Algebraic transformations of Gauss hypergeometric functions

Algebraic transformations of Gauss hypergeometric functions Raimunas Viūnas Faculty of Mathematics, Kobe University Abstract This article gives a classification scheme of algebraic transformations of Gauss hypergeometric functions, or pull-back transformations between hypergeometric ifferential equations. The classification recovers the classical transformations of egree,, 4,, an fins other transformations of some special classes of the Gauss hypergeometric function. The other transformations are consiere more thoroughly in a series of supplementing articles. Introuction An algebraic transformation of Gauss hypergeometric functions is an ientity of the form Ã, B A, B F C x = θx F C ϕx. Here ϕx is a rational function of x, an θx is a raical function, i.e., a prouct of some powers of rational functions. Examples of algebraic transformations are the following well-known quaratic transformations see [Er5, Section.], [Gou8, formulas 8, 45]: a, b a F a+b+ x = F, b a+b+ 4x x, a, b F b x = x a a F, a+ x b + x. Algebraic transformations of Gauss hypergeometric functions are usually inuce by pull-back transformations between their hypergeometric ifferential equations. General relation between these two kins of transformations is given in Lemma. here below. By that lemma, if a pull-back transformation converts a hypergeometric equation to a hypergeometric equation as well, then there are ientities of the form between hypergeometric solutions of the two hypergeometric equations. Conversely, an algebraic transformation is inuce by a pull-back transformation of the corresponing hypergeometric equations, unless the hypergeometric series on the left-han sie of satisfies a first orer linear ifferential equation. This article classifies pull-back transformations between hypergeometric ifferential equations. At the same time we essentially classify algebraic transformations of Gauss hypergeometric functions. Supporte by the Dutch NWO project -0-55, an by the Kyushu University Century COE Programme Development of Dynamic Mathematics with High Functionality of the Ministry of Eucation, Culture, Sports, Science an Technology of Japan. viunas@math.kobe-u.ac.jp

Classical fractional-linear an quaratic transformations are ue to Euler, Pfaff, Gauss an Kummer. In [Gou8] Goursat gave a list of transformations of egree, 4 an. It has been wiely assume that there are no other algebraic transformations, unless hypergeometric functions are algebraic functions. For example, [Er5, Section..5] states the following: Transformations of [egrees other than,, 4, ] can exist only if a, b, c are certain rational numbers; in those cases the solutions of the hypergeometric equation are algebraic functions. As our stuy shows, this assertion is unfortunately not true. This fact is notice in [AK0] as well. Existence of a few special transformations follows from [Ho8], [Beu0]. Regaring transformations of algebraic hypergeometric functions or more exactly, pull-back transformations of hypergeometric ifferential equations with a finite monoromy group, celebrate Klein s theorem [Kle77] ensures that all these hypergeometric equations are pull-backs of a few stanar hypergeometric equations. Klein s pull-back transformations o not change the projective monoromy group. The possible finite projective monoromy groups are: a cyclic incluing the trivial, a finite iheral, the tetraheral, the octaheral or the icosaheral groups. Transformations of algebraic hypergeometric functions that reuce the projective monoromy group are compositions of a few reucing transformations an Klein s transformation keeping the smaller monoromy group; see Remark 7. below. The ultimate list of pull-back transformations between hypergeometric ifferential equations an of algebraic transformations for their hypergeometric solutions is the following: Classical algebraic transformations of egree,, 4 an ue to Gauss, Euler, Kummer, Pfaff an Goursat. We review classical transformations in Section 4, incluing fractional-linear transformations. Transformations of hypergeometric equations with an abelian monoromy group. This is a egenerate case [Vi07]; the hypergeometric equations have rather than actual singularities. We consier these transformations in Section 5. Transformations of hypergeometric equations with a iheral monoromy group. These transformations are consiere here in Section, or more thoroughly in [Vi08a, Sections an 7]. Transformations of hypergeometric equations with a finite monoromy group. The hypergeometric solutions are algebraic functions. Transformations of hypergeometric equations with finite cyclic or iheral monoromy groups can be inclue in the previous two cases. Transformations of hypergeometric equations with the tetraheral, octaheral or icosaheral projective monoromy groups are consiere here in Section 7, or more thoroughly in [Vi08b]. Transformations of hypergeometric functions which are incomplete elliptic integrals. These transformations correspon to enomorphisms of certain elliptic curves. They are consiere in Section 8, or more thoroughly in [Vi08c]. Finitely many transformations of so-calle hyperbolic hypergeometric functions. Hypergeometric equations for these functions have local exponent ifferences /k, /k, /k, where k, k, k are positive integers such that /k + /k + /k <. These transformations are escribe in Section 9, or more thoroughly in [Vi05]. The classification scheme is presente in Section. Sections 4 through 9 characterize various cases of algebraic transformations of hypergeometric functions. We mention some three-term ientities with Gauss hypergeometric functions as well. The non-classical cases are consiere more thoroughly in separate articles [Vi08a], [Vi08b], [Vi08c], [Vi05]. Recently, Kato [Kat08] classifie algebraic transformations of the F hypergeometric series. The rational transformations for the argument z in that list form a strict subset of the transformations consiere here.

