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If x 1 we may take L to be the loop shownin Figure 2, enclosing the poles.The residue of the integrand at the pole s a n is( 1)n1·· x a n .n!Γ (b a n) x(D a)u(x)]u(x) 0.(23)1( 1)n sin(b a) Γ (1 a b n)Γ (b a n)πΓ (1 a b n)n ( 1)Γ (b a)Γ (1 a b)(1 a b)n ( 1)n.Γ (b a)Therefore, for x 1 our solution isL a 1[D(D b2 b1 ) x(D a 1 b1 )] v1 (x) 0,and similarly with b1 and b2 interchanged. Thesolution (25) must be a linear combination ofthese two. The coefficients can be determinedby looking at the residues of the integrand atthe pole s 1 b1 and at the pole s 1 b2 :Γ (a 1 b1 ) Γ (b1 b2 ) and Γ (a 1 b2 )Γ (b2 b1 )respectively. The result isRe s 0 a 2One possibility for a solution is(25)Z1u(x) Γ (a s) Γ (1 b1 s) Γ (1 b2 s) xs ds.2π i LAnother way to construct a solution is to look foru1 (x) x1 b1 v1 (x). The corresponding equationfor v1 iswhich has a solution in standard form:!a 1 b1(26)v1 (x) 1 F1x ,b2 1 b1 Xx a(1 a b)n 1 nu(x) Γ (b a) n 0n!x b a 1x a1 1 .Γ (b a)x Before proceeding to a discussion of the generalcase, we look briefly at the second order equation[(D b1 1)(D b2 1)By (18),(24)Representation by Generalized Hypergeometric Functions aIm s 0u(x) Γ (a 1 b1 ) Γ (b1 b2 ) (27)Figure 2. The contour L for (22) when x 1.Again for 0 x 1, we may integrate over acontour that encloses the poles going to the right,as in Figure 1. For x 1 we choose a contourgoing to the left, as in Figure 2, but enclosing nopoles. A calculation as in (23), (24), together withCauchy’s theorem, gives 1 b x(1 x)b a 1 ,0 x 1;Γ (b a)u(x) 0, x 1.An interesting special case is a 0, b 1, whichgives a step function:1 F11 b2 x1 F1!a 1 b2x .b1 1 b2There is one notable feature of this particularcombination of the two standard solutions v1 , v2 .Each of the vj has exponential growth as x ,but (27) does not. In fact, the integrand decaysexponentially in both directions on vertical lines,so we may deform the contour L into one thatconsists of a small circle around s a, togetherwith a vertical line Re s c Re a. The leadingterm can be computed from the residue at s a 0:(28)u(x) Γ (a 1 b1 ) Γ (a 1 b2 ) x a as x .Thus this integral form singles out the unique (upto a multiplicative constant) linear combinationof the two standard solutions that has algebraicbehavior at .General Considerations about Kernels andContoursx 6 0,where H is the Heaviside function.We leave it to the reader to consider two morepossibilities, with respective kernels Γ (a s) Γ (1 b s) and 1/[Γ (1 a s) Γ (b s)].August 2013 x!a 1 b1xb2 1 b1 Γ (a 1 b2 ) Γ (b2 b1 )As in the case p 0, we may change theintegrand, leading, for example, toZ1Γ (1 b s) su(x) x ds.2π i L Γ (1 a s)u(x) H(1 x ),1 b1For each case of equation (12), the function Φ thatsatisfies the recursion equation (15) can and willbe taken to be a quotient of products of gammafunctions in the form Γ ( cj s) or Γ (dj s).Notices of the AMS869

Therefore the poles of Φ, if any, will consist offinitely many sequences of points:(a) cj , cj 1, cj 2, . . . or (b) dj , dj 1, dj 2, . . . .The translation condition requires that any suchsequence must lie on one side of the contourL. The contour L is always chosen to separatethe sequences (a) of poles that go to the leftfrom the sequences (b) that go to the right. (Atechnical remark: We do not actually need thetranslation condition to deduce that these solutionsof (15) give rise to solutions of the generalizedhypergeometric equation; the condition that Lseparate the sequences of poles in this way issufficient.)We consider three types of contours L:L LI : beginning at i , ending at i ;L L : beginning and ending at , orientedclockwise;L L : beginning and ending at , orientedcounterclockwise.In