A new approach to hypergeometric transformation formulas
aa r X i v : . [ m a t h . C A ] S e p A NEW APPROACH TO HYPERGEOMETRICTRANSFORMATION FORMULAS
NORIYUKI OTSUBO
Abstract.
We give a new method to prove in a uniform and easy way vari-ous transformation formulas for Gauss hypergeometric functions. The key isJacobi’s canonical form of the hypergeometric differential equation. Analogyfor q -hypergeometric functions is also studied. Introduction
Recall that a Gauss hypergeometric function is defined by the power series F (cid:18) a, bc ; x (cid:19) = ∞ X n =0 ( a ) n ( b ) n ( c ) n (1) n x n , ( a ) n = n − Y i =0 ( a + i ) , which converges on the open unit disk. Here the parameters a , b , c are com-plex numbers and − c N . We know many transformation formulas among suchfunctions since Euler, Pfaff and Gauss, and some of them are quite new. Suchformulas have various aspects and to find or prove them, different techniques havebeen used. For example, elliptic functions and computation using mathematicalsoftware played important roles.In this paper, we give a new method to prove such formulas in a uniform and easyway. Recall that the hypergeometric function satisfies a linear ordinary differentialequation of order two, and hence is characterized by this equation together withthe initial values. The key of our method is the following canonical form of thehypergeometric differential equation (cid:18) ddx x c (1 − x ) e ddx − abx c − (1 − x ) e − (cid:19) y = 0where e = 1 + a + b − c (Theorem 2.2). As the author learned after writing the firstmanuscript of the present paper, this form was known by Jacobi [15] (see also [22, § a + b = c = 1,it has several advantages to the standard ones (see (2.1), (2.2)). It clarifies thesymmetry under x ↔ − x (then c ↔ e ), and behaves nicely under the change ofvariables we are to consider. Above all, it enables us to compute the differentialequation for h ( x ) F ( x ) from that for F ( x ) in a straight-forward way.Among the transformation formulas to be proved in this paper, of particularinterest are the following ones, which have strong connection to number theory. Date : September 18, 2019.2010
Mathematics Subject Classification.
Key words and phrases.
Hypergeometric functions, basic hypergeometric functions, transfor-mation formulas.
Theorem 1.1.
On a neighborhood of x = 0 , (1 + x ) a F a , a − b +12 b +12 ; x ! = F a , b b ; 1 − (cid:18) − x x (cid:19) ! , (1.1) (1 + 2 x ) a F a , a +13 a +56 ; x ! = F a , a +13 a +12 ; 1 − (cid:18) − x x (cid:19) ! , (1.2) (1 + 3 x ) a F a , a +24 a +56 ; x ! = F a , a +24 a +23 ; 1 − (cid:18) − x x (cid:19) ! . (1.3)The quadratic formula (1.1) with two free parameters a and b is very classical,due to Gauss [10, p. 225, formula 101]. The special case where a = b = 1 reducesto the Landen transformation formula(1 + x ) K ( x ) = K (cid:18) √ x x (cid:19) for the elliptic integral of the first kind K ( x ) := Z π dθ √ − x cos θ = π F (cid:18) , x (cid:19) . From this follows the formula F (cid:18) , − x (cid:19) = 1 M ( x ) , where M ( x ) denotes the arithmetic-geometric mean of 1 and x (0 < x ≤ a = 1 was first found by Ramanujan [23, secondnotebook, p. 258] in his study of elliptic functions to alternative bases. It wasrediscovered and proved by Borwein-Borwein [4, p. 694] in their study of a cubicanalogue of the arithmetic-geometric mean. The general case of (1.2) is due toBerndt-Bhargava-Garvan [3, Theorem 2.3].The quadratic formula (1.3) for a = 1 was also found by Ramanujan [23, p. 260].This is also related with a generalized arithmetic-geometric mean, and a proof wasgiven implicitly by Borwein-Borwein [4, Theorem 2.6] and explicitly by Berndt-Bhargava-Garvan [3, Theorem 9.4]. The general case of (1.3) was found recentlyby Matsumoto-Ohara [21, Corollary 3] (see Section 3.4).One may expect that our method is useful not only for proving formulas, butalso for finding new ones. In fact, the author found (1.3) independently beforeknowing [21]. For other transformation formulas which were found more recently,see for example Vid¯unas [24].This paper is constructed as follows. In Section 2, we derive the canonical formof the hypergeometric equation and explain our general method for transforma-tion formulas. In Section 3, we give a short proof of Theorem 1.1, and discusstransformation formulas for multivariable hypergeometric functions of Appell andLauricella. In Sections 4–6, we give proofs of other quadratic, cubic and quarticformulas and discuss their relations with the previous formulas. In the last Section7, we study the analogy for q -hypergeometric functions (basic hypergeometric func-tions) φ . We give a canonical form of the q -hypergeometric difference equation(Theorem 7.5), and use it to give a new proof of Heine’s transformation formula(Theorem 7.6). NEW APPROACH TO HYPERGEOMETRIC TRANSFORMATION FORMULAS 3 Generalities
Hypergeometric differential equation.
Let us write differential operatorsas ∂ = ∂ x = ddx , D = D x = x∂. One sees easily from Dx n = nx n that F (cid:16) a,bc ; x (cid:17) satisfies the differential equation(2.1) (cid:0) ( D + a )( D + b ) − x − D ( D + c − (cid:1) y = 0 . Since D = x ∂ + x∂ , (2.1) is equivalent to(2.2) (cid:0) x (1 − x ) ∂ + ( c − (1 + a + b ) x ) ∂ − ab (cid:1) y = 0 . Further, it can be written as(2.3) (cid:18) ∂ + (cid:18) cx − e − x (cid:19) ∂ − abx (1 − x ) (cid:19) y = 0 , where we define e by 1 + a + b = c + e. Then it is obvious that F (cid:16) a,be ; 1 − x (cid:17) is another solution ( − e N assumed). Remark . One cannot expect such a symmetry for the differential equation sat-isfied by a generalized hypergeometric function p F p − ( x ) in general for p >
2. Thiscan be seen from the asymmetry of the Riemann scheme.The key observation of this paper is the following.
Theorem 2.2.
