz β should be understood as a certain operator offractional integrodifferentiation. Thus for each ν in therange 1 / < ν ≤ ∞ the family of functions E ν,α ( z ) isclosed under the operation of integrodifferentiation.For noninteger ν the expression (1.3) is the solution ofthe heat equation in which the nonlocal operator ( − ∆) ν should be understood as a pseudodifferential operator de-fined by the Fourier transform [63]. The correspondingequations are called fractional diffusion equations andhave been widely discussed in [49, 64–67]. However, inthese papers fractional equations are usually consideredin (1 + 1)-dimensional ( τ, x )-space , i.e. the case of d = 1in our notations. − . . . . . . x K , ( τ , x ) ν = 5, d = 1 τ . . FIG. 4. Graphs of the function K , ( τ, x ) for different valuesof the proper time τ .
4. INTEGER POWER OF LAPLACIAN ANDSEMICLASSICAL EXPANSION
As we see, the asymptotic behavior of GEF E ν,α ( − z )at z → ∞ is critically different for noninteger and in-teger values of ν . It is power-law for noninteger ν cor-responding to the nonlocal operator ( − ∆) ν and quasi-exponential O ( z −∞ ) for integer ν corresponding to localdifferential operators of order 2 ν . But for the heat kernel(1.3) this limit is associated with the semiclassical limit x / τ /ν → ∞ , the proper time τ → (cid:126) . On the other hand, semiclassical approximationfor the solution of the heat equation (or the Schr¨odingerequation in the imaginary time) ∂ τ K F ( τ | x , y ) = − F ( ∇ x ) K F ( τ | x , y ) (4.1)begins with the Pauli–Van Vleck (or WKB) ansatz [68] (cid:115) det (cid:20) − π ∂ S ( τ | x , y ) ∂x a ∂y b (cid:21) exp [ − S ( τ | x , y ) ] , (4.2)where S ( τ | x , y ) is the principal Hamilton-Jacobi func-tion, i.e. the solution of the Hamilton-Jacobi equationwith the Hamiltonian F ( p ) − ∂S∂τ + F (cid:18) − ∂S∂ x (cid:19) = 0 . (4.3) The operator F ( ∇ ) with the space-time gradients replaced bythe canonical momenta p . We work in Euclidean time, whichexplains the absence of imaginary i -factor. The derivation fora generic Hamiltonian not necessarily polynomial in derivatives(momenta) can be found in Appendix C of [69]. For the power of Laplacian this Hamiltonian equals F ( p ) = ( − p ) ν , and the solution of this equation readilyexpresses as S ( τ | x , y ) = ( − ν − ν − (2 ν − (cid:18) ( x − y ) ν τ /ν (cid:19) ν ν − . (4.4)This leads to the absolute value of the preexponentialfactor1(2 π ) d/ (cid:12)(cid:12)(cid:12)(cid:12) det ∂ S ( τ | x , y ) ∂x a ∂y b (cid:12)(cid:12)(cid:12)(cid:12) / = 1(4 πτ /ν ) d/ (cid:18) ( x − y ) τ /ν (cid:19) − d ν − ν − ν − d ν − √ ν − . (4.5)Regarding its phase factor, which should be determinedhere by the correct choice of the branch for the fractionalpowers, the Pauli–Van Vleck algorithm does not give thisinformation in contrast to the trivial case of ν = 1. Nei-ther does it prescribe a definite linear combination ofthese branches in the heat kernel asymptotics. Below allthis will be attained by two different methods — appli-cation of the general technique of Fox H -functions andby the steepest descent approximation.Note now that in view of the discussion in the previ-ous section this quasiexponential behavior is completelyimpossible for noninteger ν with a power-law behavior,so that standard semiclassical expansion seems to breakdown for nonlocal operators ( − ∆) ν . Therefore in whatfollows we consider only the case of positive integer pow-ers. To underline this, we will further denote the powerof the Laplacian by N , when it is integer, and by ν , whenit can be either integer or noninteger. The goal of thissection is to find the heat kernel asymptotic expansionfor this higher-derivative case by an alternative methodwhich allows to get correct complex branches and to com-pare them with the semiclassical result of the above type. Asymptotic expansion of the integral (3.1) is a part ofthe general theory of Fox-Wright Ψ - and Fox H -functions,briefly outlined in Appendix A. The idea of this expan-sion