NEW ASYMPTOTIC EXPANSION AND ERROR BOUND FOR STIRLING FORMULA OF RECIPROCAL GAMMA FUNCTION

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1 M athematical Inequalities & Applications Volume, Number 4 8), doi:.753/mia NEW ASYMPTOTIC EXPANSION AND ERROR BOUND FOR STIRLING FORMULA OF RECIPROCAL GAMMA FUNCTION PEDRO J. PAGOLA Communicated by N. Elezović) Abstract. Studying the problem about if certain probability measures are determinate by its moments [4, 8, ] is useful to know the asymptotic behavior of the probability densities for large values of argument. This requires, previously, the knowledge of the asymptotic epansion of reciprocal Gamma function /Γz) when Rz is large and Iz is fied [8]. Then, the well known Stirling formula for large z of the Gamma function Γz) or its reciprocal /Γz) is not appropriate for this problem. So, the main aim of this paper is to obtain a new asymptotic epansion for reciprocal Gamma function valid for large Rz and establish a new eplicit error bound for the first term of this epansion, that is, the Stirling formula.. Introduction It s well known that, when z in the sector argz < π, the reciprocal Gamma function has the following asymptotic epansion Γz) z z+ e z π n= γ n z n, ) where γ s are the Stirling coefficients. This epansion is frequently used in standard tetbooks to illustrate the Saddle Point method see for eample [9, p. 69]). The first term of this epansion is the well known Stirling formula for reciprocal Gamma function: Γz) z z+ e z. π In [3], Boyd derived error bounds for the remainder of epansion ) by using a resurgence formula for the Gamma function arising in a general theory for comple Laplace-type integrals developed by Berry and Howls []. More recently in [7], G.Nemes has derived a better error bound for the remainder in ) when Rz > that modifies and improves Boyd s formula: Γz) = ez z z+/ π ) N γ n n= z n + R N z), ) Mathematics subject classification ): 33B5, 4A6. Keywords and phrases: Reciprocal gamma function, asymptotic epansions, error bounds. c Ð, Zagreb Paper MIA

2 958 P. J. PAGOLA with γn R N z) z N + γ ) { π N+ cscθ) if 4 < θ < π, z N+ if θ 4 π. On the other hand, in the study of the problem of whether the random variable t A t := e B s ds is determinate by its moments [8, ] B s, s) denotes a real valued Brownian motion starting from ) is convenient to know the asymptotic behavior of the probability densities f t ) of A t e π 8t f t ) = PA t d) = π t 3 e cosh u) e u t cos π u ) coshu)du,, t >, t 3) for large positive. The knowledge of the eact behavior of these densities when is essential to decide if the measures dτ t = f t )d are determinate by its moments. It is shown in [8] that f t ) may be written in terms of Φ n ) := gy) Γ n+ dy 4) + +iy) where n N and gy) verifies certain conditions to allow the integral converges. Therefore, we need to analyze the asymptotic behavior of Φ n ) when. To this end we can replace, in the right side of 4), the reciprocal Gamma function by it asymptotic epansion or at least the first term) and interchange sum and integral. In order to make the calculations as simple as possible, we need an asymptotic epansion of reciprocal Gamma function when real part of argument in inverse powers of here, formula ) does not work). In order to be controlled the error, we also need to know an eplicit epression for the remainder of this asymptotic epansion. So, this work has two main purposes: on the one hand, in section, by using a modified Saddle Point method [6], we derive a new asymptotic epansion of /Γz) for large positive Rz > with an etra property: the eplicit formula for its coefficients. Although for our purpose to know the behavior of Φ n ) is enough with the first term, we calculate the complete epansion because is new in the literature. On the other hand, in section 3, we establish a new error estimate for the first term of this epansion, that is, the Stirling formula for large positive Rz >. An asymptotic epansion of /Γz) for large Rz We start considering the contour integral representation of reciprocal Gamma function [, eq. 5.9.]: Γz) = +) e s s z ds, 5) πi