Put ϕ ( x ) = x c (1 − x ) e where e = 1 + a + b − c . Then F (cid:16) a,bc ; x (cid:17) ( resp. F (cid:16) a,be ; 1 − x (cid:17) ) is the unique solution of the differential equation (2.4) (cid:18) ∂ϕ ( x ) ∂ − ab ϕ ( x ) x (1 − x ) (cid:19) y = 0 such that y (0) = 1 , y ′ (0) = abc (cid:18) resp. y (1) = 1 , y ′ (1) = − abe (cid:19) . Proof.
For any function f ( x ) regarded as a multiplication operator, the identity ofoperators ∂f ( x ) = f ( x ) ∂ + f ′ ( x )holds. Therefore, ∂ϕ ( x ) =( x c ∂ + cx c − )(1 − x ) e = x c ((1 − x ) e ∂ − e (1 − x ) e − ) + cx c − (1 − x ) e = ϕ ( x ) (cid:18) ∂ + cx − e − x (cid:19) . Hence follows the equivalence of (2.3) and (2.4). The initial values are immediatefrom the definition and the uniqueness is evident. (cid:3)
Remark . In fact, the differential equation is regular singular at x = 0, and F (cid:16) a,bc, ; x (cid:17) (resp. F (cid:16) a,be ; 1 − x (cid:17) ) is the unique holomorphic solution with y (0) =1 (resp. y (1) = 1). NORIYUKI OTSUBO
Transformation.
We consider differential operators of the form D = ∂f ( x ) ∂ − g ( x ) . If F ( x ) is a solution of the differential equation D y = 0, we say for brevity that F ( x ) is a solution of D , and that D is a differential operator for F ( x ). This typeof differential equation is stable under a change of variables. Lemma 2.4.
Let F ( x ) be a solution of ∂f ( x ) ∂ − g ( x ) and z ( x ) be a non-constantholomorphic function. Then F ( z ( x )) is a solution of ∂f ( z ( x )) z ′ ( x ) − ∂ − z ′ ( x ) g ( z ( x )) . Proof.
Immediate from ∂ z = z ′ ( x ) − ∂ . (cid:3) Consider the differential operator as in (2.4) D = ∂ϕ ( x ) ∂ − ab ϕ ( x ) x (1 − x ) . The following examples of z ( x ) are important. First, for a positive integer s , let z ( x ) = x s . Then D becomes by Lemma 2.4(2.5) ∂x sc − s +1 (1 − x s ) e ∂ − s abx sc − (1 − x s ) e − . Secondly, for a positive integer r , let z ( x ) = 1 − x r − x . Note that the map x z is an involution since(1 + ( r − x )(1 + ( r − z ) = r. We have 1 − z = rx r − x , ∂ z = − r (1 + ( r − x ) ∂, so D becomes by Lemma 2.4(2.6) ∂x e (1 − x ) c ρ ( x ) − c − e +2 ∂ − rabx e − (1 − x ) c − ρ ( x ) − c − e , where we put ρ ( x ) = 1 + ( r − x. Finally, consider z ( x ) = (cid:18) − x r − x (cid:19) s = (cid:18) − xρ ( x ) (cid:19) s . This is the composition of the two substitutions as above. Then,(2.7) ∂ z = − rs (1 − x ) − s +1 ρ ( x ) s +1 ∂. In general, the resulting differential operator is not so simple. For ( r, s ) = (2 , ,
3) and (4 , − z = 4 x (1 + x ) , − z = 9 x (1 + x + x )(1 + 2 x ) = 9 x (1 − x )(1 − x )(1 + 2 x ) , NEW APPROACH TO HYPERGEOMETRIC TRANSFORMATION FORMULAS 5 − z = 8 x (1 + x )(1 + 3 x ) = 8 x (1 − x )(1 − x )(1 + 3 x ) . Hence the term z c (1 − z ) e is a product of powers of x , 1 − x , 1 − x s and ρ ( x ).This explains why, in each formula of Theorem 1.1, the differential equation for theright-hand side is of a manageable form.2.3. Comparison.
The formulas we prove are of the form h ( x ) F ( x ) = F ( x ) , where F i ( x ) is a solution of a differential operator D i of order 2 of the form D i = ∂f i ( x ) ∂ − g i ( x ) . Suppose that F ( x ), F ( x ) and h ( x ) are holomorphic at x = x , h ( x ) = 0 and( hF )( x ) = F ( x ) , ( hF ) ′ ( x ) = F ′ ( x ) . Then the equality h ( x ) F ( x ) = F ( x ) holds on a neighborhood of x if and only if h ( x ) F ( x ) is also a solution of D . Lemma 2.5.
The identity of operators h ( x ) D h ( x ) = D holds if and only if f ( x ) = f ( x ) h ( x ) ,g ( x ) = g ( x ) h ( x ) − ( f ( x ) h ′ ( x )) ′ h ( x ) . (2.8) Proof.
Using ∂h − h∂ = h ′ , we have an equality of differential operators h∂f ∂h = ( ∂h − h ′ ) f ( h∂ + h ′ ) = ∂f h ∂ + ∂f h ′ h − f h ′ h∂ − f h ′ = ∂f h ∂ + ( f h ′ h ) ′ − f h ′ = ∂f h ∂ + ( f h ′ ) ′ h. Hence h D h = ∂f h ∂ + ( f h ′ ) ′ h − g h , and the lemma follows. (cid:3) If the condition (2.8) holds, then D h ( x ) F ( x ) = h ( x ) − D F ( x ) = 0 , hence h ( x ) F ( x ) is a solution of D near x . Conversely, if one seeks a transforma-tion formula between F ( x ) and F ( x ), one is led to find a function h ( x ) satisfying(2.8).2.4. Linear Transformations.
As easy examples of our method, let us prove thefollowing formulas, respectively due to Euler and Pfaff. Later, we prove a q -analogueof the former in a similar manner (see Theorem 7.6). Theorem 2.6.
