AN ELEMENTARY APPROACH TO THE GAUSS HYPERGEOMETRIC FUNCTION

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1 AN ELEMENTARY APPROACH TO THE GAUSS HYPERGEOMETRIC FUNCTION TOSHIO OSHIMA Abstract. We give an introduction to the Gauss hypergeometric function, the hypergeometric equation and their properties in an elementary way. Moreover we explicitly and uniformly describe the connection coefficients, the reducibility of the equation and the monodromy group of the solutions.. Introduction The Gauss hypergeometric function is the most fundamental and important special function and it has long been studied from various points of view. Many formulae for the function have been established and they are contained in the books on special functions such as [WW], [EMO], [WG], [SW] etc. In this paper we show and prove the fundamental formulae in an elementary way. We first give local solutions of the Gauss hypergeometric equation for every parameter, recurrent relations among three consecutive functions and contiguous relations. Then we show the Gauss summation formula, the connection formula and the monodromy group which is expressed by an explicit base of the space of the solutions depending holomorphically on the parameters of the hypergeometric equation. Our results are valid without an exception of the value of the parameter. The author recently shows in [O] and [O2] that it is possible to analyze solutions of general Fuchsian linear ordinary differential equations and get the explicit formulae as in the case of the Gauss hypergeometric equations, in particular, in the case when the equation has a rigid spectral type. Theorem 8 with Remark 9 may contain a new result but most results in this paper are known. The author hopes that this paper will be useful for the reader to understand the Gauss hypergeometric functions and moreover the analysis on general Fuchsian differential equations in [O2]. In this paper we will not use the theory of integrals nor gamma functions even for the connection formula and for the expression of the monodromy groups in contrast to [MS]. For example, the Liouville theorem is known to be proved by the Cauchy integral formula, whose generalization is the Fuchs relation on Fuchsian linear ordinary differential equations, is proved without the theory of integrals as follows. Let ux n0 a nx n be a function on C defined by a power series whose radius of convergence is. Suppose there exists a non-negative integer N such that + x N ux is bounded. Suppose moreover that ux is not a polynomial. N Replacing ux by x ux N+ n0 a nx n, we may assume lim x ux 0. Then there exists c C satisfying ux uc 0 for all x C. Replacing ux by Cuax + c with a certain complex numbers a and C, we may assume that there Key words and phrases. Gauss hypergeometric function, monodromy representations. 200 Mathematics Subject Classification. Primary 33C05; Secondary 32S40, 34M35. Supported by Grant-in-Aid for Scientific Researches A, No , Japan Society of Promotion of Science.

2 2 TOSHIO OSHIMA exists a positive integer m such that ux + x m + n b nx m+n and ux u0 x C. But if 0 < ε, we have n b n ε n 2 and uε + 2 εm. 2. Gauss hypergeometric series and hypergeometric equation For complex numbers α, β and γ, Euler studied that the Gauss hypergeometric series α n β n x n F α, β, γ; x γ n n! + αβ x αα + ββ + x γ! γγ + 2! + with 2.2 n0 n a n : a + ν aa + a + n a C ν0 gives a solution of the Gauss hypergeometric equation 2.3 x xu + γ α + β + x u αβu 0. Here we note that 2.4 F α, β, γ; x F β, α, γ; x. We will review this and obtain all the solutions of the equation around the origin. We introduce the notation : d dx, ϑ : x and then the Gauss hypergeometric equation is 2.5 P α,β,γ u 0 with the linear ordinary differential operator 2.6 P α,β,γ : x x 2 + γ α + β + x αβ. Since ϑ 2 x x x ϑ, we have xp α,β,γ x γx x x α + β + x + αβ 2.7 the equation 2.3 is equivalent to ϑ 2 ϑ + γϑ x ϑ 2 + α + βϑ + αβ ϑϑ + γ xϑ + αϑ + β, 2.8 ϑϑ + γ u xϑ + αϑ + βu. Putting u n0 c nx n and comparing the coefficients of x n in the equation 2.8, we have 2.9 nn + γ c n n + αn + βc n c 0, n 0,,... and therefore α + n β + n c n c n γ + n n α + n α + n 2β + n β + n 2 γ + n γ + n 2nn which shows that 2.0 u [α,β,γ] x : F α, β, γ; x is a solution of 2.3 if 2. γ / {0,, 2,...}. c n 2 α nβ n c 0, γ n n!