consists in using the Mellin transform of the gammafunction12 πi w + i ∞ (cid:90) w − i ∞ ds Γ( µs − K ) z − s = 1 µ z − K/µ exp (cid:16) − z /µ (cid:17) , (4.6)which for µ > K > z → ∞ . How-ever, what is integrated in (3.1) is not just a gamma func-tion of this type, but rather a nontrivial ratio of those.This ratio ε N,α ( s ), which is given by Eq. (3.2), can nev-ertheless be converted to the series of gamma functionterms of the above type, so that the s -integration can besuccessfully done. For this purpose we use, first of all,the Euler reflection formulaΓ( x )Γ(1 − x ) = π sin( πx ) (4.7)to provide positive coefficients of the integration param-eter s in the arguments of all gamma functions (just likein (4.6)), ε ν,α ( s ) = 1 ν sin π ( s − α )sin πν ( s − α ) ˜ ε ν,α ( s ) , (4.8)˜ ε ν,α ( s ) = ν Γ( s )Γ( s − α )Γ (cid:0) s − αν (cid:1) . (4.9)This makes the sequence of gamma function poles run-ning to the left of the complex plane of s .For integer N the ratio of sines reduces to the sum ofcomplex exponential functionssin( N φ )sin φ = N − (cid:88) j =0 e i (2 j +1 − N ) φ , φ = πN ( s − α ) , (4.10)After the substitution of (4.8) into the Mellin transform(3.1) this leads to the sum E N,α ( − z ) = 1 N N − (cid:88) j =0 e iω j α ˜ E N,α ( − e iω j z ) , (4.11) ω j = π − N + 2 jN , (4.12)with the phase factors both in the coefficients of this sumand in the arguments of the new functions ˜ E ν,α ( − e iω j z )which can be called the generalized exponential functionsof the second kind . They read˜ E ν,α ( − z ) = ν πi (cid:90) C ds Γ( s )Γ( s − α )Γ (cid:0) s − αν (cid:1) z − s . (4.13)Critical point of the derivation is that now one canapply Eq. (A.7) of the Appendix A to expand the ratioof gamma functions in the asymptotic series of anotherset of gamma functions of decreasing arguments. Withthe parameters p = 1, A = 1 /ν , a = − α/ν , q = 2, B = B = 1, b = 0, b = − α , which give rise tothe parameters (A.5), (A.6) and (A.8) defined in thisAppendix, µ = 2 ν − ν , a = α ν − ν , (4.14) C = (2 ν − α ν − ν + ν α , (4.15)Eq. (A.7) generates the asymptotic expansion˜ ε ν,α ( s ) = (2 ν − a − µs +1 / ν α − s +1 ∞ (cid:88) m =0 E m Γ( µs − a − m ) , (4.16) with the coefficients E m independent of the integrationvariable s . These coefficients start with E = 1 andthey are systematically calculable by the procedure ofAppendix B. The essence of this expansion is that forlarge s it runs over ever decreasing terms, each term beingsmaller than the preceding one in view of the obviousrelation Γ( µs − a − m −
1) = Γ( µs − a − m ) / ( µs − a − m − E ν,α ( − z ) = ν ( νz − ν ) − α ν − √ ν − (cid:20) − (2 ν − (cid:16) zν (cid:17) ν ν − (cid:21) × ∞ (cid:88) m =0 E m (2 ν − m (cid:18) ν z (cid:19) mν ν − . (4.17)In contrast to the GEF of the first kind E ν,α ( z ), whichhas no singularities, the function ˜ E ν,α ( z ) is singular at z = 0. However, it always has a simple exponential (i.e.,not a power-law) asymptotic behavior (4.17) for z → ∞ .Second, it is monotonic for −∞ < z < N series of terms E N,α ( − z ) = N − α N − z − α N − N − √ N − × N − (cid:88) j =0 exp (cid:20) − (2 N − e iϕ j (cid:16) zN (cid:17) N N − + iϕ j α (cid:21) × ∞ (cid:88) m =0 E m (2 N − m (cid:18) N z (cid:19) mN N − e − iϕ j m , (4.18)where both the amplitudes and phases depend on thephases ϕ j of the complex factors (4.12) ϕ j = N N − ω j = π − N + 2 j N − , j = 0 , , . . . , N − . (4.19)Consequently, the expression in (1.3) with z = x / τ /N gives the heat kernel asymptotics for fixed τ and | x | → ∞ or for fixed | x | and τ → K N,d ( τ, x ) = 1(4 πτ /N ) d/ (cid:18) x τ /N (cid:19) − d N − N − N − d N − √ N − × ∞ (cid:88) m =0 E m (2 N − m (cid:18) N τ /N x (cid:19) mN N − × N − (cid:88) j =0 exp (cid:34) − (2 N − e iϕ j (cid:18) x N τ /N (cid:19) N N − (cid:35) e iϕ j ( d − m ) . (4.20)The phase factors e iϕ j here coincide with the set of frac-tional powers ( − ( N − / (2 N − in the coefficient of theHamilton-Jacobi function (4.4). This confirms the semi-classical ansatz (4.2) along with establishing a concretechoice of the linear combination of its complex branches.The leading order term of this expansion consists oftwo complex conjugated branches corresponding to themaximal real part of the exponential with j = 0 and j = N − ∓ iϕ ), ϕ = ϕ N − = − ϕ = π N − N − , (4.21) K N,d ( τ, x ) (cid:39) πτ /N ) d/ (cid:18) x τ /N (cid:19) − d N − N − N − d N − √ N − × exp (cid:34) − (2 N − e iϕ (cid:18) x N τ /N (cid:19) N N − + iϕd (cid:35) + c . c . (4.22)and the corrections in the form of growing fractional pow-ers of τ / (2 N − → Alternatively this result can be reproduced by thesteepest descent method which is the basis of the semi-classical approximation with the small parameter τ → p = τ /N k , y = τ ( N − /N x ,converts the momentum integral (2.2) to the form K N,d ( τ, x ) = 1(2 πτ /N ) d (cid:90) d d p e − S ( p ) /τ , (4.23)where S ( p ) = ( p ) N − i py . (4.24)Its short time τ → S ( p ) at which this action isstationary, ∂S ( p ) /∂p a = 0. These 2 N − p j = (cid:104) i N − (cid:105) j (cid:32) τ N − N | x | N (cid:33) N − x | x | , (4.25)where (cid:104) i N − (cid:105) j , j = 0 , , ..., N −
2, are the 2 N − i . The leading order contribution ofeach such saddle point is given by the standard expression K ( j ) N,d ( τ, x ) = e − S ( p ) /τ (2 πτ /N ) d (cid:18) det 12 πτ ∂ S ( p ) ∂p a ∂p b (cid:19) − / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p = p j , (4.26)which gives K ( j ) N,d ( τ, x ) = 1(4 πτ /N ) d/ (cid:18) x τ /N (cid:19) − d N − N − N − d N − √ N − × exp (cid:34) − (2 N − e iϕ j (cid:18) x N τ /N (cid:19) N N − + iϕ j d (cid:35) , (4.27) where the set of phase factors is determined by the rela-tions e iϕ j = − i (cid:104) i N − (cid:105) j , ϕ j = π − N + 2 j N − , (4.28) j = 0 , , . . . , N − , and in fact extends the range (4.19)up to 2 N − ϕ j < j ≥ N , contribute the terms exponentially growingwith | x | , which contradicts the known | x | −∞ falloff at | x | → ∞ . Therefore only the remaining N points with j ≤ N − H -functions. Semiclassical expansion (4.22) clearly shows a princi-pal difference of higher-order operators from the second-order case N = 1. For N > τ →
0, does not stand the limit | x | → | x | . In particular, itdoes not maintain the initial condition K ν,d (0 , x ) = δ ( x ).The exact heat kernel of course satisfies this condition,because for any smooth test function f ( x ) (cid:90) d d x K ν,d ( τ, x ) f ( x )= 1(4 π ) d/ (cid:90) d d y E ν, d (cid:18) − y (cid:19) f ( τ /ν y ) −−−→ τ → f (0) , (4.29)since 1(4 π ) d/ (cid:90) d d y E ν,α (cid:18) − y (cid:19) = 1Γ( d/ ∞ (cid:90) dz z d − E ν,α ( − z ) = Γ (cid:16) α − d/ ν (cid:17) ν Γ (cid:0) α − d (cid:1) −−−→ α → d , (4.30)where we used the relation (3.2). Interestingly, in therecovery of this result for integer N each of the N termsof the decomposition (4.12) contributes one and the same01 /N th part of it, because their dependence on ω j dropsout in the limit α → d/ π ) d/ (cid:90) d d y e iαω j N ˜ E ν,α (cid:18) − e iω j y (cid:19) = Γ (cid:0) α − d (cid:1) Γ (cid:16) α − d/ N (cid:17) e iω j ( α − d ) −−−−→ α → d/ N . (4.31)At the same time, if we try to reproduce the sameresult by using N branches (4.27) of the semiclassicalexpansion, corresponding to different terms of the abovedecomposition K N,d ( τ, x ) = N − (cid:88) j =0 K ( j ) N,d ( τ, x ) (cid:104) O (cid:16) τ N − (cid:17) (cid:105) , (4.32)then the result will be critically different. Each K ( j ) N,d ( τ, x ) is singular at x = 0, but this singularity isintegrable. However, the result of this integration is dif-ferent from (4.31) (cid:90) d d x K ( j ) N,d ( τ, x ) = N d − (2 N − d − . (4.33)Even more striking discrepancy between the exact heatkernel and its asymptotics is that, while all the terms ofthe latter are singular in the limit x →
0, the GEF (1.4)and the exact K ν,d ( τ, x ) are both well defined in this limit K ν,d ( τ,