On a neighborhood of x = 0 , (1 − x ) a + b − c F (cid:18) a, bc ; x (cid:19) = F (cid:18) c − a, c − bc ; x (cid:19) , (2.9) (1 − x ) a F (cid:18) a, bc ; x (cid:19) = F (cid:18) a, c − bc ; xx − (cid:19) . (2.10) Proof. (2.9). Put h = (1 − x ) a + b − c , F = F (cid:16) a,bc ; x (cid:17) and F = F (cid:16) c − a,c − bc ; x (cid:17) .Using the notations of Section 2.3, we have by Theorem 2.2 D = ∂x c (1 − x ) c − a − b +1 ∂ − ( c − a )( c − b ) x c − (1 − x ) c − a − b , D = ∂x c (1 − x ) a + b − c +1 ∂ − abx c − (1 − x ) a + b − c . NORIYUKI OTSUBO
The first equality of (2.8) is obvious and for the second,( f h ′ ) ′ h − g h =( a + b − c ) cx c − (1 − x ) a + b − c − ( c − a )( c − b ) x c − (1 − x ) a + b − c = abx c − (1 − x ) a + b − c = g . Since F (0) = ( hF )(0) = 1 and F ′ (0) = ( hF ) ′ (0) = ( c − a )( c − b ) c , we have hF = F .(2.10). Put h = (1 − x ) a , F = F (cid:16) a,bc ; x (cid:17) and F = F (cid:16) a,c − bc ; xx − (cid:17) . Then,by Theorem 2.2 and Lemma 2.4, D = ∂x c (1 − x ) − a + b − c +1 ∂ + a ( c − b ) x c − (1 − x ) − a + b − c − , and D is the same as (2.9). The first equality of (2.8) is obvious. For the second,since f h ′ = − ax c (1 − x ) b − c , we have using the logarithmic derivatives( f h ′ ) ′ h = f h ′ (cid:18) cx − b − c − x (cid:19) h = − ax c − (1 − x ) a + b − c − ( c − bx ) . Hence (2.8) follows. The comparison of the initial values is easy, and (2.10) follows. (cid:3) Proof of Theorem 1.1
Here we give direct proofs of the formulas (1.1), (1.2) and (1.3). Alternativeproofs are given respectively in Section 4, Section 5 and Section 4.3.1.
Proof of (1.1) . Let F be the right-hand side and F be the F ( x ) in theleft-hand side. Here, with the notations of Section 2.2, r = s = 2, ρ = 1 + x , and z = (1 − x ) ρ , − z = 4 xρ , ∂ z = − ρ − x ∂. By Theorem 2.2 and Lemma 2.4, we have with the notations of Section 2.3 D = ∂x b (1 − x ) a − b +1 ρ − a − b +1 ∂ − abx b − (1 − x ) a − b +1 ρ − a − b − = ∂x b (1 − x ) a − b +1 ρ − a ∂ − abx b − (1 − x ) a − b +1 ρ − a − . On the other hand, we have by (2.5) D = ∂x b (1 − x ) a − b +1 ∂ − a ( a − b + 1) x b (1 − x ) a − b . Letting h = ρ a , we have f h = f . Since f h ′ = ax b (1 − x ) a − b +1 ρ − a − , we have( f h ′ ) ′ h = f h ′ (cid:18) bx − a − b + 1) x − x − a + 11 + x (cid:19) h = ax b − (1 − x ) a − b ρ − ( b − ( a + 1) x − ( a − b + 1) x ) . Hence the condition (2.8) is verified. Since F (0) = ( hF )(0) = 1 and F ′ (0) =( hF ) ′ (0) = a , we obtain hF = F . (cid:3) Remark . The original proof of Gauss also compares the differential equations.In Erd´elyi et. al. [8, p. 111, (5)], another proof is suggested but not explicitly.
NEW APPROACH TO HYPERGEOMETRIC TRANSFORMATION FORMULAS 7
Proof of (1.2) . Let F be the right-hand side and F be the F ( x ) in theleft-hand side. Here, r = s = 3, ρ = 1 + 2 x , and z = (1 − x ) ρ , − z = 9 x (1 − x )(1 − x ) ρ , ∂ z = − ρ (1 − x ) ∂. By Theorem 2.2 and Lemma 2.4, we have D = ∂x a +12 (1 − x ) a +12 ρ − a ∂ − a ( a + 1) x a − (1 − x ) (1 − x ) a − ρ − a − , D = ∂x a +12 (1 − x ) a +12 ∂ − a ( a + 1) x a +32 (1 − x ) a − . Letting h = ρ a , we have( f h ′ ) ′ h = a ( a + 1) x a − (1 − x ) a − ρ − (1 − x − x − x ) . One easily verifies the condition (2.7) and the coincidence of the initial values.Hence we obtain hF = F . (cid:3) Remark . See Section 3.4 for another proof. When a = 1, other proofs are givenby Borwein-Borwein-Garvan [5, Corollary 2.4], Chan [6, Sections 5 and 6], Cooper[7, Theorem 5.3] and Maier [20, Corollary 6.2].3.3. Proof of (1.3) . Let F be the right-hand side and F be the F ( x ) in theleft-hand side. Here, r = 4, s = 2, ρ = 1 + 3 x and z = (1 − x ) ρ , − z = 8 x (1 − x )(1 − x ) ρ , ∂ z = − ρ − x ∂. By Theorem 2.2 and Lemma 2.4, we have D = ∂x a +23 (1 − x ) a +23 ρ − a ∂ − a ( a + 2)2 x a − (1 − x )(1 − x ) a − ρ − a − , D = ∂x a +23 (1 − x ) a +23 ∂ − a ( a + 2)4 x a +23 (1 − x ) a − . Letting h = ρ a , we have( f h ′ ) ′ h = a ( a + 2)4 x a − (1 − x ) a − ρ − (2 − x − x − x ) . Then, the rest is just as above. (cid:3)
Remark . See Section 3.4 for another proof. When a = 1, other proofs are givenby Cooper [7, Theorem 5.3] and Maier [20, Corollary 6.2].An investigation of our proofs suggests that, contrary to (1.1), the formulas (1.2)and (1.3) have no generalization to a formula with two free parameters.3.4. Multivariable hypergeometric functions.