Preliminaries The hypergeometric ifferential equation is [AAR99, Formula..5]: z z yz z + C A+B+ z yz z A B yz = 0. 4 This is a Fuchsian equation with regular singular points z = 0, an. The local exponents are: 0, C at z = 0; 0, C A B at z = ; an A, B at z =. A basis of solutions for general equation 4 is A, B F + A C, + B C C z, z C F C z. 5 For basic theory of hypergeometric functions an Fuchsian equations see [Beu0], [vw0, Chapters, ] or [Tem9, Chapters 4, 5]. We use the approach of Riemann an Papperitz [AAR99, Sections.,.9]. A rational pull-back transformation of an orinary linear ifferential equation has the form z ϕx, yz Y x = θx yϕx, where ϕx an θx have the same meaning as in formula. Geometrically, this transformation pullbacks a ifferential equation on the projective line P z to a ifferential equation on the projective line P x, with respect to the finite covering ϕ : P x P z etermine by the rational function ϕx. Here an throughout the paper, we let P x, P z enote the projective lines with rational parameters x, z respectively. A pull-back transformation of a Fuchsian equations gives a Fuchsian equation again. In [AK0] pull-back transformations are calle RS-transformations. We introuce the following efinition: an irrelevant singularity for an orinary ifferential equation is a regular singularity which is not logarithmic, an where the local exponent ifference is equal to. An irrelevant singularity can be turne into a non-singular point after a suitable pull-back transformation with ϕx = x. For comparison, an apparent singularity is a regular singularity which is not logarithmic, an where the local exponents are integers. Recall that at a logarithmic point is a singular point where there is only one local solution of the form x λ + α x + α x +..., where x is a local parameter there. For us, a relevant singularity is a singular point which is not an irrelevant singularity. We are intereste in pull-back transformations of one hypergeometric equation to other hypergeometric equation, possibly with ifferent parameters A, B, C. These pull-back transformations are relate to algebraic transformations of Gauss hypergeometric functions as follows. Lemma.. Suppose that pull-back transformation of hypergeometric equation 4 is a hypergeometric equation as well with the new ineterminate x. Then, possibly after fractional-linear transformations on P x an P z, there is an ientity of the form between hypergeometric solutions of the two hypergeometric equations.. Suppose that hypergeometric ientity hols in some region of the complex plane. Let Y x enote the left-han sie of the ientity. If Y x/y x is not a rational function of x, then the transformation converts the hypergeometric equation 4 into a hypergeometric equation for Y x. Proof. We have a two-term ientity whenever we have a singular point S P x of the transforme equation above a singular point Q P z of the starting equation. Using fractional-linear transformations on P x an P z we can achieve S is the point x = 0 an that Q is the point z = 0. Then ientification of two hypergeometric solutions with the local exponent 0 an the value at respectively x = 0 an z = 0 gives a two-term ientity as in. If all three singularities of the transforme equation o not

lie above {0,, } P z, they are apparent singularities. Then the transforme equation has trivial monoromy, while the starting hypergeometric equation has a finite monoromy group. As we will consier explicitly in Sections 7 an 5,, the pull-back transformations reucing the monoromy group an the pull-back transformations keeping the trivial monoromy group can be realize by two-term hypergeometric ientities. This is recape in Remark 7. below. For the secon statement, we have two secon-orer linear ifferential equations for the left-han sie of : the hypergeometric equation for Y x, an a pull-back transformation of the hypergeometric equation 4. If these two equations are not Cx-proportional, then we can combine them linearly to a first-orer ifferential equation Y x = rx Y x with rx Cx, contraicting the conition on Y x/y x. If we have an ientity without a pull-back transformation between corresponing hypergeometric equations, the left-han sie of the ientity can be expresse as terminating hypergeometric series up to a raical factor; see Kovacic algorithm [Kov8], [vps0, Section 4..4]. In a formal sense, any pair of terminating hypergeometric series is algebraically relate. We o not consier these egenerations. Remark. We also o not consier transformations of the type F ϕ z = θz F ϕ z, where ϕ z, ϕ z are rational functions of egree at least. Therefore we miss transformations of some complete elliptic integrals, such as where x + y = an Kx = + y K Kx = π F /, / x = y + y 0, 7 t t x t. Ientity 7 plays a key role in the theory of arithmetic-geometric mean; see [AAR99, Chapter.]. Other similar example is the following formula, prove in [BBG95, Theorem.]: c, c + F c+5 x = + x c c, c+ F c+ x +x. 8 The case c = / was foun earlier in [BB9]. A pull-back transformation between hypergeometric equations usually gives several ientities like between some of the 4 Kummer s solutions of both equations. It is appropriate to look first for suitable pull-back coverings ϕ : P x P z up to fractional-linear transformations. As we will see, suitable pull-back coverings are etermine by appropriate transformations of singular points an local exponent ifferences. Once a suitable covering ϕ is known, it is convenient to use Riemann s P -notation for eriving hypergeometric ientities with the argument ϕx. Recall that a Fuchsian equation with singular points is etermine by the location of those singular points an local exponents there. The linear space of solutions is etermine by the same ata. It can be efine homologically without reference to hypergeometric equations as a local system on the projective line; see [Kat9], [Gra8, Section.4]. The notation for it is P α β γ a b c z, 9 a b c where α, β, γ P z are the singular points, an a, a ; b, b ; c, c are the local exponents at them, respectively. Recall that secon orer Fuchsian equations with relevant singularities are efine uniquely 4