3 ASYMPTOTIC AND BOUND FOR RECIPROCAL GAMMA FUNCTION 959 where the integration contour C begins at, circles the origin once in the positive direction, and returns to as is shown in [, fig. 5.9.]. We write z = +α + iy and change the integration variable s = t ; we obtain: Γα + +iy) = e α iy +) e ft) gt)dt, 6) πi with ft) = t logt) and gt)= t α iy. In order to derive an asymptotic epansion of 6) for large positive we apply the modified Saddle Point method see [6, Theorem ] for more details) instead the standard Saddle Point: unlike the standard method, the modified provides eplicit formulas for the coefficients. We have that the unique saddle point of the phase function ft) is t = and f) =, f ) =, f ) = and f ) ; also g) =. Then, following the idea introduced in [6], we rewrite the phase function ft) in the form of a Taylor polynomial at the saddle point t = ft) = t ) + f t). From [6], we know that it is not necessary to compute the steepest descent paths of ft) at t, but only the steepest descent paths of the quadratic part of ft) at those points, that are simpler: they are nothing but straight lines. In this case, the steepest descent path of the quadratic part of ft) at t is the straight line Γ = { + iz z, )}. From [6] we also know that it is not necessary to deform the path C into Γ. It is enough to deform C into a new integration Γ path that contains a portion of Γ that includes t. In this case we define Γ = Γ ε Γ t with Γ ε = {z ± iπ z,]} and Γ t = {+iz z [ π,π]} In order to apply the method introduced in [6], we need to show that the contribution of the integral on the paths Γ ε is negligible compared with the contribution of Γ t. We have that R ft)) = t lnt + π ) is increasing on,] and then attains its maimum at t =, that is, R ft)) M with M = ln+π ) >. Therefore Γ ε e ft) gt)dt = Oe M ). 7) On the other hand, and using the standar Saddle Point method, it s also easy to prove that ) e ft) π gt)dt gt ) Γ t f t ) e ft) = O. 8) Therefore, apart from the eponentially negligible term 7), the integral 6) over the path C equals the integral over the path Γ t : +) π e ft) gt)dt e ft) gt)dt = e f+it) g+it)dt Γ t π

4 96 P. J. PAGOLA Now, we can split the integrand e ft) gt) = e t ) e f t) gt) with f t) = ft) t ). Putting the Taylor epansion of e f t) gt) at t into above integral and interchanging sum by integral we can obtain the desired asymptotic epansion see [6] for more details): Γα + +iy) e / α iy π n= with a n ) := n k= n k=n ) k k k! k ) k+n a n+k) ) Γk+n+/) Γ/) n k j= ) α iy j n k j j!. ) k+n 9) Remark that, in the second sumatory of 9), when k takes the value of n with n natural must be understood as zero k = ). The following tables illustrate the approimation of /Γα + + i y) supplied by the epansions given by formula 9) for diferents values of α, and y n Table : Relative error for y =, α = / and several values of and the number of terms n of the approimation given by 9). n Table : Relative error for y =, α = 3/ and several values of and the number of terms n of the approimation given by 9).

5 ASYMPTOTIC AND BOUND FOR RECIPROCAL GAMMA FUNCTION Error bound for Stirling formula We start with the following lemma such will be usefull later. LEMMA. Let f : R C be a twice diferentiable function in a,b) R. Then there eist c,c a,b) such that fb) fa) b a) f c ) ) fb) fa) b a) f a) + Proof. Put z := fb) fa) and define ϕt) = z ft) a t b) b a) f c ) ) where denotes the scalar product in the Hilbert space structure of C. Clearly ϕt) is a real-valued function on [a,b] which is two times differentiable in a,b). Using the Taylor s theorem with Lagrangian remainder a) degree Taylor polynomial ϕb) = ϕa)+b a)ϕ c) ϕb) ϕa) = b a)z f c) for some c a,b). On the other hand ϕb) ϕa) = z fb) z fa) = z z = z The Cauchy Schwarz inequality gives z = b a) z f c) b a) z f c) Henze z b a) f c), the ) inequality desired. b) degree Taylor polynomial ϕb) = ϕa)+b a)ϕ b a) a)+ ϕ c) ϕb) ϕa) = b a)z f b a) a)+ z f c) for some c a, b). The Cauchy Schwarz inequality now gives Henze z b a) f a) + z b a) z f b a) a) + z f c) b a) z f b a) a) + z f c) b a) f c), the ) inequality desired.

6 96 P. J. PAGOLA We are now ready to state the main result of this section THEOREM. Main) Let >, α, y R. Then: Γ+α + iy) = e α iy+/ + Rα,,y) ) ) π where the remainder Rα,, y) verifies: Rα,,y) Hα,y) with Hα,y) := e y π α + iy +α + α + iy + ). 3) Proof. Our starting point is the Binet representation [9, eq. 3.] of the reciprocal Gamma function: Γz) = z z+/ e z e µz), π where µz) can be epressed: with µz) := A first substitution z = α + +iy gives ): t t e t + t ) e zt dt. 4) Γα + +iy) = e α iy+/ [ + Rα,,y) ], 5) π Rα,,y) = + α + iy ) α iy+/ e α+iy e µα++iy). Defining X := / we have Rα,X,y) Rα,/X,y) = +Xα + iy)) α X iy+/ e α+iy e µ X +α+iy). In order to derive 3) we apply the property ) to the function Rα,X,y) in the range [,X], considering α and y fied and Rα,,y) = lim Rα,X,y) = X + e α iy e α+iy e =, to find: Rα,X,y) X R X α,x,y) with C,X) 7) X=C Now, we write R α,x,y) in the form X R X α,x,y) = +Xα + iy)) α X iy / e α+iy e µ X +α+iy) [ α + iy log+xα + iy)) Xα + iy) ++Xα + iy)) X ++Xα + iy))µ α + )] X + iy 6)