The formula (1.2) (resp. (1.3))is obtained as a specialization of a transformation formula for a hypergeometricfunction of two (resp. three) variables. Recall Lauricella’s hypergeometric functionof m variables [19] F ( m ) D ( a, b , . . . , b m ; c ; x , . . . , x m )= X n ,...,n m ≥ ( a ) n + ··· + n m ( b ) n · · · ( b m ) n m ( c ) n + ··· + n m (1) n · · · (1) n m x n · · · x n m m . It converges on the open unit polydisk { ( x , . . . , x m ) ∈ C m | ∀ i, | x i | < } . When m = 1, this is the Gauss function and when m = 2, this is Appell’s function F [1]. NORIYUKI OTSUBO
For m = 2 and m = 3, we have the following transformation formulas.(1 + x + y ) a F (2) D (cid:18) a , a + 16 , a + 16 ; a + 56 ; x , y (cid:19) = F (2) D (cid:18) a , a + 16 , a + 16 ; a + 12 ; 1 − u , − v (cid:19) , (3.1)where u = 1 + ωx + ω y x + y , v = 1 + ω x + ωy x + y ( ω = e πi ) . (1 + x + y + z ) a F (3) D (cid:18) a , a + 212 , a + 212 , a + 212 ; a + 56 ; x , y , z (cid:19) = F (3) D (cid:18) a , a + 212 , a + 212 , a + 212 , a + 23 ; 1 − u , − v , − w (cid:19) , (3.2)where u = 1 − x − y + z x + y + z , v = 1 − x + y − z x + y + z , w = 1 + x − y − z x + y + z . The formula (3.1) is due Koike-Shiga [17, Proposition 2.5] for a = 1 and Matsumoto-Ohara [21, Theorem 1] in general. The formula (3.2) is due Kato-Mastumoto [16,Proposition 1] for a = 1 and Matsumoto-Ohara [21, Theorem 3] in general.Then we obtain (1.2) (resp. (1.3)) from (3.1) (resp. (3.2)) by letting x = y (resp. x = y = z ), using the multinomial formula( a + · · · + a m ) n (1) n = X i + ··· + i m = n ( a ) i · · · ( a m ) i m (1) i · · · (1) i m . Note ( a ) n / (1) n = ( − n (cid:0) − an (cid:1) .All the proofs in [16], [17] and [21] use mathematical software to compare thetwo systems of partial differential equations. One might be able to give simplerproofs by extending the method of this paper. Put ∂ i = ∂∂x i , D i = x i ∂ i , D = m X i =1 D i . Then, F ( m ) D is a solution of the system of linear partial differential equations (see[13, Chapter 3, 9.1]) (cid:0) ( D + a )( D i + b i ) − x − i D i ( D + c − (cid:1) y = 0 ( i = 1 , . . . , m ) , (cid:0) ( D i + b i ) x − j D j − ( D j + b j ) x − i D i (cid:1) y = 0 ( i, j = 1 , . . . , m ) . In fact, this system is of rank m + 1. Using Theorem 2.2, we easily obtain thefollowing. Theorem 3.4.
Put ϕ i ( x ) = x c (1 − x ) a + b i − c ( i = 1 , . . . , m ) , ψ ( x ) = x c − (1 − x ) a − c . Then, F ( m ) D ( a, b , . . . , b m ; c ; x , . . . , x m ) is the unique solution of ∂ i ϕ i ( x i ) ∂ i − ab i ϕ i ( x i ) x i (1 − x i ) + ψ ( x i ) ∂ i (1 − x i ) b i X j = i x j ∂ j y = 0 ( i = 1 , . . . , m ) , NEW APPROACH TO HYPERGEOMETRIC TRANSFORMATION FORMULAS 9 (cid:16) ∂ i x b i i x b j − j ∂ j − ∂ j x b j j x b i − i ∂ i (cid:17) y = 0 ( i, j = 1 , . . . , m ) , such that y (0 , . . . ,
0) = 1 , ( ∂ i y )(0 , . . . ,
0) = ab i c ( i = 1 , . . . , m ) . Quadratic Formulas
The formulas (1.1) and (1.3) are two different combinations of the followingformulas, the former due to Kummer [18, p. 78, Formula 44], and the latter due toRamanujan (cf. [2, p. 50, Chap. 11, Entry 4]). These are in fact equivalent as isclear from the proof below.
Theorem 4.1.
On a neighborhood of x = 0 , (1 + x ) a F a , a +12 b + ; x ! = F (cid:18) a, b b ; 1 − − x x (cid:19) , (4.1) (1 + x ) a F (cid:18) a, ba − b + 1 ; x (cid:19) = F a , a +12 a − b + 1 ; 1 − (cid:18) − x x (cid:19) ! . (4.2) Proof. (4.1). Let F be the right-hand side and F be the F ( x ) in the left-handside. By (2.6) with r = 2, the differential operator for F is D = ∂x b (1 − x ) a − b +1 ρ − a − b +1 ∂ − abx b − (1 − x ) a − b ρ − a − b − . = ∂x b (1 − x ) a − b +1 ρ − a − abx b − (1 − x ) a − b ρ − a − ∂, where ρ = 1 + x . On the other hand, by (2.5) with s = 2, the differential operatorfor F is D = ∂x b (1 − x ) a − b +1 ∂ − a ( a + 1) x b (1 − x ) a − b . Letting h = ρ a so that f h = f (with the notations of Section 2.3), we have( f h ′ ) ′ h = ax b − (1 − x ) a − b ρ − (2 b − ( a + 1) x − ( a + 1) x ) , and the condition (2.8) holds. Comparing the initial values, we obtain (4.1).(4.2). If we replace x with − x x , the formula becomes (cid:18) x (cid:19) a F (cid:18) a , a +12 a − b + 1 ; 1 − x (cid:19) = F (cid:18) a, ba − b + 1 ; 1 − x x (cid:19) . By Theorem 2.2, the both sides satisfy the same differential equation as those of(4.1). Comparing the initial values at x = 1, we obtain the equality above, hence(4.2). (cid:3) Now, let us deduce (1.1) and (1.3) from Theorem 4.1. Rewrite (4.1) and (4.2) as(1 + u ) a F a , a +12 b + ; u ! = F (cid:18) a, b b ; 1 − x (cid:19) , (4.3) F (cid:18) a , a +12 a − b ′ + 1 ; 1 − v (cid:19) = (1 + y ) a F (cid:18) a, b ′ a − b ′ + 1 ; y (cid:19) , (4.4)where (1 + x )(1 + u ) = (1 + y )(1 + v ) = 2 . First, if we let x = ξ , y = η and (1 + ξ )(1 + η ) = 2, then u + v = (cid:18) − ξ ξ (cid:19) + − ( − ξ ξ ) − ξ ξ ) ! = 1 . Letting b ′ = a − b + and equating the left-hand sides of (4.3) and (4.4), we obtain(1 + u ) a (1 + y ) a F a, a − b + b + ; y ! = F (cid:18) a, b b ; 1 − x (cid:19) . Since (1 + u )(1 + y ) = (1 + η ) , it becomes (1.1) (in variable η ) after a suitablechange of parameters.Secondly, if we let x + y = 1, then v = 1 − y y = x − x = − u u − − u u = 1 − u u . Letting b = b ′ = a +13 and equating the right-hand sides of (4.3) and (4.4), we obtain(1 + y ) a (1 + u ) a F a , a +122 a +56 ; u ! = F a , a +122 a +23 ; 1 − v ! . Since (1 + y )(1 + u ) = 1 + 3 u , it becomes (1.3) (in variable u ) after a suitable changeof parameters.Let us give short proofs of two other important formulas. First, the following isdue to Gauss [10, p. 226, Formula 102]. Theorem 4.2.