by their singularities an local exponents, unlike general Fuchsian equations with more than singular points. Our approach can be entirely formulate in terms of local systems, without reference to hypergeometric equations an their pull-back transformations. By Papperitz theorem [AAR99, Theorem..] we must have a + a + b + b + c + c =. We are looking for transformations of local systems of the form 0 0 P 0 0 Ã x C C Ã B B = θx P 0 0 A C C A B B ϕx. 0 The factor θx shoul shift local exponents at irrelevant singularities to the values 0 an, an it shoul shift one local exponent at both x = 0 an x = to the value 0. In intermeiate computations, Fuchsian equations with more than singular points naturally occur, but those extra singularities are irrelevant singularities. We exten Riemann s P -notation an write α β γ S... S k P a b c e... e k a b c e +... e k + to enote the local system of solutions of a Fuchsian equation with irrelevant singularities S,..., S k. This notation makes sense if a local system exists i.e., if the local exponents sum up to the right value; then it can be transforme to a local system like 9. For example, if none of the points γ, S,..., S k is the infinity, local system can be ientifie with z S e... z S k e k z γ e+...+e k P z α β γ a b c + e +... + e k z. a b c + e +... + e k Here is an example of computation with local systems leaing to quaratic transformation : P 0 0 0 a b b a a+ t = P = P = x a P 0 0 0 0 a b b a b a a + 0 0 0 0 a b b a b a a + 0 0 0 a b b a b t x = t t + To conclue, one has to ientify two functions with the local exponent 0 an the value at t = 0 an x = 0 in the first an the last local systems respectively, like in the proof of part of Lemma.. Once a hypergeometric ientity is obtaine, it can be compose with Euler s an Pfaff s fractionallinear transformations; we recall them in formulas 8 below. Geometrically, these transformations permute the singularities, on P z or P x an their local exponents. Besies, simultaneous permutation of the local exponents at x = 0 an z = 0 usually implies a similar ientity to, as presente in the following lemma. x. 5

Lemma. Suppose that a pull-back transformation inuces ientity in an open neighborhoo of x = 0. Then ϕx C Kx ec as x 0 for some constant K, an the following ientity hols if both hypergeometric functions are well-efine: F + Ã C, + B C C x = θx ϕx C K x e C F + A C, + B C C ϕx. Proof. The asymptotic relation ϕx C Kx ec as x 0 is clear from the transformation of local exponents. We are ensure that θ0 =. Further, we have relation 0 an the relations 0 P 0 0 A C C A B B 0 P 0 0 Ã C C Ã B B ϕx = ϕx C P x = x ec P From here we get the right ientification of local systems for. 0 C 0 A + C 0 C A B B + C 0 C 0 Ã + C 0 C Ã B B + C A general pull-back transformation converts a hypergeometric equation to a Fuchsian ifferential equation with several singularities. To fin proper caniates for pull-back coverings ϕ : P x P z, we look first for possible pull-back transformations of hypergeometric equations to Fuchsian equations with at most relevant singularities. These Fuchsian equations can be always transforme to hypergeometric equations by suitable fractional-linear pull-back transformations, an vice versa. Relevant singular points an local exponent ifferences for the transforme equation are etermine by the covering ϕ only. Here are simple rules which etermine singularities an local exponent ifferences for the transforme equation. Lemma.4 Let ϕ : P x P z be a finite covering. Let H enote a Fuchsian equation on P z, an let H enote the pull-back transformation of H uner. Let S P x, Q P z be points such that ϕs = Q.. The point S is a logarithmic point for H if an only if the point Q is a logarithmic point for H.. If the point Q is non-singular for H, then the point S is not a relevant singularity for H if an only if the covering ϕ oes not branch at S.. If the point Q is a singular point for H, then the point S is not a relevant singularity for H if an only if the following two conitions hol: The point Q is not logarithmic. The local exponent ifference at Q is equal to /k, where k is the branching inex of ϕ at S. Proof. First we note that if the point S is not a relevant singularity, then it is either a non-singular point or an irrelevant singularity. Therefore S is not a relevant singularity if an only if it is not a logarithmic point an the local exponent ifference is equal to. Let p, q enote the local exponents for H at the point Q. Let k enote the branching orer of ϕ at S. Then the local exponent ifference at S is equal to kp q. To see this, note that if m C is the orer of θx at S, the local exponents at S are equal to kp + m an kq + m. This fact is clear if Q is not logarithmic, when the local exponents can be rea from solutions. In general one has to use the inicial polynomial to etermine local exponents. The first statement is clear, since local solutions of H at S can be pull-backe to local solutions of H at Q, an local solutions of H at Q can be push-forware to local solutions of H at S. x ϕx.,

If the point Q is non-singular, the point S is not logarithmic by the first statement, so S is a not a relevant singularity if an only if k =. If the point Q is singular, then the local exponent ifference at S is equal to if an only if the local exponent ifference p q is equal to /k. The following Lemma gives an estimate for the number of points S to which part of Lemma.4 applies, an it gives a relation between local exponent ifferences of two hypergeometric equations relate by a pull-back transformation an the egree of the pull-back transformation. In this paper we make the convention that real local exponent ifferences are non-negative, an complex local exponent ifferences have the argument in the interval π, π]. Lemma.5 Let ϕ : P x P z be a finite covering, an let enote the egree of ϕ.. Let enote a set of points on P z. If all branching points of ϕ lie above, then there are exactly + istinct points on P x above. Otherwise there are more than + istinct points above.. Let H enote a hypergeometric equation on P z, an let H enote a pull-back transformation of H with respect to ϕ. Suppose that H is hypergeometric equation as well. Let e, e, e enote the local exponent ifferences for H, an let e, e, e enote the local exponent ifferences for H. Then e + e + e = e + e + e. Proof. For a point S P x let or S ϕ enote the branching orer of ϕ at S. By Hurwitz formula [Har77, Corollary IV..4] we have = + D, where D = or S ϕ. S P x Therefore D =. The number of points above is or S ϕ D = +. ϕs We have the equality if an only if all branching points of ϕ lie above. Now we show the secon statement. For a point S P z or S P x, let les enote the local exponent ifference for H or H respectively at S. The following sums make sense: les les S P x = Q P z = Q P z ϕs=q leq ϕs=q leq + D. = Q P z The first sum is equal to e + e + e. The last expression is equal to e + e + e +. The classification scheme The core problem is to classify pull-back transformations of hypergeometric equations to Fuchsian equations with at most relevant singular points. By Lemma.4, a general pull-back transformation gives a Fuchsian equation with quite many relevant singular points, especially above the set {0,, } P z. 7