order to see which contours are available in agiven case, we note here some basic facts aboutasymptotics.First, Stirling’s formula,(29)s 2π z z1Γ (z) 1 Oas z ,zezis valid uniformly in the right half-plane Re z 0.It implies that Γ grows faster than exponentiallyto the right along any horizontal line and decaysexponentially in both directions along any verticalline in the right half-plane.Second, the reflection formula (18), combinedwith (29), gives the asymptotic behavior in theleft half-plane Re z 0: Γ decays faster thanexponentially to the left along any horizontalray other than the negative real axis and decaysexponentially in either direction along any verticalline in the left half-plane as well. These facts implythat, for any fixed a, b,Γ (a s) s a bΓ (b s)(30)as s along any ray that avoids the zeros and poles ofthe quotient.Meijer G–functionsHere we consider the general hypergeometricequation (12) but with a difference from thestandard notation: the indices {bj }, {aj } arereplaced by the reflections {1 bj }, {1 aj }: qpYY(31) (D bj ) x(D 1 aj ) u(x) 0.j 1870j 1We assume again that p q. The previous considerations lead us to a solution which, in Meijer’s0,pnotation, is Gp,q :!a1 , . . . , a p0,pGp,qxb1 , . . . , b q(32)Z Qp1j 1 Γ (1 aj s) sQq x ds.2π i L j 1 Γ (1 bj s)Meijer [10] introduced a family of solutions ofequation (31), denoted by!a10 , . . . , ap0m,nm n pGp,q( 1)x,b10 , . . . , bq00 m q, 0 n p,where the {aj0 } and {bj0 } are permutations of theoriginal indices {aj } and {bj } respectively. Theupper index m indicates that the first m factorsin the denominator of the integrand have beenchanged to factors Γ (bj s) in the numerator andthe last p n factors in the numerator have beenchanged to factors Γ (aj s) in the denominator.This results in a total of m n factors in thenumerator.Taking into account the invariance of theintegrand under permutations of the indices ajand bj (separately) in the numerator and also inthe denominator, one can obtain 2p q solutionsof (31). However they are not necessarily distinct.If p q, then the 2p solutions with m 0 vanishidentically, as in our first solution (17) in the caseq 1, p 0. If p q, only the solution withm n 0 vanishes identically.The choice of contours depends first on p and q.If p q (resp. p q), the gamma function kernel Φhas faster than exponential decay to the right (resp.left) on horizontal lines. Thus, for p q, one cantake a contour L L as in Figure 1, and for p qa contour L L as in Figure 2. In either casethe contour is chosen to separate the sequences ofpoles going to from those going to .If there are more gamma factors in the numeratorthan in the denominator, i.e., m n (p q)/2,then Φ decays exponentially along vertical linesand the contour can be deformed into LI from i to i . If m n (p q)/2, Φ has algebraicbehavior at by (30), and again the contour canbe deformed, but this may require interpreting theintegral in the sense of distributions.When p q, the kernel Φ itself has algebraicbehavior as s , just as in the case p q 1above, so the choice of contour differs accordingto whether 0 x 1 or x 1.A striking advantage of this method of producingsolutions is the ease of finding solutions withprescribed behavior as x 0 or x . Assumingagain that p q, the standard solutions of theform xbj p Fq 1 that are xbj [1 (O(x)] as x 0 areNotices of the AMSVolume 60, Number 7