On a neighborhood of x = 0 , (4.5) F (cid:18) a, b a + b +12 ; x (cid:19) = F a , b a + b +12 ; 1 − (1 − x ) ! . Proof.
Letting z = (1 − x ) , we have 1 − z = 4 x (1 − x ), ∂ z = − − x ) ∂ . ByTheorem 2.2 and Lemma 2.4, the differential operator for the right-hand side is ∂x a + b +12 (1 − x ) a + b +12 ∂ − abx a + b − (1 − x ) a + b − , which coincides with the differential operator for the left-hand side. Comparing theinitial values, we obtain (4.5). (cid:3) The following is due to Kummer [18, Formula 53].
Theorem 4.3.
On a neighborhood of x = 0 , (4.6) (1 + x ) a F a, a − b +12 a + b +12 ; − x ! = F a , b a + b +12 ; 1 − (cid:18) − x x (cid:19) ! . Proof.
Let z = ( − x x ) as in the proof of (1.1). By Theorem 2.2 and Lemma 2.4,the differential operator for the right-hand side is D = ∂x a + b +12 (1 + x ) − a − b +1 ∂ − abx a + b − (1 + x ) − a − b − . Similarly, the differential operator for the F ( − x ) in the left-hand side is D = ∂x a + b +12 (1 + x ) a − b +1 ∂ + a ( a − b + 1)2 x a + b − (1 + x ) a − b . NEW APPROACH TO HYPERGEOMETRIC TRANSFORMATION FORMULAS 11
Letting h = (1 + x ) a , one verifies (2.8). Comparing the initial values, we obtain(4.6). (cid:3) Remark . In fact, (4.5) and (4.6) are equivalent to each other; apply Pfaff’sformula (2.10) to the left-hand side of (4.5) and then replace x with x x , to obtain(4.6). 5. Cubic Formulas
A cubic analogue of Theorem 4.1 is the following formulas due to Goursat [11,p. 140, (127) and (126)]. Here we give short proofs and see that two differentcombinations of (5.1) and (5.2) give the formulas (1.2) and (5.3).
Theorem 5.1.
On a neighborhood of x = 0 , (5.1) (1 + 8 x ) a F a , a +134 a +56 ; x ! = F a , a +134 a +56 ; 64 x (cid:18) − x x (cid:19) ! . On a neighborhood of x = 1 , (5.2) (cid:18) x (cid:19) a F a , a +134 a +12 ; 1 − x ! = F a , a +134 a +56 ; 64 x (cid:18) − x x (cid:19) ! . Proof.
If we let z = 64 x (cid:0) − x x (cid:1) , then1 − z = τ ρ , ∂ z = − ρ − x ) τ ∂, where we put ρ = 1 + 8 x, τ = 1 − x − x . By Theorem 2.2 and Lemma 2.4, the differential operator for the right-hand sidesof (5.1) and (5.2) is D = ∂x a +56 (1 − x ) a +12 ρ − a ∂ − a ( a + 1)9 x a − (1 − x ) a +32 ρ − a − . Letting h = ρ a , we have( f h ′ ) ′ h = 4 a x a − (1 − x ) a − ρ − (4 a + 5 − a + 1) x − a ) x ) , and then g h − ( f h ′ ) ′ h = 4 a (4 a + 1)9 x a − (1 − x ) a − . Therefore, by Theorem 2.2 and (2.8), h D h is nothing but the differential operatorfor F ( x ) (resp. for F (1 − x )) in the left-hand side of (5.1) (resp. (5.2)). Com-paring the initial values at x = 0 (resp. x = 1), we obtain (5.1) (resp. (5.2)). (cid:3) As was found by Chan [6, Section 6], (1.2) is deduced from Theorem 5.1 asfollows. If we let x = ξ , y = η and (1 + 2 ξ )(1 + 2 η ) = 3, then x (cid:18) − x x (cid:19) = y (cid:18) − y y (cid:19) , x )1 + 8 y = (1 + 2 ξ ) . Equating the right-hand sides of (5.1) and (5.2) and letting a → a/
4, we obtain(1.2) (in variable ξ ). On the other hand, by equating the left-hand sides of (5.1) and (5.2), which isonly possible for a = , we obtain (cid:18) x )9 − x (cid:19) F , − x ) (cid:18) x − x (cid:19) ! = F , x (cid:18) − x x (cid:19) ! on a neighborhood of x = 0. In particular, the both sides satisfy the same dif-ferential equation, which remains true near x = 1. Replacing x with x x andcomparing the initial values at x = 1, we obtain the following. Corollary 5.2.
On a neighborhood of x = 0 (resp. x = 1 ), (5.3)(1 + 80 x ) F (cid:18) , x − x x (cid:19) = C F , x x (cid:18) − x x (cid:19) ! , where C = 1 (resp. C = 3 ). The author does not know if it is equivalent to a known formula.6.
Quartic formulas
Here we treat two quartic formulas. First, the following is an iteration of (1.1).
Corollary 6.1.
On a neighborhood of x = 0 , (6.1) (1 + x ) a F a , a +16 a +56 ; x ! = F a , a +162 a +13 ; 1 − (cid:18) − x x (cid:19) ! . Proof.
As in the proof of (1.1) given in Section 4, let x = ξ , y = η and(1 + ξ )(1 + η ) = (1 + x )(1 + u ) = (1 + y )(1 + v ) = 2 , so that u + v = 1. Solving ( a , b , b ) = ( a ′ , a ′ − b ′ +12 , b ′ +12 ), we have b = a +23 ,( a ′ , b ′ , b ′ ) = ( a , a +16 , a +13 ). Then by (1.1), we have(1 + ξ ) a F a , a +16 a +56 ; ξ ! = F a , a +26 a +23 ; 1 − u ! = F a , a +26 a +23 ; v ! = (cid:18) η (cid:19) a F a , a +162 a +13 ; 1 − η ! . Since η = ξ (1+ ξ ) , we obtain (6.1) (in variable ξ ). (cid:3) The following formula of Matsumoto-Ohara [21, Corollary 2] is a specializationof a transformation formula (loc. cit. Theorem 2) for Appell’s function F , whoseproof is similar to (3.1) and (3.2). We give a direct proof. Theorem 6.2.