In orer to get a Fuchsian equation with so few singular points, we have to restrict parameters or local exponent ifferences of the original hypergeometric equation, an usually we can allow branching only above the set {0,, } P z. We classify pull-back transformations between hypergeometric equations an algebraic transformations of Gauss hypergeometric functions in the following five principal steps:. Let H enote hypergeometric equation 4, an consier its pull-back transformation. Let H enote the pull-backe ifferential equation, an let T enote the number of singular points of H. Let enote the subset {0,, } of P z, an let enote the egree of the covering ϕ : P x P z in transformation. We consequently assume that exactly N {0,,, } of the local exponent ifferences for H at are restricte to the values of the form /k, where k is a positive integer. If k = then the corresponing point of is assume to be not logarithmic, as we cannot get ri of singularities above a logarithmic point.. In each assume case, use Lemma.4 an etermine all possible combinations of the egree an restricte local exponent ifferences. Let k,..., k N enote the enominators of the restricte ifferences. By part 4 of Lemma.4, T [the number of singular points above ] + [the number of non-singular points above ] N +. j= Since we wish T, we get the following restrictive inequality in integers: k j N. 4 j= To skip specializations of cases with smaller N, we may assume that maxk,..., k N. A preliminary list of possibilities can be obtaine by ropping the rouning own in 4; this gives a weaker but more convenient inequality + N j= k j k j. 5. For each combination of an restricte local exponent ifferences, etermine possible branching patterns for ϕ such that the transforme equation H woul have at most three singular points. In most cases we can allow branching points only above, an we have to take the maximal number /k j of non-singular points above the point with the local exponent ifference /k j. 4. For each possible branching pattern, etermine all rational functions ϕx which efine a covering with that branching pattern. For this can be one using a computer by a straightforwar metho of unetermine coefficients. In [Vi05, Section ] a more appropriate algorithm is introuce which uses ifferentiation of ϕx. In many cases this problem has precisely one solution up to fractionallinear transformations. But not for any branching pattern a covering exists, an there can be several ifferent coverings with the same branching pattern. For infinite families of branching patterns we are able to give a general, algorithmic or explicit characterization of corresponing coverings. For instance, if hypergeometric solutions can be expresse very explicitly, we can ientify the local systems in 0 up to unknown factor θx. Then quotients of corresponing hypergeometric solutions aka Schwarz maps can be ientifie precisely, which gives a straightforwar way to etermine ϕx. 8

5. Once a suitable covering ϕ : P x P z is compute, there always exist corresponing pull-back transformations. Two-term ientities like can be compute using extene Riemann s P -notation of Section. We have two-term ientities for each singular point S of the transforme equation such that ϕs, as in the proof of part of Lemma.. Once we fix S, ϕs as x = 0, z = 0 respectively, permutations of local exponents an other singularities give ientities which are relate by Euler s an Pfaff s transformations an Lemma.. If the transforme equation has less than actual singularities, one can consier any point above in this manner. Some of the obtaine ientities may be the same up to change of free parameters. Now we sketch explicit appliance of the above proceure. When N = 0, i.e., when no local exponent ifferences are restricte, then = by formula 5. This gives Euler s an Pfaff s fractional-linear transformations. When N =, we have the following cases: k =, =. This gives the classical quaratic transformations. See Section 4. k =, any. The z-point with the local exponent ifference /k is assume to be non-logarithmic, so the equation H has only two relevant singularities. As we show in Lemma 5. below, the two unrestricte local exponent ifferences must be equal. As it turns out, the covering ϕ branches only above the two points with unrestricte local exponent ifferences. If the triple of local exponent ifferences for H is, p, p, the triple of local exponent ifferences for H is, p, p. Formally, this case has a continuous family of fractional-linear pull-back transformations, but that oes not give interesting hypergeometric ientities. When N =, we have the following cases: If maxk, k >, the possibilities are liste in Table. Steps an of the classification scheme are straightforwar, an a snapshot of possibilities after them is presente by the first four columns of Table. The notation for a branching pattern in the fourth column gives + branching orers for the points above ; branching orers at points in the same fiber are separate by the + signs, branching orers for ifferent fibers are separate by the = signs. Step 4 of our scheme gives at most one covering up to fractional-linear transformations for each branching pattern. Ultimately, Table yiels precisely the classical transformations of egree, 4, ue to Goursat [Gou8]; see Section 4. It is straightforwar to figure out possible compositions of small egree coverings, an then ientify them with the unique coverings for Table. Degrees of constituents for ecomposable coverings are liste in the last column from right for the constituent transformation from H to left. Note that one egree covering has two istinct ecompositions; a corresponing hypergeometric transformation is given in formula 8 below. k =, k =, any. The monoromy group of H is a iheral group. The hypergeometric functions can be expresse very explicitly, see Section. The triple /, /, p of local exponent ifferences for H is transforme either to /, /, p for any, or to, p/, p/ for even. k = ; k an are any positive integers. The z-point with the local exponent ifference /k is not logarithmic, so the triple of local exponent ifferences for H must be, /k, /k. The monoromy group is a finite cyclic group. Possible transformations are outline in Section 5. When N =, we have the following three very istinct cases: /k +/k +/k >. The monoromy groups of H an H are finite, the hypergeometric functions are algebraic. The egree is unboune. Klein s theorem [Kle77] implies that any hypergeometric equation with a finite monoromy group or equivalently, with algebraic solutions is a pull-back transformation of a stanar hypergeometric equation with the same monoromy group. These are the most interesting pull-back transformations for this case. Equations with finite cyclic monoromy 9