multiples of the G–functions with m 1. Solutionsthat are xaj 1 [1 O(x 1 ] as x are multiplesof the G–functions with n 1, m 1 (p q)/2.The Mellin Transform and the ConvolutionTheoremThe Mellin transform is the multiplicative versionof the Laplace transform. We normalize it asZ dxMf (s) f (x) x s.x0The inverse transform is thenZ c i 1M 1 g(x) g(s) xs ds.2π i c i This suggests, correctly, that if a G–functionZ1G(x) Φ(s) xs ds2π i Lhas (in some suitable sense) a Mellin transform,then that transform is the kernel Φ. As an exercise,one can check this with (24): change the variable ofintegration to y 1/x to find that the Mellin transform is a beta function and therefore expressiblein terms of gamma functions.It is not difficult to verify that the Mellintransform takes multiplicative convolution to aproduct: in the notation of (5),M(G1 G2 ) MG1 · MG2 .If G1 and G2 are G–functions with MGj Φj , thenthe product Φ1 Φ2 is itself a quotient of productsof gamma functions. Ifmj ,njGj Gpj ,qj ,j 1, 2,j 1j 1p q.As we noted earlier, under the changes of variablesx 1/x the operator D goes to D. Therefore ifwe set ( 1)p qu(x) v,xequation (33) is equivalent to equation (31). Thisexplains the departure in (31) from the usual indexnotation (12).We cannot help noting that the remark is oflimited explanatory value (apart from the questionof notation), since p Fq 1 , with p q and a givenset of indices, denotes a single solution to (9), not2p q solutions.Meijer’s original definition in 1936 [10] is thegeneral version of (27): represent the functionas a linear combination of the standard basisof solutions of (31). (This definition tacitly rulesout the cases with index m 0, which give thetrivial solution.) Only in [11] did Meijer presentthe integral formulation above. The 1936 papertranslates many formulas for special functions ofthe formf (x) h(x) k(x)orf (x) h1 (x) k1 (x) h2 (x) k2 (x)into a form likeG(x) G1 (x) G2 (x),then G1 G2 is a G–function of typem m2 ,n1 n2Gp11 p2 ,q1 q2 .Some bookkeeping will identify the indices {aj },{bj } for G in terms of the indices {aj,1 }, {bj,1 } forG1 and {aj,2 }, {bj,2 } for G2 .This explains the convolution result (5). Weleave verification of the closure under (1), (2), (3),and (4) to the reader.We note in passing that the most commonlyused integral transforms, such as the Laplace,Hankel, and fractional integral transforms, as wellas the Mellin transform itself, can be viewed asspecial cases or variants of the general G–functiontransformZ T f (x) G(xt) f (t) dt.0Conversely, the range of the G–function transformcan be characterized in terms of the special casesjust mentioned; see [17].August 2013Discussion and HistoryThe one bit of motivation that we have found in theliterature is the remark in [4] that “the G–functionprovides an interpretation of the symbol p Fq whenp q 1.” Let us start from the equation pqYY(33) (D aj 1) x(D bj ) v(x) 0,and many integral formulas into the convolutionformG(x) G1 G2 (x).It appears that Meijer may simply have done somany calculations that he could see exactly whatlinear combinations would make these formulaswork out so neatly. (He may also have beenguided by known examples that yield specifiedasymptotics, as in the example (27) above.) Onlylater did Meijer establish the integral formulaslike (32) that are now taken as the definition [10].In a lengthy series of papers he explored furtherrelations with classical special functions, relationsamong the G–functions themselves, and questionsof asymptotics [12], [13]; see also [5].The idea to represent solutions of the GHGE asintegrals of the kind we have been discussing—atype now known as Mellin–Barnes integrals—goesback to Barnes in 1908 for the Euler–Gauss caseq p 2 [2] and to Mellin in 1910 for the generalNotices of the AMS871