3 AN ELEMENTARY APPROACH TO THE GAUSS HYPERGEOMETRIC FUNCTION 3 3. Local solutions For a function hx and a linear differential operator P we put and then Thus we have 3. Ad hx P : hx P hx Ad hx h x hx, Adxλ ϑ ϑ λ λ C. Ad x γ xp α,β,γ ϑ γ + ϑ xϑ + α γ + ϑ + β γ + xp α γ+,β γ+,2 γ. Since P α,β,γ u 0 is equivalent to Ad x γ xp α,β,γ x γ u 0, we have another solution 3.2 v [α,β,γ] x : x γ F α γ +, β γ +, 2 γ; x if 2 γ / {0,, 2,...}, namely, 3.3 γ / {2, 3, 4,...}. We have linearly independent solutions u [α,β,γ] and v [α,β,γ] when γ / Z. Since u [α,β,] v [α,β,], the function 3.4 w 0 [α,β,γ] : γ u [α,β,γ] v [α,β,γ] is holomorphic with respect to γ when γ <. Then we have 3.5 w 0 [α,β,] x log x F α, β, ; x + a k x k with some a k C and 3.6 w 0 [α,β,] x d x t F α + t, β + t, + t; x F α, β, t; x. dt t0 Note that u [α,β,γ] x and w 0 [α,β,γ]x give independent solutions when γ <. Now we examine the case when γ m with a non-negative integer m. In this case the function γ +mu [α,β,γ] γ m is a solution of P α,β, m u 0 and therefore γ+mf α, β, γ; x γ m α m+β m+ m m m +! xm+ F α+m+, β+m+, m+2; x and m m m m!. Hence the solution 3.7 w m+ [α,β,γ] u [α,β,γ] α m+β m+ γ m+ m +! v [α,β,γ] m {0,, 2,...} is holomorphic with respect to γ if γ + m < and w m+ [α,β,γ] m γ m k0 α k β k m k x k k! + km+ b k x k + α m+β m+ m m m +! xm+ log x F α + m +, β + m +, m + 2; x. Then v [α,β,γ] and w m+ [α,β,γ] are independent solutions when γ + m <. k

4 4 TOSHIO OSHIMA By the analytic continuation along the path [0, 2π] t e t z, the solutions change as follows u [α,β,γ] e 2π z u [α,β,γ] z, v [α,β,γ] e 2π z e 2π γ v [α,β,γ] z, w m+ [α,β, m] z + 2π v [α,β, m] z m, w m+ [α,β, m] e2π z w m+ [α,β, m] z +2π αm+βm+ m v mm+! [α,β, m]z m 0,,.... When γ m 2 < with m {0,, 2,...}, we have independent solutions u [α,β,γ] and x γ w m+ [α γ+,β γ+,2 γ], which is obtained by the correspondence ux x γ ux. Thus we have the following pairs of independent solutions. u[α,β,γ], v [α,β,γ] γ / Z, 3.8 v[α,β,γ], w m [α,β,γ] γ m <, m {, 0,, 2,...}, u[α,β,γ], x γ w m [α γ+,β γ+,2 γ] γ m <, m {2, 3, 4,...}. Remark. It is easy to see from 2.9 that there exists a polynomial solution ux with u0 if and only if 3.9 {α, β} {0,, 2, 3,...} and γ / {0,,..., m} or m 0 with m : max { {α, β} {0,, 2, 3,...} }. Then the polynomial solution is called a Jacobi polynomial and equals m α k β k x k 3.0 γ k k!. k0 Here 3.9 is equivalent to the existence of m {0,, 2,...} such that 3. α + mβ + m 0 and α + kβ + kγ + k 0 k 0,..., m. 4. Symmetry of hypergeometric equation By the coordinate transformation T 0 : x x, we have T 0 and T 0 P α,β,γ x x 2 + γ α + β + + α + β + x αβ P α,β,α+β γ+. Then we have local solutions for x < : 4. u [α,β,α+β γ+] x F α, β, α + β γ + ; x, v [α,β,α+β γ+] x x γ α β F γ α, γ β, γ α β + ; x. By the coordinate transformation T 0 : x x, we have T 0 ϑ ϑ and x Ad x α T 0 xp α,β,γ Ad x α xϑ ϑ + γ ϑ + α ϑ + β xϑ + αϑ + α γ + ϑϑ + α β xp α,α γ+,α β+. Then we have local solutions for x > : 4.2 x α u [α,α γ+,α β+] x x α F α, α γ +, α β + ; x, x α v [α,α γ+,α β+] x x β F β, β γ +, β α + ; x. F m, β, γ; x is called a Jacobi polynomial and the Legendre polynomial, spherical polynomial and Chebyshev polynomial are special cases of this Jacobi polynomial cf. [WG] etc..