On a neighborhood of x = 0 , (6.2) (1 + x ) a F a , a +14 a +34 ; − x ! = F a , a +14 a +12 ; 1 − (cid:18) − x x (cid:19) ! . NEW APPROACH TO HYPERGEOMETRIC TRANSFORMATION FORMULAS 13
Proof.
By Theorem 2.2 and (2.7), the differential operator D for the right-handside is given by f = (cid:0) x (1 + x ) (cid:1) a +12 (1 + x ) − a ,g = a ( a + 1)2 (cid:0) x (1 + x ) (cid:1) a − (cid:18) − x x (cid:19) (1 + x ) − a . On the other hand, by Theorem 2.2 and Lemma 2.4 applied to z ( x ) = − x , thedifferential operator D for the F ( − x ) in the left-hand side is given by f = ( x (1 + x )) a +12 , g = − a ( a + 1)2 x a +12 (1 + x ) a − . Letting h = (1 + x ) a , one easily verifies (2.8). Comparing the initial values, weobtain (6.2). (cid:3) Remark . The author learned from Hiroyuki Ochiai that (6.2) is also obtainedas a combination of (1.1) and (4.6) as follows. Use the same notations as in theproof of Corollary 6.1. By (1.1) with ( a, b ) → ( a , a +12 ), F a , a +34 ; u ! = (cid:18) x (cid:19) a F a , a +14 a +12 ; 1 − x ! . By (4.6) with ( a, b ) → ( a , ), F a , a +34 ; 1 − v ! = (1 + y ) a F a , a +14 a +34 ; − y ! . Then (6.2) (in variable η ) follows similarly as (6.1).7. q -analogues We give a new canonical form of the difference equation for a q -hypergeometricseries φ which generalizes (2.4), and apply it to give a proof of Heine’s transfor-mation formula.7.1. Preliminaries.
For the moment, let α , β , γ and q be indeterminates. Recallthe q -Pochhammer symbol ( α ; q ) n = n − Y i =0 (1 − αq i ) ∈ Z [ α, q ] ∩ Z [ α ][[ q ]] ∗ . Here, for a ring R , R ∗ denotes its unit group. The q -hypergeometric series φ isdefined by φ (cid:18) α, βγ ; x (cid:19) = ∞ X n =0 ( α ; q ) n ( β ; q ) n ( γ ; q ) n ( q ; q ) n x n . This is a power series in x with coefficients in Q ( α, β, γ, q ) ∩ Z [ α, β, γ ][[ q ]] ∗ .Recall the q -number [ n ] = 1 − q n − q ( n ∈ Z ) . Note that [ n ] | q =1 = n . We write α = q a symbolically and define the number[ a ] = 1 − q a − q = 1 − α − q ∈ Q ( α, q ) ∩ Z [ α ][[ q ]] ∗ . Define a difference operator ( q -derivation) ∆ by∆ f ( x ) = f ( x ) − f ( qx ) x − qx . Following Jackson [14], define the shift operator q δ by q δ f ( x ) = f ( qx ) , and the difference operator by[ δ + a ] = 1 − q δ + a − q = 1 − αq δ − q . In particular, we have by definition [ δ ] = x ∆ . Then we have [ δ + a ] x n = 1 − αq n − q x n = [ a + n ] x n . Hence [ δ + a ] is the q -analogue of the differential operator D + a , where D = x ddx .Any function (or a power series) f ( x ) defines a multiplication operator. To avoidpossible confusion in writing operators, we write f ′ ( x ) = ∆ f ( x ) (the function f ( x ) acted by ∆).As an operator, ∆ f ( x ) means the composition of f ( x ) and ∆. Lemma 7.1.
For any f ( x ) , we have an identity of operators ∆ f ( x ) = f ( qx )∆ + f ′ ( x ) . Proof.
Since( f ( x ) g ( x )) ′ = f ( x ) g ( x ) − f ( qx ) g ( qx ) x − qx = f ( qx ) g ( x ) − g ( qx ) x − qx + f ( x ) − f ( qx ) x − qx g ( x ) = f ( qx ) g ′ ( x ) + f ′ ( x ) g ( x )for any g ( x ), the lemma follows. (cid:3) q -hypergeometric difference equation. Recall the differential equation(2.1) for F (cid:16) a,bc ; x (cid:17) . Similarly, since[ δ + a ] x n = ( α ; q ) n +1 (1 − q )( α ; q ) n x n , the series φ (cid:16) α,βγ ; x (cid:17) satisfies the difference equation(7.1) (cid:0) [ δ + a ][ δ + b ] − x − [ δ ][ δ + c − (cid:1) y = 0 , where we used symbols α = q a , β = q b and γ = q c . From this, we derive differenceequations analogous to (2.2) and (2.3). Proposition 7.2.
Put ε = qαβγ − . Then φ (cid:16) α,βγ ; x (cid:17) is a solution of the differ-ence equation (7.2) (cid:16) γx (1 − εx )∆ + (cid:0) [ c ] − ( αβ + β [ a ] + α [ b ]) x (cid:1) ∆ − [ a ][ b ] (cid:17) y = 0 , NEW APPROACH TO HYPERGEOMETRIC TRANSFORMATION FORMULAS 15 or equivalently (7.3) (cid:18) ∆ + (cid:18) [ c ] γx − αβ + β [ a ] + α [ b ] − ε [ c ] γ (1 − εx ) (cid:19) ∆ − [ a ][ b ] γx (1 − εx ) (cid:19) y = 0 . Proof.
In (7.1), substitute[ δ + a ] = 1 − αq δ − q = α − q δ − q + 1 − α − q = α [ δ ] + [ a ] . By Lemma 7.1, we have ∆ x = qx ∆ + 1 (operators), hence[ δ ] = x ∆ x ∆ = qx ∆ + x ∆ . Then, using [ c −
1] + q − γ = [ c ], we obtain (7.2), hence (7.3). (cid:3) Now, let q ∈ C with 0 < | q | <
1. Let a , b , c ∈ C , α = q a , β = q b , γ = q c andassume γ q − N . Then the power series φ (cid:16) α,βγ ; x (cid:17) ∈ C [[ x ]] defines an analyticfunction on | x | <
1. Note that, in the limit as q →
1, we have α, β, γ → a ] , [ b ] , [ c ] → a, b, c , hence φ (cid:16) α,βγ ; x (cid:17) → F (cid:16) a,bc ; x (cid:17) . Since [ δ + a ] D + a and∆ → ∂ , the equation (7.1) (resp. (7.2), (7.3)) specializes to (2.1) (resp. (2.2),(2.3)).We give a q -analogue of (2.4). The function x a satisfies(7.4) ∆ x a = [ a ] x a − . A q -analogue of the function (1 − x ) a is the following. Definition 7.3.