0) = 1 (cid:0) πτ /ν (cid:1) d/ Γ( d/ ν ) ν Γ( d/ . (4.34)The short time expansion of this coincidence limit whichis a main goal of the Seeley-Gilkey technique [17–19, 24]runs in powers of τ /ν , whereas the expansion (4.22) goesin the other fractional powers τ / (2 N − .The source of all these discrepancies is the fact that,contrary to the second order Laplacian, the heat kernelasymptotic expansion is not uniform in x →
0. Whilein this limit the expansion (1.1) for a minimal secondorder operator F ( ∇ ) = − ∆ + . . . is just a regularizationof the delta function, for N > x → x =0. Obviously, there is no such a discrepancy in the caseof N = 1 with a single j = 0 branch of the heat kernelexpansion, so that the coincidence limit y = x can bedirectly taken in the asymptotic expansion (1.1). There is additional controversy with the result (4.33)—while all K ( j ) N,d ( τ, x ) with j (cid:54) = 0 and j (cid:54) = N −
5. CONCLUSIONS
Thus we obtained the expression (1.3) for the heat ker-nel K ν,d ( τ, x ) of the operator ( − ∆) ν in the d -dimensionalflat space, which is a direct generalization of the well-known heat kernel (1.2) to local higher derivative andnonlocal (pseudodifferential) operators. This general-ization is represented in terms of the newly introducedtwo-parameter family of generalized exponential func-tions E ν,α ( z ), α = d/ z = − x /τ /ν , defined by theTaylor series (1.4) and related to Fox-Wright Ψ - and Fox H -functions. We studied various properties of these func-tions and their integral representations. They include, inparticular, the Mellin-Barnes representation which allowsone to find, by the technique of Fox H -functions, theirasymptotic expansion in the limit of z → ∞ correspond-ing to two equivalent asymptotics of the heat kernel as τ → | x | → ∞ .This expansion turns out to be critically different forinteger and noninteger values of ν . In contrast to theexponential behavior, anticipated on the ground of semi-classical considerations with τ → (cid:126) ,for noninteger ν , that is for nonlocal operators ( − ∆) ν ,this is a power-law falloff. For integer ν > ν = 1.In particular, this asymptotic expansion is not uniformfor all values of | x | → x = 0 which is, on the other hand, eas-ily accessible directly from the short time expansion for ν = 1. In addition, the heat kernel for higher-derivativeand nonlocal operators with ν (cid:54) = 1 features oscillatory be-havior in x -space contrary to the monotonic exponentialfalloff for the pure Laplacian.The coincidence limit of the heat kernel K F ( τ | x , x )and its functional traceTr e − τF = (cid:90) d d x tr K F ( τ | x , x ) (5.1)are the objects of major interest in the Schwinger–DeWitt technique for quantum effective action. Forgeneric minimal second order operators with x -dependent coefficients it is based on the expansion (1.1)for K F ( τ | x , y ) with split arguments (generically lackingtranslation invariance). This coincidence limit is alsothe subject of mathematical Seeley–Gilkey treatise byvarious functorial methods not directly appealing to off-diagonal elements of K F ( τ | x , y ). The absence of a non-singular coincidence limit in the heat kernel asymptoticexpansion for ν (cid:54) = 1 considered above seems to invali-date any attempt to use it for some generalization of theSchwinger–DeWitt method. So these expansions are notvery physically interesting in field theory applications.However, a systematic way of the short time expansionof the heat kernel, which underlies UV properties of fieldmodels, is provided by the recurrent equations for thecoefficients of the expansion (1.1), these equations heav-1ily relying on the off-diagonal K F ( τ | x , y ). The Seeley–Gilkey functorial methods are not so universal and pow-erful enough to yield everything what physicists need inquantum gravity and other applications. For example,Hoˇrava–Lifshitz gravity [36, 37] are encumbered with thenecessity of working with higher order and nonminimaloperators whose leading symbol does not reduce to thepower of Laplacian and, therefore, go outside of the scopeof functorial methods. Derivation of recurrent equationsfor the two-point coefficients of (1.1) generalized to suchoperators then becomes indispensable.The generalized exponential functions introducedabove provide fundamental building blocks of such recur-rent equations, and the lack of uniformity of their asymp-totics does not make them less efficient. Note that thecharacteristic feature of the Schwinger–DeWitt expan-sion (1.1) is a single overall exponential factor absorbingall essentially singular dependence on τ →