5 AN ELEMENTARY APPROACH TO THE GAUSS HYPERGEOMETRIC FUNCTION 5 We have local solutions at the singular points x and x by using u [α,β,γ ], v [α,β,γ ] if γ / Z. When γ Z, we have independent solutions by the pairs of functions given in 3.8. We have the Riemann scheme 4.3 P 0 0 α ; x γ γ α β β which indicates the characteristic exponents at the singular points 0, and of the equation P α,β,γ u 0 and represents the space of solutions of P α,β,γ u 0. In general a differential equation has a characteristic exponent λ at x c if it has a solution u whose singularity at x c is as follows. Under the coordinate y x c or y x according to c or c, there exists a positive integer k such that lim y 0 y λ log k y uy is a non-zero constant. The maximal integer k is the multiplicity of the characteristic exponent λ. In most cases the multiplicity is free and then k. The characteristic exponent of P α,β,γ u 0 at the origin is multiplicity free if and only if γ. Then T 0 and T 0 give x 0 P 0 0 α ; x P 0 0 α ; x γ γ α β β γ α β γ β P α 0 0 ; x β γ α β γ x α P 0 0 α ; x. β α γ α β α γ + which corresponds to the above solutions. Compositions of transformations that we have considered give x α+β γ P 0 0 α ; x γ γ α β β P 0 α + β γ γ β ; x γ 0 γ α and P 0 0 α x ; x γ γ α β β P 0 α 0 ; x γ β γ α β x α P 0 0 α ; x γ β α γ β.

6 6 TOSHIO OSHIMA Put γ γ, α α and β γ β. The local holomorphic function at the origin in the above which takes the value at the origin gives Kummer s formula 4.4 F α, β, γ; x x γ α β F γ α, γ β, γ; x 4.5 x α x F α, γ β, γ; x 4.6 x β x F β, γ α, γ; x. 5. Reccurent relations For integers l, m and n the function F α + l, β + m, γ + n; x is called the consecutive function of F α, β, γ; x and it has been shown by Gauss that among three consecutive functions F, F 2 and F 3, there exists a recurrence relation of the form 5. A F + A 2 F 2 + A 3 F 3 0, where A, A 2 and A 3 are rational functions of x. In short we put F F α, β, γ; z and F α + l, γ + n F α + l, β, γ + n; x, F α F α, β, γ; x etc. Then there is a recurrence relation among F and any two functions of 6 closed neighbors F α ±, F β ±, F γ ±, which we give in this section. There are recurrence relations of this type and they generate the recurrence relations 5.. Since we have and therefore 5.2 α + n n! α + n β n γ n n! α n n! α + n α + n α n! x n α nβ n x n β γ n n! γ Moreover since α n γ α α + n γ n γ n we have α + n, n! α + n β + n x n γ + n n! γ F α + F βxf α +, β +, γ +. γ α α n γ n α n γ n γ + n α + n γ α 0, 5.3 γ F γ αf α + γ α F 0. We obtain other recurrence relations from these two as follows. By the symmetry between α and β we have 5.4 γ F γ βf β + γ β F 0 and by the difference of the above two relations we have 5.5 αf α + βf β + α βf 0. Moreover 5.2 γ γ is γ F α +, γ αf α + γ α F βxf α +, β + and therefore γ F γ α F γ αf α βxf β + 0.