Local exponent ifferences Degree Branching pattern above Covering /k, /k, p above the regular singular points composition /, /, p /, p, p + = = + inecomposable /, /, p /, p, p 4 + = + = + inecomposable /, /, p /, p, p 4 + = + = + no covering /, /, p p, p, 4p + + = + = 4 + + /, /, p p, p, p + + = + = + + or /, /, p p, p, p + + = + = + + no covering /, /4, p p, p, p 4 + = 4 = + + /, /, p p, p, p = = + + inecomposable Table : Transformations of hypergeometric functions with free parameter groups are mentione in the previous subcase; their transformations are consiere in Section 5. Equations with finite iheral monoromy groups are consiere in Section. Equations with the tetraheral, octaheral or icosaheral projective monoromy groups are characterize in Section 7. /k + /k + /k =. Non-trivial hypergeometric solutions of H are incomplete elliptic integrals, see Section 8. The egree is unboune, ifferent transformations with the same branching pattern are possible. Most interesting transformations pull-back the equation H into itself, so that H = H ; these transformations come from enomorphisms of the corresponing elliptic curve. /k + /k + /k <. Here we have transformations of hyperbolic hypergeometric functions, see Section 9. The list of these transformations is finite, the maximal egree of their coverings is 4. Existence of some of these transformations is shown in [Ho8], [Beu0], [AK0]. The egree of transformations is etermine by formula, except in the case of incomplete elliptic integrals. If all local exponent ifferences are real numbers in the interval 0, ], the covering ϕ : P x P z is efine over R an it branches only above {0,, } P z, then it inuces a tessellation of the Schwarz triangle for H into Schwarz triangles for H, as outline in [Ho8, Beu0] or [Vi05, Section ]. Recall that a Schwarz triangle for a hypergeometric equation is the image of the upper half-plane uner a Schwarz map for the equation. The escribe tessellation is calle Coxeter ecomposition. If it exists, formula can be interprete nicely in terms of areas of the Schwarz triangles for H an H in the spherical or hyperbolic metric. Out of the classical transformations, only the cubic transformation with the branching pattern = = + + oes not allow a Coxeter ecomposition; see formula below. The following sections form an overview of algebraic transformations for ifferent types of Gauss hypergeometric functions. We also mention some three-term ientities with Gauss hypergeometric functions. Non-classical cases are consiere more thoroughly in other articles [Vi08a], [Vi08b], [Vi08c], [Vi05]. 4 Classical transformations Formally, Euler s an Pfaff s fractional-linear transformations [AAR99, Theorem..5] a, b F a, c b c z = z a F z c z c a, b = z b F z c 7 z c a, c b = z c a b F c z. 8 0

can be consiere as pull-back transformations of egree. These are the only transformations without restrictions on the parameters or local exponent ifferences of a hypergeometric function uner transformation. In a geometrical sense, they permute the local exponents at z = an z =. In general, permutation of the singular points z = 0, z =, z = an local exponents at them gives 4 Kummer s hypergeometric series solutions to the same hypergeometric ifferential equation. Any three hypergeometric solutions are linearly relate, of course. To present other classical an non-classical transformations, we introuce the following notation. Let p, q, r p, q, r schematically enote a pull-back transformation of egree, which transforms a hypergeometric equation with the local exponent ifferences p, q, r to a hypergeometric equation with the local exponent ifferences p, q, r. The orer of local exponents in a triple is irrelevant. Note that the arrow follows the irection of the covering ϕ : P x P z. The list of classical transformations comes from the ata of Table. Here is the list of classical transformations with inecomposable ϕ, up to Euler s an Pfaff s fractional-linear transformations an the conversion of Lemma.. /, p, q p, q, q. These are classical quaratic transformations. All two-term quaratic transformations of hypergeometric functions can be obtaine by composing or with Euler s an Pfaff s transformations. An example of a three-term relation uner a quaratic transformation is the following see also Remark 5. below, an [Er5,.]: a, b F a+b+ x = Γ a+b+ Γ a Γ a+ b+ Γ F, b x + x Γ a+b+ Γ Γ a Γ b F a+, b+ x. 9 /, /, p /, p, p. These are well-known Goursat s cubic transformations. Two-term transformations follow from the following three formulas, along with Euller s an Pfaff s transformations an application of Lemma. to : a, a+ F 4a+ x = x a a F, a+ 7 x x 4 4a+5 4 x, 0 a, a+ a F 4a+5 x = + x a F, a+ 7 x x 4a+5 + x, a, a+ F x = + x a a F, a+ x 9 x + x. /, /, p p, p, p These are less-known cubic transformations. Let ω enote a primitive cubic root of unity, so ω + ω + = 0. Since singular points of the transforme equation are all the same, there is only one two-term formula up to changing the parameter: a, a+ F a+ x = + ω x a a F, a+ ω+ xx a+ x + ω. A three-term formula is the following see also [Er5,.8]: a, a+ F a+ x = a +ω x a [ Γ a+ Γ a Γ Γa F +ωx Γ a+ Γ a+ a+ +ω x Γ 4 Γa F a, a+, a+ 4 x+ω x+ω x+ω ]. 4 x+ω