0. The at-tempt to directly generalize this expansion to ν (cid:54) = 1 witha single semiclassical exponential factor fails because itgenerates infinitely many negative powers of τ .On the contrary, resummation of these singular termscan be performed with the aid of the generalized expo-nential functions, but in contrast to (1.1) these functionswill not form a single overall factor, but rather comprisethe series of terms with different α -parameters. As weare going to show in the coming paper [71], for a mini-mal differential operator F of an (integer) order 2 N ina curved Riemannian space [( x − y ) / → σ ( x , y )] thefollowing generalization of the expansion (1.1) holds K F ( τ | x , y ) = 1(4 πτ /N ) d/ × ∞ (cid:88) j =0 τ j/N E N, d − j (cid:18) − σ ( x , y )2 τ /N (cid:19) a j [ F | x , y ] . (5.2)The generalized HaMiDeW-coefficients a j [ F | x , y ] satisfythe manageable chain of recurrent equations, which canbe solved for the coincidence limit x = y . Note alsothat the coincidence limit of (5.2) is well defined, eventhough the asymptotics of the underlying E N,α ( − z ) arenot uniform for z →
0. In fact, these asymptotics are notneeded for this limit. Since E N,α (0) = Γ( α/N ) /N Γ( α ) wehave the following expansion for the coincidence limit K F ( τ | x , x ) = τ − d/ N ∞ (cid:88) j =0 τ j/N A j [ F | x ] , (5.3)where A j [ F | x ] = 1(4 π ) d/ Γ (cid:16) d/ − jN (cid:17) N Γ( d/ − j ) a j [ F | x , x ] . (5.4)So GEF should be treated as entire building blocks ofthe formalism, the operations with them being based ontheir simple differentiation rule (3.11) and the value at z = 0. In our next paper [71] we will consider variousproperties of the generalized HaMiDeW-coefficients in (5.2). In particular, we will prove the generalized “func-torial property” for an arbitrary power λ of a differentialoperator F , a j [ F λ | x , y ] = a j [ F | x , y ] (5.5)(previously known only in the coincidence limit x = y [28]), and also easily reproduce and extend the resultsof [28] for higher order operators. These coefficients andthe computational methods based on them promise to bevery efficient and are likely to simplify the calculation ofbeta functions for theories with higher derivatives andHoˇrava–Lifshitz type models [36, 37]. All this makes theuse of GEF and the associated heat kernel coefficientsvery prospective.The above formalism seems equally applicable to thecase of generic noninteger ν . This case is, in particu-lar, important in superrenormalizable quantum gravitymodels [34, 35, 72, 73], within analytical regularizationof Feynman graphs [74] or for the calculation of UV coun-terterms in Hoˇrava–Lifshitz gravity models. For exam-ple, in (3+1)-dimensional Hoˇrava gravity cubic in spa-tial curvature counterterms follow from the heat kernelof the operator which is a square root of the sixth or-der nonminimal differential operator [37]. However, aswe saw above there is a number of new features (like thepower-law heat kernel asymptotics confronting their ex-ponential analogue for integer N or the mismatch withthe semiclassical expansion) which might backfire underindiscriminate extension of this method. Here we onlybriefly comment on possible modifications due to thesesubtleties.One modification follows from the recovery of the heatkernel diagonal elements by inverse Mellin transformfrom the operator zeta function F − s δ ( x , y ) | x = y used in[25]. For the operators of the form F = H ν , where H is a Laplace type (minimal second order) operator, andnoninteger