7 AN ELEMENTARY APPROACH TO THE GAUSS HYPERGEOMETRIC FUNCTION 7 Substituting x 5.5 from the above, we have 5.6 α xf α + + γ 2α + α βx F γ αf α 0. The equation 5.2 α α x 5.4 γ γ+ shows 5.7 γ xf γf α + γ βxf γ + 0 and γ α 5.7 γ 5.6 shows 5.8 γ α γ βx F αγ xf α + + γ αγ βxf γ + 0 and 5.8 γ x 5.3 shows 5.9 γ γ 2γ α β x F + γ αγ βxf γ + γγ xf γ 0. The relations 5.6, its transposition of α and β, 5.3, 5.7 and 5.5 generate other 4 relations among the nearest neighbors except for 5.9. In fact 5.6 x 5.5 gives 5.0 β xf β + + γ α βf γ αf α 0 and 5.0 α β 5.6 gives 5. α β xf + γ αf α γ βf β 0 and x gives 5.2 γ xf γ α γ β x F γ αf α 0. Then 5.3, 5.5, 5.6, 5.7, 5.8, 5.9, 5.0, 5. and 5.2 are the 5 recurrence relations. Here the sign represents two relations under the transposition of α and β. Since we have 6. Contiguous relations d α n β n x n dx γ n n! αβ γ α + n β + n x n γ + n n! d αβ 6. dxf α, β, γ; x γ F α +, β +, γ + ; x. Combining this with 5.2, we have αf α + αf + αβx γ F α +, β +, γ + x d dx + α F and then 5.3 shows x d dx + α F γ F γ + α + γf. Thus we have Since x d dx + α F α F α +, x d dx + β F β F β +, x d dx + γ F γ F γ. P α,β,γ x x + α γ α + β + x x α x αβ γ α βx αβ x γ α βxx + α α x γ α,

8 8 TOSHIO OSHIMA we have 6.5 xp α,β,γ x x + γ α βx x + α αγ α and x x + γ α βx αf α + αγ α F. Hence 6.6 x x + γ α βx F γ α F α, x x + γ β αx F γ β F β. 6.7 In the same way, since we have P α,β,γ x x + γ γ α + β + x x γ x αβ γ α β x αβ γ α β x + γ γ α γ β, 6.8 P α,β,γ x + γ α β x + γ γ α γ β and Hence 6.9 γ x + γ α β F γ γ α γ β F. Note that x d dx + γ α β γ αγ β F F γ +. γ P P P 2 c P 2 P P 2 P cp 2 and P P P P 2 P c for linear differential operators P, P and P 2 and c C. We apply this to the equations 6.5 and 6.8. Then P 2 P c equals and x + α x x + γ α βx αγ α x 2 x 2 + x 2x + α x + γ α βx βx αβx x x x 2 + γ α + β + 2x β αβ xp α+,β,γ x + γ x + γ α β γ α γ β x x 2 + x + γ α β x + γ x αβ x x 2 + γ α + β + x αβ P α,β,γ, respectively, and we have 6.0 x + αxp α,β,γ xp α+,β,γ x + α, xp α,β,γ x x + γ α βx x x + γ α βx xpα+,β,γ, x + γ P α,β,γ P α,β,γ x + γ, P α,β,γ x + γ α β x + γ α β Pα,β,γ.