4 /, /, p /, p, p. These are inecomposable Goursat s transformations of egree 4. Twoterm transformations follow from the following three formulas, if we compose them with Euller s an Pfaff s transformations an apply Lemma. to 7: 4a F, 4a+ 4a+ x = 8x a a F, a+ 4 x x 9 4a+5 9 8x, 5 4a F, 4a+ a 4a+5 x = + 8x a F, a+ 4 x x 4a+5 + 8x, 4a F, 4a+ a x = x a F, a x 8 + x 4 x. 7 As recore in Table, there are four ways to compose quaratic an cubic transformations to higher egree transformations of hypergeometric functions. This gives three ifferent pull-back transformations of egree 4 an. The composition transformations can be schematically represente as follows: /, /4, p /, /, p /, /, p /, /, p /, p, p p, p, p, /, p, p p, p, 4p, /, p, p p, p, p, /, /, p p, p, p. The last two compositions shoul prouce the same covering, since computations show that the pull-back /, /, p p, p, p is unique up to fractional-linear transformations; see [Vi05, Section ]. Inee, one may check that the ientity a, a+ a F 4a+ x = x + x a F, a+ 7 x x 4a+5 4 x x+ 8 is a composition of an, an also a composition of, an. compositions use ifferent types of cubic transformations. Note that these two 5 Hypergeometric equations with two singularities Here we outline transformations of hypergeometric equations with two relevant singularities; their monoromy group is abelian. The explicit classification scheme of Section refers to this case three times. These equations form a special sample of egenerate hypergeometric equations [Vi07]. For the egenerate cases, not all usual hypergeometric formulas for fractional-linear transformations or other classical algebraic transformations may hol, since the structure of 4 Kummer s solutions egenerates; see [Vi07, Table ]. Here we consier only the new case of pull-back transformations of the hypergeometric equations with the cyclic monoromy group. If a Fuchsian equation has the local exponent ifference at some point, that point can be a nonsingular point, an irrelevant singularity or a logarithmic point. Here is how the logarithmic case is istinguishe for hypergeometric equations. Lemma 5. Consier hypergeometric equation 4, an let P {0,, }. Suppose that the local exponent ifference at S is equal to. Then the point S is logarithmic if an only if absolute values of the two local exponent ifferences at the other two points of the set {0,, } are not equal.

Proof. Because of fractional-linear transformations, we may assume that S is the point z = 0, an the local exponents there are 0 an. Therefore C = 0. Then the point z = 0 is either a non-singular point or a logarithmic point. It is non-singular if an only if A B = 0. If B = 0, then local exponent ifferences at z = an z = are both equal to A. This lemma implies that a hypergeometric equation has at most two relevant singularities if an only if the local exponent ifference at one of the three points z = 0, z =, z = is, an the local exponent ifferences at the other two points are equal. After applying a suitable fractional-linear transformation to this situation we may assume that the point z = 0 is non-singular. Like in the proof of Lemma 5., we have C = 0 an we may take B = 0. Then we are either in the case n = m = 0 of [Vi07, Section 7 or 8], or in the case n = m = l = 0 of [Vi07, Section 9]. Most of the 4 Kummer s solutions have to be interprete either as the constant or the power function z a. The only interesting hypergeometric function up to Euler s an Pfaff s transformations is the following: + a, F z = z a, if a 0, a z 9 z log z, if a = 0. For general a, pull-back transformation of the consiere hypergeometric equation to a hypergeometric equation branches only above the points z = an z =. Inee, if the covering ϕ : P x P z branches above other point, then these branching points woul be singular by part of Lemma.4, an there woul be at least singular points above {, } P z by part of Lemma.5. To keep the number of singular points own to, the covering ϕ shoul branch only above {, }. Up to fractional-linear transformations on P x, these coverings have the form z x, or z x φ x, where φ x = x. 0 x Note that φ x is a polynomial of egree. A corresponing hypergeometric ientity is + a, F x = φ x + a, F x φ x. This transformation is obvious from the explicit expressions in 9. Formally, we aitionally have a continuous family z β + βz of fractional-linear pull-back transformations which fix the two points z = an z =. However, they o not give interesting hypergeometric ientities since Kummer s series at those two points are trivial. If a = /k for an integer k >, there are more pull-back transformations of hypergeometric equations with the local exponent ifferences, a, a. In this case, the monoromy group is a finite cyclic group, of orer k. Pull-backe equations will have a cyclic monoromy group as well, possibly of smaller orer. On the other han, the mentione Klein s theorem [Kle77] implies that any hypergeometric equation with a cyclic monoromy group of orer k is a pull-back of a hypergeometric equation with the local exponent ifferences, /k, /k. These pull-back transformations can be easily compute from explicit terminating solutions of the target ifferential equation. Accoring to [Vi07, Section 7], a general hypergeometric equation with a completely reucible but non-trivial monoromy representation has the local exponents m + n +, a, a + n m, where a Z an n, m Z are non-negative. A basis of terminating solutions is n, a m F m, a n m n z, z a F m n z.

