We derive explicit forms for the three integrals used in Kim and Wand [3] and
Wand, Ormerody, Padoan and Fr¨uhwirth [7]. The explicit forms involve known special
functions for which in-built routines are available

Saralees Nadarajah

Tibor Pogany

English

Hrčak - Portal of scientific journals of Croatia

MATHEMATICAL COMMUNICATIONS 101Math. Commun. 26(2021), 101–105Explicit forms for three integrals in Wand et al.∗Tibor K. Pogány1,2,†and Saralees Nadarajah31 Faculty of Maritime Studies, University of Rijeka, Studentska 2, HR-51 000 Rijeka,Croatia2 Institute of Applied Mathematics, Óbuda University, 1034 Budapest, Bécsi út 96/b,Hungary3 Department of Mathematics, University of Manchester, Manchester M13 9PL, UKReceived September 3, 2020; accepted January 13, 2021Abstract. We derive explicit forms for the three integrals used in Kim and Wand [3] andWand, Ormerody, Padoan and Frühwirth [7]. The explicit forms involve known specialfunctions for which in-built routines are available.AMS subject classifications: Primary 60E10; Secondary 33C60, 62G32Key words: Grünwald–Letnikov fractional derivative, Kummer’s confluent hypergeomet-ric function, parabolic cylinder function, Tricomi confluent hypergeometric function1. IntroductionArticle [3] gave an explicit form of expectation propagation for a simple statisticalmodel, while [7] studied mean field variational Bayes for elaborate distributions.Their studies involved the following three integrals. The first integral isA(p, q, r, s, t, u) =∫Rxp exp{qx− rx2}(t+ sx+ x2)u dx (1)for p ≥ 0, q, s ∈ R, r, u > 0 and s2 < 4t; see equation (2.1) on page 552 of [3]. Thesecond integral isB(p, q, r, s, t, u) =∫Rxp exp {qx− rex − s ex/ (t+ ex)}(t+ ex)u dx (2)for p, s ≥ 0, q ∈ R and r, t, u > 0; see equation (2.1) on page 552 of [3]. The thirdintegral isI(p, q, r, s) =∫Rxp exp{qx− rx2 − s e−x}dx (3)for p ≥ 0, q ∈ R and s, r > 0; see page 851 in [7].∗The research of Tibor K. Pogány has been supported in part by the University of Rijeka, Croatia;project codes uniri-pr-prirod-19-16 and uniri-tehnic-18-66.†Corresponding author. Email address: poganj@pfri.hr (T. K. Pogány),Saralees.Nadarajah@manchester.ac.uk (S. Nadarajah)http://www.mathos.hr/mc c©2021 Department of Mathematics, University of Osijek102 T. K. Pogány and S. NadarajahBoth publications [3] and [7] provided explicit forms for none of the three inte-grals. In this note, we provide explicit forms for all three integrals. The forms involveknown special functions and in-built routines for computing them are available inthe literature.The organization of this note is the following. Section 2 gives two explicit formsfor (1). Section 3 gives one explicit form for (2). Section 4 gives two explicit formsfor (3).2. The integral AWe present an approach in which we use the Grünwald-Letnikov fractional derivative.The Grünwald-Letnikov fractional derivative of order ν with respect to the argumentx of a suitable function f is defined by [6]Dνx[f ] := limh↓01hν∞∑m=0(−1)m(νm)f (x+ (ν −m)h) ,where h ↓ 0 means that in approaching zero h remains positive. As is well-known(see, for example, [4]), the Grünwald-Letnikov fractional derivative Dνx of order ν ofthe exponential function isDνx [eαx] = ανeαx. (4)Firstly, consider the well-known integralI (α, β) =∫Reαx−βx2dx =√πβexp{α24β}, <(β) > 0.Obviously, we haveDpα [I (α, β)] =∫Rxpeαx−βx2dx.On the other hand,Dpα [I (α, β)] =∫ ∞0xpeαx−βx2dx+∫ 0−∞xpeαx−βx2dx =: I+ + I−.Using equation (13), page 313 of [1],I+ = (2β)−p+12 Γ(p+ 1) exp{α28β}D−p−1(− α√2β),where Dµ(·) denotes the parabolic cylinder function of order µ (see, for example,[2]), and the constraint p + 1 > 0 should be satisfied (which is definitely a weakerassumption than the assumed p > 0 by Kim and Wand [3]). Accordingly, we haveI− = (−1)p(2β)−p+12 Γ(p+ 1) exp{α28β}D−p−1(α√2β).Explicit forms for three integrals in Wand et al. 103Therefore,Dpα [I (α, β)] = (2β)−p+12 Γ(p+ 1) exp{α28β}×[D−p−1(− α√2β)+ (−1)pD−p−1(α√2β)].Now, introduce a parameter a > 0 and specify α = q − as ∈ R and β = r + a, thelatter evidently positive; therefore nothing harms the assumption on the parameterspace of (p, q, r, s, t). Considering now the integrale−atDpq−as [I (q − as, r + a)] =∫Rxpeqx−rx2−a(t+sx+x2) dx,we conclude thatA(p, q, r, s, t, u) = (−1)u D−ua[e−atDpq−as [I (q − as, r + a)]] ∣∣∣a=0.This formula proves the following result.Proposition 1. For all p ≥ 0, q, s ∈ R, r, u > 0 and s2 < 4t, we haveA(p, q, r, s, t, u) = eiπu 2−p+12 Γ(p+ 1) lima↓0D−ua[(r + a)−p+12 exp{(q − as)28(r + a)− at}×{D−p−1(as− q√2(r + a))+ eiπp D−p−1(q − as√2(r + a))}].We now present another approach to calculating integral A. Kummer’s (or con-fluent hypergeometric) function series definition is1F1(a, c, z) =∑n≥0(a)n(c)nznn!.The parabolic cylinder function Dν is expressible in terms of the Tricomi confluenthypergeometric function, viz.U(a; c; z) =Γ(1− c)Γ(1 + a− c) 1F1(a; c; z) +Γ(c− 1)Γ(a)z1−c 1F1(1 + a− c; 2− c; z),as (see equations (2) and (4) on page 117 of [2])Dν(z) = 2ν2 e−z24 U(−ν2,12,z22)= 2ν−12 z e−z24 U(1− ν2,32,z22), (5)where in both cases −π < 2 arg(z) ≤ π. Applying the first formula in (5), we obtainProposition 2. For the same parameter space as in the previous proposition, wehaveA(p, q, r, s, t, u) = 2−(p+1) eiπu(1 + eiπp)Γ(p+ 1)× lima↓0D−ua[e−at(r + a)p+12U(p+ 12,12,(q − as)24(r + a))].104 T. K. Pogány and S. Nadarajah3. The integral BWe apply again the Grünwald-Letnikov fractional derivative (4) of the exponentialfunction. Firstly, we eliminate the denominator and the power xp in the integrand,namely,B(p, q, r, s, t, u) = (−1)u DusDpq−u[∫Rexp{(q − u)x− rex − s ext+ ex}dx].Now, we transform the inner integral I(s) and use the Maclaurin expansion withrespect to s of the appropriate exponential term to obtainB(p, q, r, s, t, u)= (−1)u Dus[e−s Dpq−u[∫Rexp{(q − u)x− rex + s tt+ ex}dx]]= (−1)u Duse−s Dpq−u∑n≥0(st)nn!∫Re(q−u)x−rex(t+ ex)n dx= (−1)u∑n≥0tnn!Dus[e−s sn Dpq−u[∫R+yq−u−1e−ry(t+ y)ndy]]= (−1)u∑n≥01n!Dus[e−s sn Dpw[Γ(w) tw U(w,w + 1− n, r t)]w=q−u], (6)where in (6) the Laplace transform formula (see equation (2.1.3.1) on page 18 of [5])∫R+e−px xα−1(x+ z)ρdx = Γ(α) zα−ρ U(α, α+ 1− ρ, p z)was used, which holds for all <(α) > 0, <(p) > 0 and | arg(z)| < π. This proves thefollowing result.Proposition 3. For all p, s ≥ 0, q ∈ R, r, t, u > 0 and q > −u, we haveB(p, q, r, s, t, u) = eiπu∑n≥01n!Dus[e−s sn× Dpq−u[Γ(q − u) tq−u U(q − u, q − u+ 1− n, r t)]]. (7)Unfortunately, our method holds true for q − u > 0 only since Γ(q − u) in (7).4. The integral IThis time it is enough to split the integration domain into positive and negative realsand take the Maclaurin expansion in both sub-integrals of the exponential expressionexp {−se−x}. We obtain the followingExplicit forms for three integrals in Wand et al. 105Proposition 4. For all p ≥ 0, q ∈ R, s, r > 0, we haveI(p, q, r, s) = Γ(p+ 1)(2r)p+12∑n≥0(−s)nn!{D−p−1(n− q√2r)+ eiπp D−p−1(q − n√2r)}. (8)Moreover, the following computable series representation holds:I(p, q, r, s) = Γ(p+ 1)(4r)p+12(1 + eiπp) ∑n≥0(−s)nn!U(p+ 12,12,(q − n)24r).Expression (8) could be deduced by some aspects of the discussion on pages 851-852 of [7]. However, there the authors’ approach to quadratures for I(p, q, r, s) wascompletely different.AcknowledgementThe authors are indebted to anonymous referees for their careful reading of an earlierversion of the manuscript and their valuable comments and corrections.References[1] A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of IntegralTransforms, volume I, McGraw-Hill, New York, 1954. (Based, in part, on notes leftby Harry Bateman).[2] A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Higher Transcen-dental Functions, volume II, McGraw-Hill, New York, 1954. (Based, in part, onnotes left by Harry Bateman).[3] A. S. I. Kim, M. P. Wand, The explicit form of expectation propagation for a simplestatistical model, Electron. J. Stat. 10(2016), 550–581.[4] M. D. Ortigueira, A coherent approach to non-integer order derivatives, Signal Pro-cess. 86(2006), 2505–2515.[5] A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and series. DirectLaplace transforms, volume 4, Gordon and Breach, New York, 1992.[6] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives:theory and applications, Gordon and Breach, New York, 1993.[7] M. P. Wand, J. T. Ormerody, S. A. Padoan, R. Frühwirth, Mean field varia-tional Bayes for elaborate distributions, Bayesian Anal. 6(2011), 847–900.

Explicit forms for three integrals in Wand et al.

Abstract

Similar works

Full text

Available Versions

Hrčak - Portal of scientific journals of Croatia