9 AN ELEMENTARY APPROACH TO THE GAUSS HYPERGEOMETRIC FUNCTION 9 Remark 2. Suppose ux is a solution of P α,β,γ u 0. Then 6.0 shows that vx x + αux is a solution of P α+,β,γ v 0. In fact, we have two linear maps 6. P 0 0 α ; x γ γ α β β x d dx +α P 0 0 α + ; x. γ γ α β β x x d dx +γ α βx If αγ α 0, 6.5 shows ux αγ α x x + γ α βx vx and hence these linear maps give the isomorphisms between the space of solutions of P α,β,γ u 0 and that of P α+,β,γ v 0. Suppose αγ α 0. Then x x + γ α βx x + αux 0. Since the kernels of these maps in 6. are of dimension 0 or, they are non-zero maps. Hence the dimensions of the kernels should be. For example, x α belongs to the left Riemann scheme in 6.. The same argument as above is valid for the linear maps 6.2 P 0 0 α ; x γ γ α β β x d dx +γ x d dx +γ α β They are bijective if and only if γ α γ β 0. P 0 0 α ; x. 2 γ γ α β β 7. Gauss summation formula When Re γ α β > 0 and γ / {0,, 2,...}, we have the Gauss summation formula α n β n 7. F α, β, γ; Cγ α, γ β; γ, γ α β. γ n n! Here we put n0 7.2 Cα, α 2 ; β, β 2 C α α 2 α + nα 2 + n β β 2 : β n0 + nβ 2 + n Γβ Γβ 2 Γα Γα 2 with α + α 2 β + β 2. Since the solution of P α,β,γ u 0 on the open interval 0, is spanned by 2 u [α,β,α+β γ+] x and v [α,β,α+β γ+] x when 7.3 γ α β / Z, 2 Let p C \ {0, }. Putting x p after applying n to 2.3, we see that u n+2 p is determined by u 0 p,..., u n+ p for n 0,,... and therefore u n+2 p is determined by up and u p, which implies that the dimension of local analytic solutions at p is at most 2. It is easy to show that any local solution can be analytically continued along any path which does not go through the singular point of the equation.

10 0 TOSHIO OSHIMA there exist constants C α,β,γ and C α,β,γ such that 7.4 F α, β, γ; x C α,β,γ F α, β, α + β γ + ; x + C α,β,γ x γ α β F γ α, γ β, γ α β + ; x. Since the hypergeometric series F α, β, γ; x is holomorphic with respect to the parameters α, β and γ under the condition 7.5 γ / {0,, 2,...}, C α,β,γ and C α,β,γ are holomorphic with respect to the parameters under the conditions 7.3 and 7.5. Hence under these conditions, the equation 7.4 implies 7.6 lim x 0 F α, β, γ; x C α,β,γ if 7.7 Re γ α β > 0. We prepare the following lemma to prove the absolute convergence of 7.. Lemma 3. i For any positive number b we have + b2 n 2 <. n ii Let α, β and γ C satisfying γ / {0,, 2,...}. Fix a non-negative integer N such that {Re α+n, Re β +N, Re γ +2N},. Then there exits a positive number C > 0 satisfying α n β n Re α + N γ < C n Re β + N n n {0,, 2,...}. Re γ + 2N n Proof. i For a positive integers N m > b 2 + we have N + b2 n 2 nm n N b2 n 2 nm N nm b 2 n 2 b2 m + b2 N > m b2 +, m N nm b 2 nn which implies the claim i. ii We may assume α, β / {0,, 2,...}. The number α n β n γ α + n β + n n γ + 2 αβγ + nγ + n + α + nβ + nγγ + n converges to αβ γγ+ when n, which implies that if N > 0, then α, β, γ, N may be replaced by α +, β +, γ + 2, N for the proof of ii. Hence we may assume that Re α, Re β and Re γ are larger than and N 0. Putting α a + bi with a, we have 2 n αn a + k 2 + b 2 n Re α n a + k 2 + b2 k 2 + b2 k 2 <. k0 k We have the similar estimate for β and hence the claim ii. k