MATHEM AT IC SAT TH EN AT I O N AL S EC U R IT Y A GE N CYFar from OrdinaryReferencesThe U.S. is under cyber attack, every minute of every day. That’swhy cyberspace has become today’s new front line. What youknow can make a difference at the National Security Agency.Whether it’s collecting foreign intelligence or preventing foreignadversaries from accessing U.S. secrets, you can protect thenation by putting your intelligence to work. Explore technologythat’s years ahead of the private sector. Plus exciting careerfields, paid internships, co-op and scholarship opportunities. Seehow you can be a part of our tradition of excellence and help thenation stay a step ahead of the cyber threat.KNOWINGMATTERSExcellent CareerOpportunities forExperts in the Following:Number Theoryn Probability Theoryn Group Theoryn Finite Field Theoryn Combinatoricsn Linear Algebra Plus otheropportunitiesnwhere intelligence goes to work Search NSA to DownloadU.S. citizenship is required. NSA is an Equal Opportunity Employer.87212BWNS-03 3.25x9.5.indd 1case [14]. Indeed, Mellin has a thorough discussionof the recursion equation (15) and its solutions.[1] G. E. Andrews, R. Askey, and R. Roy, Special Functions,Cambridge Univ. Press, Cambridge, 1999.[2] E. W. Barnes, A new development of the theory of thehypergeometric function, Proc. London Math. Soc. (2)6 (1908), 141–177.[3] M. Bertola, M. Gekhtman, and J. Szmigielski,Cauchy–Laguerretwo–matrixmodeland the Meijer–G random point field, 2012,http://arxiv.org/abs/1211.5369[math.PR].[4] A. Erdélyi, W. Magnus, F. Oberhettinger, and F.Tricomi, Higher Transcendental Functions, Vol. 1,Krieger, Melbourne, FL, 1981.[5] J. L. Fields, The asymptotic expansion of the MeijerG-function, Math. Comp. 26 (1972), 757–765.[6] D. Lichtblau, Symbolic definite Conferences/5832/.[7] Y. L. Luke, The Special Functions and Their Approximations, Vol. 1, Academic Press, New York,1969.[8] O. I. Marichev, Handbook of Integral Transform ofHigher Transcendental Functions, Wiley, New York,1983.[9] A. M. Mathai and R. K. Saxena, Generalized Hypergeometric Functions with Applications in Statistics andPhysical Sciences, Springer, Berlin, 1973.[10] C. Meijer, Über Whittakersche bezw. Besselsche Funktionen und deren Produkten, Nieuw Arch. Wisk. 18(1936), 10–39.[11], Multiplikationstheoreme für die Funktionm,nGp,q (z), Nederl. Akad. Wetensch., Proc. Ser. A 44(1941), 1062–1070.[12], On the G–functions, I–VII, Nederl. Akad.Wetensch., Proc. Ser. A 49 (1946), 344–356, 457–469,632–641, 765–772, 936–943, 1063–1072, 1165–1175.[13], Expansion theorems for the G–function, I–XI,Nederl. Akad. Wetensch., Proc. Ser. A 55 (1952), 369–379, 483–487; 56 (1953), 43–49, 187–193, 349–357,57 (1954), 77–82, 83–91, 273–279; 58 (1955), 243–251,309–314; 59 (1956), 70–82.[14] Hj. Mellin, Abriss einer einheitlichen Theorie derGamma—und der hypergeometrischen Funktionen,Math. Ann. 68 (1910), 305–337.[15] F. Olver, D. Lozier, R. Boisvert, and C. Clark,NIST Handbook of Mathematical Functions, CambridgeUniv. Press, Cambridge, 2010; NIST Digital Library ofMathematical Functions, http://dlmf.nist.gov.[16] A. P. Prudnikov, Yu. A. Brychkov, and O. I.Marichev, Integrals and Series, 5 vols., Gordon andBreach, Newark, NJ, 1986–1992.[17] P. G. Rooney, On integral transforms with G–functionkernels, Proc. Royal Soc. Edinburgh A 93 (1982/83),265–297.[18] N. M. Temme, Special Functions, an Introduction tothe Classical Functions of Mathematical Physics, Wiley,New York, 1996.[19] S. Wolfram, Festschrift for Oleg Marichev, specialfunctions/intro.html.[20] Wolfram Research, MeijerG, ns/MeijerG/.Notices of the AMS6/25/13 9:22 AMVolume 60, Number 7

Meijer G-Functions: A Gentle Introduction Richard Beals and Jacek Szmigielski T he Meijer G-functions are a remarkable family Gof functions of one variable, each of them determined by finitely many indices. Although each such func-tion is a linear combination of certain special functions of standard type, they seem not