The monoromy group is finite cyclic if a = l/k with co-prime positive k, l Z. The terminating solutions can be written as terminating hypergeometric series at z = as well: n, a m F m n z = + a n m! n, a m F m + n! + a z, etc. The quotient of two solutions in efines a Schwarz map for the hypergeometric equation. In the simplest case n = m = 0, a = /k, the Schwartz map is just z /k. Klein s pull-back transformation for, /k, /k m+n+, l/k, l/k +n m is obtaine from ientification of the two Schwarz maps. The pull-back covering is efine by / n, l/k m k z x l F m, l/k n m n x k F m n x. The Schwarz maps or pairs of hypergeometric solutions are ientifie here by the corresponing local exponents at x = place above z = an the same value at x = 0 place above x = 0. The egree of the transformation is equal to maxnk + l, mk, by formula as well. Besies, z Ox n+m+ at x = 0 by the require branching pattern. In particular, / x x = x l/k + Ox n+m+ 4 F m, l/k n m n F n, l/k m m n at x = 0, hence the quotient of two hypergeometric polynomials is the Paé approximation of x l/k of precise egree m, n. For example, the Pae approximation of x of egree, is 4 x/4 x. Hence the following pullback must give a transformation, /, /, /, /: z x x 4 x 4. A corresponing hypergeometric ientity is /, F 4 x = 4 /, 4 x F x x 4. 5 Transformation is Klein s pull-back transformation if gck, l =. Otherwise the transforme hypergeometric equation has a smaller monoromy group. These transformations must factor via 0 with = gck, l, an Klein s transformation between equations with the smaller monoromy group. Even l/k Z can be allowe if the transforme equation has no logarithmic points. The conition for that is l/k > m; see [Vi07, Corollary. part ]. Uner this conition, one may even allow k = an consier transformations,, l+n m + n +, l, l + n m. All hypergeometric equations with the trivial monoromy group can be obtaine in this way, by Klein s theorem. Solutions of these hypergeometric equations are analyze in [Vi07, Section 8]. A hypergeometric equation with the local exponent ifferences,, can be transforme to y = 0 by fractional-linear transformations. We unerscore that transformation specializes nicely even for k = if only logarithmic solutions are not involve; the corresponing two-term hypergeometric ientities are trivial. Remark 5. Algebraic transformations of Gauss hypergeometric functions often hol only in some part of the complex plane, even after stanar analytic continuation. For example, formula is obviously false at x =. Formula hols when Rex < /, as the stanar z-cut, is mappe into the line Rex = / uner the transformation z = 4x x. An extreme example of this kin is the following transformation of a hypergeometric function to a rational function: /, F 4 x x x + x x = x x +. 4

This ientity hols in a neighborhoo of x = 0, but it certainly oes not hol aroun x = or x =. Apparently, stanar cuts for analytic continuation for the hypergeometric function isolate the three points x = 0, x =, x =. Note that F /, z = z/z is a two-value algebraic function on P z. Its composition in with the egree rational function apparently consists of two isjoint branches. The secon branch is the rational function x /x, which is the correct evaluation of the left-han sie of aroun the points x =, x = check the power series. Many ientities like can be prouce for hypergeometric functions of this section with /a Z. The pull-backe hypergeometric equations shoul be Fuchsian equations with the trivial monoromy group. More generally, any algebraic hypergeometric function can be pull-backe to a rational function. Other algebraic hypergeometric functions are consiere in the following two sections. Three-term hypergeometric ientities may also have limite region of valiity. But it may happen that branch cuts of two hypergeometric terms cancel each other in a three-term ientity. For example, stanar branch cuts for the hypergeometric functions on the right-han sie of 9 are the intervals [, an, 0] on the real line. But ientity 9 is vali on C \ [,, if we agree to evaluate the right-han consistently on the interval, 0]: either using analytic continuation of both terms from the upper half-plane, or from the lower half-plane. Diheral functions Hypergeometric equations with infinite or finite iheral monoromy group are characterize by the property that two local exponent ifferences are rational numbers with the enominator. By a quaratic pull-back transformation, these equations can be transforme to Fuchsian equations with at most 4 singularities an with a cyclic monoromy group. Explicit expressions an transformations for these functions are consiere thoroughly in [Vi08a]. Here we look at transformations of hypergeometric equations which have two local exponent ifferences equal to /. The explicit classification scheme of Section refers to this case twice. The starting hypergeometric equation for new transformations has the local exponent ifferences /, /, a. Hypergeometric solutions of such an equation can be written explicitly. In particular, quaratic transformation with b = a + implies Other explicit formulas are a F, a+ F a+, a+ a F, a+ a + z z z = a + z. 7 = z a + + z a, 8 z a + z a = a, if a 0, z z log + 9 z z, if a = 0. General iheral Gauss hypergeometric functions are contiguous to these F functions. As shown in [Vi08a], explicit expressions for them can be given in terms of terminating Appell s F or F series. For example, generalizations of 7 8 are a F, a+ + l a + k + l + z a k l + z = z k/ k +, l + ; k, l F z a + k + l + z, 5 z, 40

a+ n n a F, a+ + n m z = + z a + z a a; m, n F z m, n + z, + z a; m, n F z m, n, z z. 4 Here m, n are assume to be non-negative integers. For general a, there are two types of transformations: /, /, a /, /, a. These are the only transformations to a iheral monoromy group as well, as there is a singularity above the point with the local exponent ifference a. Ientification of explicit Schwarz maps gives the following recipe for computing the pull-back coverings ϕ : P x P z. Expan + x in the form θ x + θ x x with θ x, θ x C[x]. Then ϕx = x θx/θ x gives a pull-back transformation of iheral hypergeometric equations. Explicitly, θ x = θ x = / k=0 / k=0 x k = F k k+, x, x k = F, x. A particular transformation of hypergeometric functions is the following: a F, a+ a x = θ x a F, a+ x θ x θ x. 4 It is instructive to check this transformation using 8. Other transformations from the same pullback covering are given in [Vi08a, Section ]. Particularly interesting are the following formulas; they hol for o or even, respectively: F a, a F a, a x x The branching pattern of ϕx is = F a, a = F a, a x F, + / x, x x F, + / x. + + +... + = = + + +... +, if is o, + + + +... + = = + +... +, if is even. /, /, a l, la, l a, an = l is even. These are transformations to hypergeometric equations of Section 5. They are compositions of the mentione quaratic transformation an the transformations /, /, a /, /, a or, a, a, a, a escribe above. If a = /k with k a positive integer, the monoromy group is the finite iheral group with k elements, an hypergeometric solutions are algebraic. Klein s theorem [Kle77] implies that any hypergeometric equation with a finite iheral monoromy group is a pull-back from a hypergeometric equation with the local exponent ifferences /, /, /k an the same monoromy group. The pull-back transformation can be compute by the similar metho: ientification of explicit Schwarz maps, using the mentione explicit evaluations with terminating Appell s F or F series. That leas to expressing a polynomial in x in the form θ x + xθ x as above.