11 AN ELEMENTARY APPROACH TO THE GAUSS HYPERGEOMETRIC FUNCTION Proposition 4. If the conditions 7.5 and 7.7 are valid, the hypergeometric series F α, β, γ; x converges absolutely and uniformly to a continuous function on the closed unit disk D : {x C x }. The function is also continuous for the parameters α, β and γ under the same conditions and lim F α, β, γ + m; x uniformly on x D. m Proof. If we use Starling s formula 3 for the Gamma function, we easily prove the first claim of the proposition but here we don t use it. Lemma 3 assures that we may assume that α, β and γ are positive real numbers satisfying γ > α + β. Fix ϵ 0 such that 0 < γ α β ϵ / Z. Then the hypergeometric series F α, β + ϵ, γ; t with t 0, is a sum of non-negative numbers and defines a monotonically increasing function on 0, and it satisfies n0 α n β n γ n x n n! F α, β, γ; x F α, β + ϵ, γ; x C α,β+ϵ,γ < for x D and therefore the proposition is clear. Suppose 7.5 and 7.7. If Re γ α + β > 0, then F, F γ + and F α + in 5.8 are continuous functions on D and we have 4 γγ α βf α, β, γ; + γ αγ βf α, β, γ + ; 0 by putting x. This equality is valid by holomorphic continuation for γ under the conditions 7.5 and 7.7. Hence F α, β, γ; γ αγ β γγ α β F α, β, γ+; γ α mγ β m γ m γ α β m F α, β, γ+m; for m, 2,.... Putting m in the above, we have the Gauss summation formula A connection formula Suppose 7.3 and 7.5. We have proved C α,β,γ Cγ α, γ β; γ, γ α β in 7.4. Then 4.4 shows F α, β, γ; x x γ α β F γ α, γ β, γ; x C γ α,γ β,γ x γ α β F γ α, γ β, γ α β + ; x + C γ α,γ β,γf α, β, α + β γ + ; x. Comparing this equation with 7.4, we have C α,β,γ C γ α,γ β,γ Cα, β; γ, α + β γ and the connection formula F α, β, γ; x C F α, β, α + β γ + ; x 8. + C α β γ α+β γ γ α γ β γ γ α β x γ α β F γ α, γ β, γ α β + ; x. This is valid when γ / {0,, 2,...} and γ α β / Z. If F α, β, γ ; x with γ Z appears in the above, we use w γ [α,β,γ ] to give one of the local solutions as in 3.8 and we get a connection formula from 8. with 3.4 or Note that αn Γα+n. Stirling s formula says Γz 2πe Γα z z z 2 when z and π Argz is larger than a fixed positive number. We can also prove the claim by the estimates t 2 log + t t for 0 t 2 and logn + n m + log n. m 4 Differentiating the first equalities in 8 by x, we also get this equality because the equality 6. shows αβ γ C γ α,γ β,γ+ α + β γcγ α, γ β, γ. 5 Other proofs and generalizations of the formula are given in [O2].

12 2 TOSHIO OSHIMA 9. Symmetric expression of the equation By applying Ad x λ x µ to P α,β,γ, the corresponding Riemann scheme of the equation P ũ 0 with P : x x Ad x λ x µ P α,β,γ equals 9. with P λ 0, λ, λ, ; x λ 0,2 λ,2 λ,2 x 0 P λ µ α λ µ ; x γ + λ γ α β + µ β λ µ { λ0, λ, λ, µ, λ, α λ µ, λ 0,2 γ + λ, λ,2 γ α β + µ, λ,2 β λ µ, α λ 0, + λ, + λ,, β λ 0, + λ, + λ,2, γ λ 0, λ 0,2 + and ũx x λ 0, x λ, ux is a solution of P ũ 0 for the solution ux of P α,β,γ u 0. The Riemann scheme 9. represents the solutions of P ũ 0 and it is called the Riemann P -function. Here we have the Fuchs relation 9.4 λ 0, + λ 0,2 + λ, + λ,2 + λ, + λ,2 and P x x x x λ x + µ x 2 + γ α + β + x λ x + µ x x 2 x λ x + 2µ x + λ x + µ 2 x + λ2 2 x 2λµ 2 x x + µ2 x 2 + γ α + β + x x x λ x + µx αβx x x 2 x x x 2λ x + 2µx + γ α + β + x + λ 2 + λ x 2 2λµx x + µ 2 + µx 2 + γ α + β + x λ + µx λ + αβx 2 x x 2 x 2 2 x x α + β 2λ 2µ + x + 2λ γ + λ 2 + λ + 2λµ + µ 2 + µ α + β + λ + µ + αβ x 2 + 2λ 2 2λ 2λµ + γλ + µ + λα + β + αβ x + λ 2 + λ γλ x 2 x 2 2 x x λ, + λ,2 + x + λ 0, + λ 0,2 + λ, λ,2 x 2 + λ, λ,2 λ 0, λ 0,2 λ, λ,2 x + λ 0, λ 0,2 x 2 x 2 2 λ0,+λ0,2 λ 0, λ 0,2 +λ, λ,2 λ, λ,2 x x x. + λ,+λ,2 x + λ0,λ0,2 x + λ,λ,2 2 x + 2 αβ Suppose λ p, λ p,2 / Z for p 0, and. Let u λp,ν p x denote the normalized local solutions of P ũ 0 at x p such that x p 0, 9.5 up λp,ν x y λp,ν ϕ p,ν y, y x p, x p