ν this method leads to additional terms [25] K F ( τ | x , x ) = τ − d/ ν ∞ (cid:88) j =0 τ j/ν A j ( x ) + ∞ (cid:88) k =1 τ k B k ( x ) . (5.6)While the coefficients A j ( x ) are in one-to-one corre-spondence with the HaMiDeW-coefficients a j [ H | x , x ] ofthe operator H and are local quantities, the coefficients B k ( x ) are determined through the values of the zeta func-tion at certain values of s . Rather than being expressedin terms of a j [ F | x , x ], they turn out to be nonlocal andirrelevant to UV renormalization because they do notcontribute to UV divergences in view of analytic expan-sion in τ starting with a linear term. However, accordingto our method, just in the case of operators of the form F = H ν , such additional terms do not arise.Another modification known from mathematical stud-ies [23, 25] is the origin of logarithmic terms in the propertime expansion of the heat kernel. For special values ofnoninteger ν leading to − α = n + kν (cid:54) = m [with positiveinteger k , n and m —see the discussion after (3.9)] the2functions E ν,α ( z ) in the expansion (1.2) are not definedbecause of gamma function singularities. This excep-tional case can occur, in particular, for even order roots ofthe Laplace type operator in odd spacetime dimensions.In this case the logarithmic terms appear [23, 25] K F ( τ | x , x ) = τ − d/ ν ∞ (cid:88) j =0 τ j/ν A j ( x ) + ∞ (cid:88) k =1 τ k log τ C k ( x ) . (5.7)which are again unrelated to renormalization of UV di-vergences.All these modifications can apparently be attributed tothe fact that zeta-function approach of [23, 25] actuallyrepresents a regularization which for nonlocal theories(corresponding to noninteger values of ν ) leads to dif-ferent results . Absence of uniformity of the asymptoticsmall time expansion in the vicinity of the heat kernel di-agonal discussed above shows up for noninteger values of ν . The search for an asymptotic expansion of GEF andheat kernel that would be uniform for all x and y (anal-ogous, for example, to the uniform WKB asymptoticexpansion of Legendre functions [75]) apparently couldhave resolved the problem of these discrepancies. Thishowever goes beyond the scope of this paper, in particu-lar, because nonuniformity of the asymptotic expansionof GEF is harmless in the calculation of the functionaltrace (5.1) if one uses generalized exponential functionsas building blocks of the expansion and takes their exactvalues at zero argument.To summarize, the generalized exponential functionscan serve as a very efficient tool in quantum field theoryand quantum gravity. Moreover, their connection withfractional calculus opens the prospect of applying theobtained heat kernels far beyond the area of QFT. Thisincludes the theory of fractional differential equationswhich can be effectively used to construct phenomenolog-ical models of fractal media, systems with memory andnonlocal interaction. Numerous applications of fractionalcalculus to physical problems are discussed, for example,in [76] and references therein. ACKNOWLEDGEMENTS
The authors are grateful to A. E. Kazantsev,A. A. Lobashev and M. M. Popova for numerous fruit-ful discussions, and especially to O. I. Marichev for thehelp with the asymptotic expansion of the Fox–Wright Ψ -functions. We also thank the anonymous referee for Note that the method of derivation of (5.6) in [25] can be inter-preted as zeta-function regularization, because it operates withthe regularized (and therefore finite) expression for the coinci-dence limit of the Green’s function of the operator F s . On thecontrary, our expansion is done for separate arguments of theheat kernel which of course renders this coincidence limit singu-lar and invokes point separation or dimensional regularization. constructive suggestions on the extension of our results.This work was supported by the RFBR Grant No.17-02-00651 and by the Foundation for the Advancement ofTheoretical Physics and Mathematics “BASIS.” Appendix A: Wright Ψ -functions and Fox–Wright H -functions The Fox–Wright Ψ -functions p Ψ q [( a, A ); ( b, B ); z ] arelabeled by two sets of parameters A k , a k , r = 1 , . . . , p ,and B j , b j , j = 1 , . . . , q , among which A k and B j are realand positive. These functions are defined by their Taylorseries p Ψ q [( a, A ); ( b, B ); z ] = ∞ (cid:88) k =0 p (cid:81) j =1 Γ( a j + A j k ) q (cid:81) i =1 Γ( b i + B i k ) z k k ! . (A.1)They represent one of the possible further extensionsof the generalized hypergeometric series, p F q [ a ; b ; z ] = p Ψ q [( a, b, z ]Γ( b ) / Γ( a ), and have applications, inparticular, in fractional calculus [63–67, 77–79]. Theywere introduced by E. M. Wright, who studied theirasymptotic behavior [51, 52].In their turn the Fox–Wright Ψ -functions form a spe-cial case of more general Fox H -functions H m,np,q (cid:104) z (cid:12)(cid:12) ( a,A )( b,B ) (cid:105) .They are defined by the Mellin–Barnes integral H m,np,q (cid:104) z (cid:12)(cid:12)(cid:12) ( a,A )( b,B ) (cid:105) = 12 πi (cid:90) C h m,np,q [ s ] z − s ds, (A.2) h m,np,q [ s ] = m (cid:81) i =1 Γ( b i + B i s ) n (cid:81) j =1 Γ(1 − a j − A j s ) q (cid:81) i = m +1 Γ(1 − b i − B i s ) p (cid:81) j = n +1 Γ( a j + A j s ) , (A.3)also with real and positive A i and B j . The poles l i,k of the gamma functions Γ( b i + B i s ), i = 1 , . . . , m , enu-merating the poles index k being integer, run to theleft of the complex plane of s , whereas the poles r j,k of Γ(1 − a j − A j s ), j = 1 , . . . , n , run to the right. It isassumed that the parameters A j , a j , B i and b i are suchthat these poles do not match, l i,k (cid:54) = r j,l . Then the con-tour of integration C is chosen to pass from − i ∞ to i ∞ and to separate the poles l i,k and r j,k .The Fox H -functions are related to Fox–Wright Ψ -functions in exactly the same way as the well-knownMeyer G -functions to generalized hypergeometric func-tions. Obviously p Ψ q (cid:104) ( a,A )( b,B ) (cid:12)(cid:12)(cid:12) z (cid:105) = H ,pp,q +1 (cid:104) − z (cid:12)(cid:12)(cid:12) (1 − a,A )(0 , , (1 − b,B ) (cid:105) . (A.4)The general theory of H -functions and H -transformscan be found in [53–57]. Here we briefly sketch theirmain properties and the way of handling their asymptotic3behavior. This behavior is characterized by the followingthree combinations of their parameters µ = q (cid:88) j =1 B j − p (cid:88) k =1 A k , β = p (cid:81) k =1 A A k kq (cid:81) j =1 B B j j , (A.5) a = p (cid:88) k =1 a k − q (cid:88) j =1 b j + 12 ( q − p − , (A.6)Note that the structure of the expression (A.3) allowsone to relocate gamma functions between the numeratorand the denominator using the Euler reflection formula(4.7). Under this operation only the parameters m and n change, while the parameters p , q , µ , β and a , as it iseasy to see, remain intact.The main result, based on the use of the Stirling for-mula, is as follows: for µ > C in (A.2)can be closed on the left of the complex plane, then m series obtained by summing the residues at the poles l i,k will converge absolutely on the whole complex z plane,defining, generally speaking, a multivalued function withan essentially singular point at z = ∞ . If in this case weformally close the contour C on the right, then the sumof the residues at the poles r j,k will determine the asymp-totic (divergent) power series as z → ∞ . For µ < r j,k will absolutely converge at z (cid:54) = 0, andthe divergent series of residues at the poles l i,k will deter-mine the asymptotic behavior of the function at z → µ = 0 the se-ries obtained by closing the contour C on the left willconverge inside the circle | z | < β − , and the series ob-tained by closing the contour C on the right will convergeoutside of it.Exponential asymptotic behavior for z → ∞ appearswhen h m,np,q [ s ] does not have any rightgoing poles r j,k ,i.e., when in the expression (A.3) either the functionsΓ(1 − a j − A j s ) in the numerator are completely absent( n = 0), or all their poles are canceled with