Theorem. Let k, l, m, n be positive integers, an suppose that k, gck, l =. Let us enote m +, n + ; m, n Gx = x m/ F x + + l/k x, + x. This is a polynomial in x. We can write + x l Gx k = Θ x + x m+ Θ x, so that Θ x an Θ x are polynomials in x. Then the rational function Φx = x m+ Θ x /Θ x efines Klein s pull-back covering /, /, /k m + /, n + /, l/k. The egree of this rational function is equal to m + nk + l. Proof. This is Theorem 7. in [Vi08a]. The conition gck, l = can be replace by the weaker conition l/k Z, but then the transforme hypergeometric equation has a smaller iheral monoromy group, an it factors via the transformation in 4 with = gck, l. Even more, l/k Z can be allowe, if the transforme equation has no logarithmic solutions. Sufficient an necessary conitions for that are given in [Vi08a, Theorem.]. The branching pattern for all these coverings has the following pattern: Above the two points with the local exponent ifference /, there are two points with the branching orers m +, n +, an the remaining points are simple branching points. Above the point with the local exponent ifference /k, there is one point with the ramification orer l, an m + n points with the ramification orer k. Any covering /, /, /k m + /, n + /, l/k is unique up to fractional-linear transformations, as Schwarz maps are ientifie uniquely. Transformations from the local exponent ifferences /, /, /k to hypergeometric equations with finite cyclic monoromy groups are either the mentione egeneration l/k Z, or compositions with the quaratic transformation /, /, /k, /k, /k. Other transformations involving iheral Gauss hypergeometric functions are special cases of classical transformations. For the purposes of Theorem., the function Gx can be alternatively efine as follows: m+n+l l/k m n; m, n + x F k x m, n + x, +. 4 x The two efinitions iffer by a constant multiple. The F an F sums are relate by reversing the orer of summation in both irections in the rectangular sums, as note in [Vi08a]. For an example, consier the case n =, m = 0, l = of Theorem.. To compute the transformation /, /, /k k+ /, /, /k we nee to expan k x + x = θ x + x / θ 4 x. k Straightforwar computation shows that k θ x = F, k+ x / k, θ 4 x = k k k F, k x 5/ k. 44 A transformation of hypergeometric functions is k F, k x = θ x /k k F, k 7 x θ 4 x θ x. 45

On the other han, F k, k k z = k z k + z /k by formula 40. Note that the construction in 44 breaks own if k = ; a hypergeometric equation with the local exponent ifferences /, /, has logarithmic solutions. As compute in [Vi08a, Section 7], the polynomials Θ x, Θ x of Theorem. in the case n =, m = 0, l = can be expresse as terminating F series. 7 Algebraic Gauss hypergeometric functions Algebraic Gauss hypergeometric functions form a classical subject of mathematics. These functions were classifie by Schwarz [Sch7]. Recall that a Fuchsian equation has a basis of algebraic solutions if an only if its monoromy group is finite. Finite projective monoromy groups for secon orer equations are either cyclic, or iheral, or the tetraheral group isomorphic to A 4, or the octaheral group isomorphic to S 4, or the icosaheral group isomorphic to A 5. An important characterization of secon orer Fuchsian equations with finite monoromy group was given by Klein [Kle77, Kle78]: all these equations are pullbacks of a few stanar hypergeometric equations with algebraic solutions. In particular, this hols for hypergeometric equations with finite monoromy groups. The corresponing stanar equation epens on the projective monoromy group: Secon orer equations with a cyclic monoromy group are pull-backs of a hypergeometric equation with the local exponent ifferences, /k, /k, where k is a positive integer. Klein s transformations to general hypergeometric equations with a cyclic monoromy group are consiere in Section 5 above. Secon orer equations with a finite iheral monoromy group are pull-backs of a hypergeometric equation with the local exponent ifferences /, /, /k, where k. Klein s transformations to general hypergeometric equations with a iheral monoromy group are consiere in Section above. Secon orer equations with the tetraheral projective monoromy group are pull-backs of a hypergeometric equation with the local exponent ifferences /, /, /. Hypergeometric equations with this monoromy group are contiguous to hypergeometric equations with the local exponent ifferences /, /, / or /, /, /. Secon orer equations with the octaheral projective monoromy group are pull-backs of a hypergeometric equation with the local exponent ifferences /, /, /4. Hypergeometric equations with this monoromy group are contiguous to hypergeometric equations with the local exponent ifferences /, /, /4 or /, /4, /4. Secon orer equations with the icosaheral projective monoromy group are pull-backs of a hypergeometric equation with the local exponent ifferences /, /, /5. Hypergeometric equations with this monoromy group are contiguous to hypergeometric equations with the local exponent ifferences /, /, /5, /, /, /5, /, /5, /5, /, /, /5, /, /, /5, /, /5, /5, /, /5, /5, /, /5, /5, /5, /5, 4/5 or /5, /5, /5. A general algorithm for computation of Klein s coverings is given in [vhw05]. The algorithm is base on fining semi-invariants of the monoromy group by solving appropriate symmetric powers of the given secon orer ifferential equation. A more effective algorithm specifically for hypergeometric equations with finite monoromy groups is given in [Vi08b]. This algorithm is base on ientification of explicit Schwarz maps for the given an the corresponing stanar hypergeometric equation. 8