7 ASYMPTOTIC AND BOUND FOR RECIPROCAL GAMMA FUNCTION 963 and finally, we are going to find a bound for the modulus of each factor in the above epression for X > : a) +Xα + iy)) α X iy / = +Xα) +Xy) ) X α+ 4 e yθ with θ = Arg+Xα + iy)) π, π ). We have +Xα) +Xy) and X α + α ). Then 4 +Xα + iy)) α X iy / e π y /. b) e α+iy = e α. c) We have the inequality e µα+ X +iy) from [7, pp. 8]. d) We denote GX) = +Xα + iy))log+xα+ iy)) Xα + iy)) and applying ) in the range [,X] GX) = GX) G) X G C) = X α + iy + α + iy)c +α + iy)c with C,X) we obtain + iy)) Xα + iy) +Xα + iy))log+xα α + iy. X e) Finally ) +Xα + iy))µ X + α + iy = X t t e t + t ) ) X + α + iy e α+ X +iy)t dt Integrating by parts we find that the above integral reads X t e t ) e t ) e α+ X +iy)t dt X t e t e t ) e α+ X )t dt. Using the inequality [5, eq. 8)]: t e t e t ), t, ),

8 964 P. J. PAGOLA we find X t e t e t ) e α+ X )t dt e α+ X )t dt X = e ) α+ X )t +Xα) = +Xα). Using the bounds a) e) and replacing X by in 7) we obtain the desired result 3). As a direct consequence of Theorem, taking the particular case α =, we can obtain the Stirling formula for reciprocal Gamma function /Γz) when real part of z = + i y is large and establish a new eplicit error bound with COROLLARY. For >, Γ+iy) = e iy+/ + R,y) ), y R, 8) π R,y) ) e y π y + y +. 9) If we set y = in Corollary, that is, the argument of reciprocal Gamma function is real and positive, we obtain the well known formula Γ) = e +/ + R,) ) with R,) π. Acknowledgements. This research was supported by the Spanish Ministry of Economía y Competitividad, project MTM The Universidad Pública de Navarra is acknowledged by its financial support. R E F E R E N C E S [] R. A. ASKEY, R. ROY, NIST Handbook of Mathematical Functions: Chapter 5, Gamma Function, NIST and Cambridge Univ. Press, New York,. [] M. V. BERRY AND C. J. HOWLS, Hyperasymptotics for integrals with saddles, Proc. Roy. Soc. London 434, 99), [3] W. G. C. BOYD, Gamma function asymptotics by an etension of the method of steepest descents, Proc. Roy. Soc. London 447, 994), [4] P. HÖRFELT, The moment problem for some wiener functionals:corrections to previous proofs with an appendi by H. L. Pedersen), J. Appl. Prob. 4, 5),

9 ASYMPTOTIC AND BOUND FOR RECIPROCAL GAMMA FUNCTION 965 [5] A. Q. LIU, G. F. LI, B. N. GUO, F. QI, Monotonicity and logarithmic concavity of two functions involving eponential function, Internat. J. Math. Ed. Sci. Tech. 39, 5 8), [6] J. L. LÓPEZ, P. J. PAGOLA AND E. PÉREZ SINUSÍA, A systematization of the saddle point method. Application to the Airy and Hankel functions, J. Math. Anal. Appl. 354, 9), [7] G. NEMES, Error bounds and eponential improvements for the asymptotic epansions of the Gamma function and its reciprocal, Proc. Roy. Soc. Edinburgh 45, 3 5), [8] P. J. PAGOLA, Asymptotic behaviour of the density function of the integral of a geometric Brownian motion, Submitted. [9] N. M. TEMME, Special Functions: an Introduction to the Classical Functions of Mathematical Physics, John Wiley and Sons, New York, 996. [] M. YOR, On some eponential functionals of Brownian motion, Adv. Appl. Prob. 4, 99), Received August 3, 6) Pedro J. Pagola Dpto. de Estad istica, Informática y Matemáticas Universidad Pública de Navarra pedro.pagola@unavarra.es Mathematical Inequalities & Applications mia@ele-math.com