For α = q a , define a function by φ α ( x ) = ( x ; q ) ∞ ( αx ; q ) ∞ , where ( x ; q ) ∞ = Q ∞ i =0 (1 − xq i ). It converges on | x | < F (cid:0) a ; x (cid:1) = (1 − x ) − a . Similarly, we have the q -binomial theorem φ (cid:16) α ; x (cid:17) := ∞ X n =0 ( α ; q ) n ( q ; q ) n x n = φ α ( x ) − (see for example [9, (1.3.2)]). Moreover, one sees easily the following. Lemma 7.4.
We have φ q ( x ) = 1 − x,φ αβ ( x ) = φ α ( βx ) φ β ( x ) = φ α ( x ) φ β ( αx ) ,φ ′ α ( x ) = − [ a ] φ α ( x )1 − x = − [ a ] φ q − α ( qx ) . Our canonical form of the difference equation for φ is the following. Theorem 7.5.
Put ϕ ( x ) = x c φ ε ( x ) where ε = qαβγ − . Then φ (cid:16) α,βγ ; x (cid:17) is theunique solution of the difference equation (7.5) (cid:18) ∆ ϕ ( x )∆ + (1 − q )[ a ][ b ] ϕ ( x )1 − x ∆ − [ a ][ b ] ϕ ( x ) x (1 − x ) (cid:19) y = 0 such that y (0) = 1 , (∆ y )(0) = [ a ][ b ][ c ] . Proof.
By Lemma 7.1, (7.4) and Lemma 7.4, we have identities of operators∆ ϕ ( x ) = ( γx c ∆ + [ c ] x c − ) φ ε ( x )= γx c ( φ ε ( qx )∆ + φ ′ ε ( x )) + [ c ] x − ϕ ( x )= γϕ ( x ) (cid:18) − εx − x ∆ − [ e ]1 − x + [ c ] γx (cid:19) = γϕ ( x ) 1 − εx − x (cid:18) ∆ + [ c ] γx − [ e ] γ (1 − εx ) (cid:19) . Then we obtain (7.5) from (7.3), using[ e ] − (1 − q )[ a ][ b ] = αβ (1 − q ) + β (1 − α ) + α (1 − β ) − ε (1 − γ )1 − q = αβ + β [ a ] + α [ b ] − ε [ c ] . The initial condition follows by∆ φ (cid:18) α, βγ ; x (cid:19) = [ a ][ b ][ c ] φ (cid:18) qα, qβqγ ; x (cid:19) , and the uniqueness is evident. (cid:3) Transformation.
The following transformation formula due to Heine [12, p.325, XVIII] is a q -analogue of Euler’s formula (2.9). We give an analogous proofusing Theorem 7.5. Theorem 7.6.
We have (7.6) φ αβγ − ( x ) φ (cid:18) α, βγ ; x (cid:19) = φ (cid:18) α − γ, β − γγ ; αβγ − x (cid:19) . Before the proof, we introduce another notation.
Definition 7.7.
For α = q a , define an operator α δ = q aδ by α δ f ( x ) = f ( αx ) . Then, one easily shows the following.
Lemma 7.8.
There are identities of operators ( αβ ) δ = α δ β δ , ∆ α δ = αα δ ∆ ,α δ g ( x ) α − δ = g ( αx ) . Proof of Theorem 7.6.
Put s = a + b − c and σ = q s = αβγ − . By Theorem7.5, φ (cid:16) α − γ,β − γγ ; x (cid:17) and φ (cid:16) α,βγ ; x (cid:17) are solutions of the difference operators,respectively, D =∆ x c φ qσ − ∆ + (1 − q )[ c − a ][ c − b ] x c φ σ − ( qx )∆ − [ c − a ][ c − b ] x c − φ σ − ( qx ) . D =∆ x c φ qσ ∆ + (1 − q )[ a ][ b ] x c φ σ ( qx )∆ − [ a ][ b ] x c − φ σ ( qx ) . The right-hand side of (7.6) is a solution of D σ − δ . We show the identity ofoperators(7.7) σ − c φ σ ( qx ) σ δ D σ − δ φ σ ( x ) = D . NEW APPROACH TO HYPERGEOMETRIC TRANSFORMATION FORMULAS 17
Then it follows that the left-hand side of (7.6) is also a solution of D σ − δ . Com-paring the initial values, which is easy, we obtain the theorem.We compute the left-hand side of (7.7). For the first term, using Lemmas 7.1,7.4, 7.8 and (7.4), we have σ − c φ σ ( qx ) σ δ ∆ x c φ qσ − ( x )∆ σ − δ φ σ ( x )= σ − c φ σ ( qx )∆ σ δ x c φ qσ − ( x ) σ − δ ∆ φ σ ( x )= φ σ ( qx )∆ x c φ qσ − ( σx )∆ φ σ ( x )=(∆ φ σ ( x ) − φ ′ σ ( x )) x c φ qσ − ( σx )( φ σ ( qx )∆ + φ ′ σ ( x ))=∆ x c φ σ ( x ) φ qσ − ( σx ) φ σ ( qx )∆ + ∆ x c φ σ ( x ) φ qσ − ( σx ) φ ′ σ ( x ) − x c φ ′ σ ( x ) φ qσ − ( σx ) φ σ ( qx )∆ − x c φ ′ σ ( x ) φ qσ − ( σx )=∆ x c φ qσ ( x )∆ − [ s ]∆ x c φ σ ( x ) + [ s ] x c φ σ ( qx )∆ − [ s ] x c φ q − σ ( qx )=∆ x c φ qσ ( x )∆ − [ s ] { ( qx ) c φ σ ( qx )∆ + ( x c φ σ ( x )) ′ } + [ s ] x c φ σ ( qx )∆ − [ s ] x c φ q − σ ( qx )=∆ x c φ qσ ( x )∆ + [ s ](1 − γ ) x c φ σ ( qx )∆ − [ s ]( x c φ σ ( x )) ′ − [ s ] x c φ q − σ ( qx )=∆ x c φ qσ ( x )∆ + [ s ](1 − γ ) x c φ σ ( qx )∆ − [ s ] (1 − γ ) x c φ q − σ ( qx ) − [ s ][ c ] x c − φ σ ( x ) . For the second term, we have similarly σ − c φ σ ( qx ) σ δ x c φ σ − ( qx )∆ σ − δ φ σ ( x )= σ − c φ σ ( qx ) σ δ x c φ σ − ( qx ) σ − δ ∆ φ σ ( x )= σx c φ σ ( qx ) φ σ − ( qσx )( φ σ ( qx )∆ + φ ′ σ ( x ))= σx c φ σ ( qx )∆ − [ s ] σx c φ q − σ ( qx )) . Finally for the last term, we have σ − c φ σ ( qx ) σ δ x c − φ σ − ( qx ) σ − δ φ σ ( x )= σx c − φ σ ( qx ) φ σ − ( qσx ) φ σ ( x ) = σx c − φ σ ( x ) . Then, using [ s ](1 − γ ) + (1 − q )[ c − a ][ c − b ] σ = (1 − q )[ a ][ b ] ,φ σ ( qx ) = (1 − σx ) φ q − σ ( qx ) = 1 − σx − x φ σ ( x ) , we obtain (7.7), hence the theorem. (cid:3) Acknowledgement
The author would like to thank Ryojun Ito, Hiroyuki Ochiai, Nobuki Takayamaand Raimundas Vid¯unas for helpful discussions. This work is supported by JSPSGrant-in-Aid for Scientific Research: 18K03234.
