HYPERDIRE: HYPERgeometric functions DIfferential REduction: MATHEMATICA based packages for differential reduction of generalized hypergeometric functions pFq, F1,F2,F3,F4
HYPERDIRE is a project devoted to the creation of a set of Mathematica based programs for the differential reduction of hypergeometric functions. The current version includes two parts: one, pfq, is relevant for manipulations of hypergeometric functi…
Authors: Vladimir V. Bytev (Dubna, JINR), Mikhail Yu. Kalmykov (Hamburg U.
DESY 13–071 ISSN 0418–9 833 Ma y 2013 HYPERDIRE HYPERgeometric functions DIfferen t ial REdu c tion: MA THEMA TICA based p ack ages for differen t ial reduction of generalized hyp ergeometric functions p F p − 1 , F 1 , F 2 , F 3 , F 4 Vladimir V. Bytev a,b , Mikhail Yu. Kalmyk o v a,b , Bernd A. Kniehl a a I I. Institut f ¨ ur Theoretisc he Ph ysik, Univ ersit¨ at Ham burg, Lurup er Chaussee 149, 22761 Hamburg, German y b Join t Institute for Nuclear Researc h, 141980 Dubna (Mosco w Region), Russia Abstract HYPERDIRE is a pro j ect devo ted to the creation of a set of Mat hematica based programs for the differentia l reduction of h yp ergeometric functions. The current v er- sion includes tw o parts: one, pfq , is relev an t for manipu lations of h yp ergeometric functions p +1 F p , and the seco nd one, App ellF1F4 , for manipulations with App ell h yp er geometric functions F 1 , F 2 , F 3 , F 4 of tw o v ariables. P A C S num b ers: 02.30. Gp, 02.30.Lt, 11.15.Bt , 12.38. Bx Keyw ords: Hyp ergeometric fu n ctions; Diffe rentia l reduction PR OGRAM SUMMA R Y Title of pr o gr am : HYPERDIRE V ersion : 1.0.0 R ele ase : 1.0 .0 Catalo gue numb er : Pr o gr am obtaine d fr om https:/ /sites.google.c om/site/loopcalculatio ns/home : E-mail: bvv @jinr.ru Lic ens i n g terms : GNU General Public Licence Computers : a ll computers running Mathematica Op er ating systems : op erating systems running Mathematica Pr o gr ammin g language : Mathematica Keywor ds : Generalized Hyp ergeometric functions, App ell functions, F eynman integrals. Natur e of the pr oblem : Reduction of h ypergeometric functions p F p − 1 , F 1 , F 2 , F 3 , F 4 to sets of basis functions. Metho d of solution : Differen tial reduction R estriction on the c omplexity of the pr oble m : none T ypic al running time : Dep ending on the complexit y of problem. 2 LONG WRITE-UP 1 In tro d uction Multiple h yp ergeometric functions [1 – 4] pla y an imp or t a n t role in many bra nc hes of sci- ence. In par t icular, a large class of F ey nman diagrams are expressed in terms of Horn-type h yp ergeometric functions [5]. Let us consider a m ultiple series: H ( ~ γ ; ~ σ ; ~ x ) = ∞ X m 1 ,m 2 , ··· ,m r =0 Π K j =1 Γ ( P r a =1 µ j a m a + γ j ) Γ − 1 ( γ j ) Π L k =1 Γ ( P r b =1 ν k b m b + σ k ) Γ − 1 ( σ k ) ! x m 1 1 · · · x m r r , (1) with µ ab , ν ab ∈ Z , γ j , σ k ∈ C . The sequences ~ γ = ( γ 1 , · · · , γ K ) and ~ σ = ( σ 1 , · · · , σ L ) are called upp e r and lower para meters of the hy p ergeometric function, respective ly . Let ~ e j = (0 , · · · , 0 , 1 , 0 , · · · , 0) denote the unit v ector with unit y in its j th en try , a nd let us define ~ x ~ m = x m 1 1 · · · x m r r for an y integer m ulti-index ~ m = ( m 1 , · · · , m r ). Tw o functions of t yp e (1) with sets of parameters shifted b y unit y , H ( ~ γ + ~ e c ; ~ σ ; ~ x ) and H ( ~ γ ; ~ σ ; ~ x ), are related b y a linear differential op erato r : H ( ~ γ + ~ e c ; ~ σ ; ~ x ) = 1 γ c r X a =1 µ ca x a ∂ ∂ x a + γ c ! H ( ~ γ ; ~ σ ; ~ x ) ≡ U + [ γ c → γ c +1] H ( ~ γ , ~ σ , ~ x ) . (2) Similar relations a lso exist for the lo w er parameters: H ( ~ γ ; ~ σ − ~ e c ; ~ x ) = 1 σ c − 1 r X b =1 ν cb x b ∂ ∂ x b + σ c − 1 ! H ( ~ γ ; ~ σ ; ~ x ) ≡ L − [ σ c → σ c − 1] H ( ~ γ ; ~ σ ; ~ x ) . (3) The linear differen tial op erato r s U + γ c → γ c +1 , L − σ c → σ c − 1 are called the step-up and step-down op erators for the upp er and lo w er indices, resp ectiv ely . If additional step-do wn and step-up op erators U − γ c , L + σ c satisfying U − [ γ c +1 → γ c ] U + [ γ c → γ c +1] H ( ~ γ , ~ σ , ~ x ) = L + [ σ c − 1 → σ c ] L − [ σ c → σ c − 1] H ( ~ γ , ~ σ , ~ x ) = H ( ~ γ , ~ σ , ~ x ) ( i.e. , the inv erses of U + γ c , L − σ c ) are constructed, w e can combine these op erat o rs to shift the parameters of the hypergeometric function b y an y in teger. This pro cess of applying U ± γ c , L ± σ c to shift the parameters b y integers is called differen tial reduction of a h yp ergeometric function. In this w ay , t he Horn-type structure pr ovides an opp o r t unit y to reduce h yp ergeometric functions to a set of basis functions with parameters differing from the orig ina l v alues b y in teger shifts: P 0 ( ~ x ) H ( ~ γ + ~ k ; ~ σ + ~ l ; ~ x ) = P | k i | + P | l i | X m 1 , ··· ,m p =0 P m 1 , ··· ,m r ( ~ x ) ∂ ∂ x 1 m 1 · · · ∂ ∂ x r m r H ( ~ γ ; ~ σ ; ~ x ) , (4) 3 where P 0 ( ~ x ) and P m 1 , ··· ,m p ( ~ x ) are p olynomials with respect to ~ γ , ~ σ , and ~ x , and ~ k , ~ l are lists of in tegers. Algebraic relations b et w een the functions H ( ~ γ , ~ σ ; ~ x ) with parameters shifted b y integers are called con t iguous relations . The dev elopmen t of systematic tec hniques for the solution of con tiguous relations has a long history . It w as started by Gauss, who describ ed the reduction for the 2 F 1 h yp ergeometric function in 1823 [1]. Numerous pa p ers hav e since b een published [6 – 8] o n this problem. An alg orithmic solution was found b y T ak a y ama in Ref. [9], and those metho ds ha v e b een extended la ter in a series of publications [10, 11] (see also Refs. [12 – 14]). Let us recall that an y h yp ergeometric function can b e considered to b e the solutio n of a prop er system of partial differen tial equations (PDEs). In particular, for a Horn-type h yp ergeometric function, the syste m of PD Es can b e deriv ed from the co efficien ts of the series H = X ~ m C ( ~ m ) ~ x ~ m . In this case, the ratio of tw o co efficien ts can b e represen ted as a ratio of t w o p olynomials, C ( ~ m + e j ) C ( ~ m ) = P j ( ~ m ) Q j ( ~ m ) = Π K j =1 Γ ( P r a =1 µ j a m a + µ j a δ ai + γ j ) Γ ( P r a =1 µ j a m a + γ j ) Π L k =1 Γ ( P r b =1 ν k b m b + σ k ) Γ ( P r b =1 ν k b m b + ν k b δ bi + σ k ) , (5) so that t he Horn-type h yp ergeometric function satisfies the follow ing sy stem of differen tial equations: 0 = D j ( ~ γ , ~ σ , ~ x ) H ( ~ γ , ~ σ , ~ x ) = " Q j r X k =1 x k ∂ ∂ x k ! 1 x j − P j r X k =1 x k ∂ ∂ x k !# H ( ~ γ , ~ σ , ~ x ) , (6) where j = 1 , . . . , r . It w as p o in ted out in sev eral publications [15 – 17] that (i) the differential reduction algorithm, Eq . (4), can be applied to the reduction of F eynman diagrams to some subsets of basis h ypergeometric functions with w ell-kno wn analytical prop erties [1 5, 16]; (ii) the system of differen tial equations, Eq. (6 ), can b e also used for the construction of so-called ε expansions of h ypergeometric functions ab out rational v alues of parameters via the direct solution o f the systems of differen tial equations [17]. This is another motiv a t io n f or creating a pac k age for the manipulation of the parameters of Horn-type hypergeometric functions. The aim of this paper is to presen t the Mathematic a [18] base d pac k age HYPERD I RE for t he differen tial reduction o f the Horn-type hy p ergeometric function with arbitrary v alues of parameters to a set of basis functions. The curren t v ersion consists of tw o part s: one, pfq , for the manipulation of hy p ergeometric functions, p +1 F p , a nd the se cond one, App ellF1F4 , for the manipulation of Appell functions, F 1 , F 2 , F 3 , F 4 . The algorithm of differen t ial reduc- tion for other f unctions can b e implemen ted as an additiv e mo dule. In con trast to the recen t pro grams written b y mem b ers of computational particles ph ysics comm unit y [19 – 24], the aim of our pac k age is the manipulation of hypergeometric functions without the construction of ε expansions [2 5, 26]. The preliminary v ersion of pfq was presen ted in Ref. [27] and is av ailable in Ref. [28]. The latest v ersion is a v ailable in Ref. [3 4]. 4 2 Differen tial-red uction algorithm for generaliz e d h y- p ergeometric function p +1 F p 2.1 General consideration Let us consid er the generalized h yp ergeometric function, p F q ( a ; b ; z ), defined aro und z = 0 b y a series p F q ( ~ a ; ~ b ; z ) ≡ p F q ~ a ~ b z = ∞ X k =0 z k k ! Π p i =1 ( a i ) k Π q j =1 ( b j ) k , (7) where ( a ) k is a P o c hhammer sym b ol, ( a ) k = Γ( a + k ) / Γ( a ). The sequences ~ a = ( a 1 , · · · , a p ) and ~ b = ( b 1 , · · · , b q ) are called the upp er and lo w er parameters of hypergeometric functions, resp ectiv ely . In terms of the op erator θ : θ = z d dz , (8) the differen tial equation for the h ypergeometric function p F q can b e written as [ z Π p i =1 ( θ + a i ) − θ Π q i =1 ( θ + b i − 1)] p F q ( ~ a ; ~ b ; z ) = 0 . (9) 2.2 Differen tial reduction The differen tial reduction for these functions w as analyzed in details in Ref. [16]. Here w e recall some of the main relations relev a n t to our program. The univ ersal differen tial op erators, Eqs. (2) and ( 3), ha v e t he follo wing fo r m: p F q ( a 1 + 1 , ~ a ; ~ b ; z ) = B + a 1 p F q ( a 1 , ~ a ; ~ b ; z ) = 1 a 1 ( θ + a 1 ) p F q ( a 1 , ~ a ; ~ b ; z ) , (10) p F q ( ~ a ; b 1 − 1 , ~ b ; z ) = H − b 1 p F q ( ~ a ; b 1 , ~ b ; z ) = 1 b 1 − 1 ( θ + b 1 − 1) p F q ( ~ a ; b 1 , ~ b ; z ) , (11) where the op erators B + a 1 ( H − b 1 ) are called t he step-up (step-down) op erat ors for the upp er (lo w er) parameters of h yperg eometric functions. This t yp e of op erators we re explicitly con- structed for the h yp erg eometric function p +1 F p b y T ak ay ama in Ref. [10]. F or completeness, w e repro duce his result here: p +1 F p ( a i − 1 , ~ a ; ~ b ; z ) = B − a i p +1 F p ( a i , ~ a ; ~ b ; z ) , p +1 F p ( ~ a ; b i + 1 , ~ b ; z ) = H + b i p +1 F p ( ~ a ; b 1 , ~ b ; z ) , (12) 5 where B − a i = − a i c i [ t i ( θ ) − z Π j 6 = i ( θ + a j )] | a i → a i − 1 , c i = − a i Π p j =1 ( b j − 1 − a i ) , t i ( x ) = x Π p j =1 ( x + b j − 1 ) − c i x + a i = p X j =0 P ( p ) p − j ( { b r − 1 } ) [ x j +1 − ( − a i ) j +1 ] x + a i = p X j =0 P ( p ) p − j ( { b r − 1 } ) j X k =0 x j − k ( − a i ) k , (13) H + a i = b i − 1 d i d dz Π j 6 = i ( θ + b j − 1 ) − s i ( θ ) b i → b i +1 , d i = Π p +1 j =1 (1 + a j − b i ) , s i ( x ) = Π p +1 j =1 ( x + a j ) − d i x + b i − 1 p +1 X j =0 P ( p +1) p +1 − j ( { a r } ) [ x j − (1 − b i ) j ] x − (1 − b i ) = p X j =0 P ( p +1) p − j ( { a r } ) j X k =0 x j − k (1 − b i ) k . (14) There | a → a +1 means substitution of a b y a + 1, and the p o lynomials P ( p ) j ( r 1 , · · · , r p ) are defined as p Y k =1 ( z + r k ) = p X j =0 P ( p ) p − j ( r 1 , · · · , r p ) z j ≡ p X j =0 P ( p ) p − j ( ~ r ) z j ≡ p X j =0 P ( p ) j ( ~ r ) z p − j . (15) P ( p ) s ( ~ r ) is a p olynomial o f order s with resp ect to the v ariables r , and P ( p ) 0 ( ~ r ) = 1 , P ( p ) j ( ~ r ) = p X i 1 , ··· ,i r =1 Y i 1 < ···
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