the poles ofthe functions Γ(1 − b i − B i s ) in the denominator (as it hap-pens in the case of the functions E ν,α ( z ) for an integer ν ).The general recipe for finding exponential asymptotics,which is explained in detail in [53], is the following: onefirst needs to use the Euler reflection formula to relocategamma functions so that they have only leftgoing poles,i.e. to convert the expression (A.3) to the form, whenall gamma functions with the coefficients A j are in thedenominator, and all gamma functions with the coeffi-cients B i are in the numerator. After that, one needsto use the asymptotic expansion for the ratio of gammafunction products, q (cid:81) j =1 Γ( B j s + b j ) p (cid:81) k =1 Γ( A k s + a k ) = C ( βµ µ ) − s ∞ (cid:88) m =0 E m Γ( µs − a − m ) , (A.7) which is derived by the method sketched in Appendix B.Here | s | → ∞ , | π − arg s | > (cid:15) , the parameters µ , β and a are defined as above in (A.5) and (A.6), C = (2 π ) ( q − p − / µ a +1 / p (cid:89) k =1 A / − a k k q (cid:89) j =1 B b j − / j , (A.8) E = 1 and other coefficients E m are systematically cal-culable by the method also sketched below in AppendixB. Finally, application of the inverse Mellin transform(4.6) yields the required asymptotic expansion. This pro-cedure is used in Sec. 4 for the derivation of the large z expansion of the generalized exponential function and theassociated heat kernel. Appendix B: The ratio of gamma function products
Here we briefly sketch the details of a special asymp-totic expansion at s → ∞ of the ratio of two productsof gamma functions in Eq.(A.7). If we denote this ratioby R ( s ) and divide it by Γ( µs − a ), then in view of theStirling expansionΓ( s + x ) = √ π e − s s s + x − / × exp (cid:34) ∞ (cid:88) k =1 ( − k +1 k ( k + 1) B k +1 ( x ) s − k (cid:35) , (B.1)( B k ( x ) are Bernoulli polynomials) the result will read as R ( s )Γ( µs − a ) = C ( βµ µ ) − s exp (cid:34) ∞ (cid:88) n =1 D n s − n (cid:35) , (B.2)where the parameters µ , β , a , and C are defined byEqs. (A.5), (A.6) and (A.8) and the coefficients D n equal D n = ( − n +1 n ( n + 1) q (cid:88) j =1 B n +1 ( b j ) B nj − p (cid:88) k =1 B n +1 ( a k ) A nk − B n +1 ( − a ) µ n (cid:33) . (B.3)The factor of gamma function Γ( µs − a ) was especiallyadded in the left-hand side of (B.2) in order to cancel thepowers of s and s s .Now the exponential factor in (B.2) can be reexpandedin inverse powers of s to giveexp (cid:34) ∞ (cid:88) n =1 D n s − n (cid:35) = 1 + ∞ (cid:88) n =1 C n s − n , (B.4)where each coefficient C k is uniquely determined by thefirst k coefficients D , . . . , D k , C = D , C = D + D / C = D + D D + D / /s -expansion herein terms of the inverse Pochhammer symbols for a special4choice of the argument x = a + 1 − µs composed of theparameters s , µ and a ,Γ( x )Γ( x + k ) = k (cid:89) n =1 x + n − . (B.5)To begin with, this symbol can be expanded in inversepowers of s ,Γ( a + 1 − µs )Γ( a + 1 − µs + k ) = ( − k Γ( µs − a − k )Γ( µs − a )= ( − k ∞ (cid:88) j =1 d kj s − j , (B.6)where the coefficients of the infinite lower-triangular ma-trix [ d kj ] depend on µ and a , d kj = 0 for j < k , d kk = µ − k . Inversion of this relation allows one to ex-pand s − k in terms of the sequence of such symbols s − k = ∞ (cid:88) j =1 d − kj Γ( µs − a − j )Γ( µs − a ) , (B.7)where d − kj are the coefficients of the inverse matrix, d − kj = 0 for j < k , d − kk = µ k . Using the relations (B.6)and (B.7) we can trade the expansion in powers of 1 /s for the expansion in Γ( µs − a − j ) / Γ( µs − a ), ∞ (cid:88) k =1 C k s − k = ∞ (cid:88) j =1 E j Γ( µs − a − j )Γ( µs − a ) , E j = ∞ (cid:88) k =1 C k d − kj . (B.8)Then using (B.4) and (B.8) in (B.2) and multiplyingthe result by Γ( µs − a ) we finally get the expansion (A.7). [1] K. S. Stelle, “Renormalization of higher-derivative quan-tum gravity,” Phys. Rev. D16 , 953–969 (1977).[2] G. W. Gibbons, “Quantum field theory in curved space-time,” in
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