References [1] P. Appell and J. Kamp´e de F´eriet,
Fonctions hyperg´eom´etriques et hypersph´eriques , Gau-thier Villars, Paris, 1926.[2] B. C. Berndt,
Ramanujan’s Notebooks, Part II , Springer-Verlag, New York, 1989.[3] B. C. Berndt, S. Bhargava and F. G. Garvan, Ramanujan’s theories of elliptic functions toalternative bases, Trans. Amer. Math. Soc. , No. 11 (1995), 4163–4244.[4] J. M. Borwein and P. B. Borwein, A cubic counterpart of Jacobi’s identity and the AGM,Trans. Amer. Math. Soc. (1991), No. 2, 691–701.[5] J. M. Borwein, P. B. Borwein and F. G. Garvan, Some cubic modular identities of Ramanu-jan, Trans. Amer. Math. Soc. (1994), No. 1, 35–47.[6] H. H. Chan, On Ramanujan’s cubic transformation formula for F ( , ; 1; z ), Math. Proc.Cambridge Philos. Soc. (1998), No. 2, 193–204.[7] S. Cooper, Inversion formulas for elliptic functions, Proc. London Math. Soc. (3) (2009),461–483.[8] A. Erd´elyi et al. ed., Higher transcendental functions, Vol. 1 , McGrow-Hill, New York, 1953.[9] G. Gasper and M. Rahman,
Basic Hypergeometric Series , Second ed., Cambridge Univ.Press, 2004.[10] C. F. Gauss, Determinatio seriei nostrae per aequationem differentialem secundi ordinis; in:
C. F. Gauss, Werke, Band III , K¨oniglichen Gesellschaft der Wissenschaften, G¨ottingen,1876, 207–230.[11] ´E. Goursat, Sur l’equation diff´erentielle lin´eaire, qui admet pour int´egrale la s´erie hy-perg´eom´etrique, Ann. Sci. ´Ecole Norm. Sup. (1881), 3–142.[12] E. Heine, Untersuchungen ¨uber die Reihe 1 + (1 − q α )(1 − q β )(1 − q )(1 − q γ ) · x + (1 − q α )(1 − q α +1 )(1 − q β )(1 − q β +1 )(1 − q )(1 − q )(1 − q γ )(1 − q γ +1 ) · x + · · · , J. Reine Angew. Math. (1847), 285–328.[13] K. Iwasaki, H. Kimura, S. Shimomura and M Yoshida, From Gauss to Painlev´e: a moderntheory of special functions; dedicated to Professor Tosihusa Kimura , Springer, 1991.[14] F. H. Jackson, q -difference equations, Amer. J. Math. , No. 4 (1910), 305–314.[15] C. G. J. Jacobi, Untersuchungen ¨uber die Differentialgleichung der hypergeometrischenReihe, J. reine angew. Math. , 149–165; in: C. G. J. Jacobi’s Gesammelte Werke, Vol.6 , Cambridge Univ. Press, 2013, 184–202.[16] T. Kato and K. Matsumoto, The common limit of a quadruple sequence and the hypergeo-metric function F D of three variables, Nagoya Math. J. (2009), 113–124.[17] K. Koike and H. Shiga, Isogeny formulas for the Picard modular form and a three termsarithmetic geometric mean, J. Number Theory (2007), 123–141.[18] E. E. Kummer, ¨Uber die hypergeometrische Reihe 1 + α · β · γ x + α ( α +1) β ( β +1)1 · · γ ( γ +1) x + α ( α +1)( α +2) β ( β +1)( β +2)1 · · · γ ( γ +1)( γ +2) x + · · · , J. Reine Angew. Math. (1836), 39-83, 127–172. in: Collected Papers II , Springer-Verlag, Berlin, 1975, 75–166.[19] G. Lauricella, Sulle Funzioni ipergeometrche a piu variabili, Rend. Circ. Mat. Palermo (1893), 111–158.[20] R. S. Maier, Algebraic hypergeometric transformations of modular origin, Trans. Amer.Math. Soc. (2007), No. 8, 3859–3885.[21] K. Matsumoto and K. Ohara, Some transformation formulas for Lauricella’s hypergeometricfunctions F D , Funkcialaj Ekvacioj (2009), 203–212.[22] E. G. C. Poole, Introduction to the Theory of Differential Equations , Oxford Univ. Press,1936.[23] S. Ramanujan,
Notebooks (2 volumes), Tata Institute of Fundamental Research, Bombay,1957.[24] R. Vid¯unas, Algebraic transformations of Gauss hypergeometric functions, Funkcialaj Ek-vacioj, (2009), 139–180. E-mail address : [email protected]@math.s.chiba-u.ac.jp