13 AN ELEMENTARY APPROACH TO THE GAUSS HYPERGEOMETRIC FUNCTION 3 and ϕ p,ν t are holomorphic if t < and satisfies ϕ p,ν 0. In fact, 4. and 4.2 imply 9.6 ϕ 0,ν x x λ,i F λ 0,ν + λ,i + λ,, λ 0,ν + λ,i + λ,2, λ 0,ν λ 0, ν + ; x, ϕ,ν x x λ 0,i F λ 0,i + λ,ν + λ,, λ 0,i + λ,ν + λ,2, λ,ν λ, ν + ; x, ϕ,ν x x λ,if x λ0, + λ,i + λ,ν, λ 0,2 + λ,i + λ,ν, λ,ν λ, ν + ; x. For indices i, j, ν {, 2}, we put ī 3 i, j 3 j and ν 3 ν. Then for i and 2, we have the connection formula 2 u λ 0,i λ0,i +λ 0 x C, ν +λ, λ 0,i +λ, ν +λ,2 λ 0,i λ 0,ī + λ, ν λ,ν u λ,ν {ī 3 i, x ν 3 ν 9.7 ν 2 ν Γλ 0,i λ 0,ī + Γλ, ν λ,ν Γλ 0,i + λ, ν + λ, Γλ 0,i + λ, ν + λ,2 uλ,ν x when λ 0,i λ 0,ī / {, 2,...} and λ,ν λ, ν / Z. Since C λ0,+λ,2+λ, λ 0,+λ,2+λ,2 λ 0, λ 0,2 + λ,2 λ, C γ α γ β γ γ α β and the connection coefficients in the right hand side of 9.7 are invariant under the transformation u ũ x λ x µ u, 8. implies that the coefficient of u λ, x is as given in 9.7. Hence 9.7 is valid because of the symmetries λ, λ,2 and λ 0, λ 0,2. First note that P λ 0, λ, λ, ; x P λ 0,2 λ, λ, ; x etc. λ 0,2 λ,2 λ,2 λ 0, λ,2 λ,2 and P is invariant under the transpositions λ p, λ p,2 for p 0, and. Moreover for λ C we have x x λ P λ 0, λ, λ, ; x P λ 0, + λ λ, λ, λ ; x, λ 0,2 λ,2 λ,2 λ 0,2 + λ λ,2 λ,2 λ 9.9 x λ P λ 0, λ, λ, ; x P λ 0, λ, + λ λ, λ ; x, λ 0,2 λ,2 λ,2 λ 0,2 λ,2 + λ λ,2 λ 9.0 P λ 0, λ, λ, ; x P λ, λ 0, λ, ; x, λ 0,2 λ,2 λ,2 λ,2 λ 0,2 λ,2 9. P λ 0, λ, λ, ; x P λ, λ, λ 0, ; x. λ 0,2 λ,2 λ,2 λ,2 λ,2 λ 0,2 By the transformation x x in 9.7, we have 9.2 u λ,i x 2 ν 2 ν λ,i +λ C 0, ν +λ, λ,i +λ 0, ν +λ,2 λ,i λ,ī + λ 0, ν λ 0,ν u λ 0,ν 0 x Γλ,i λ,ī + Γλ 0, ν λ 0,ν Γλ,i + λ 0, ν + λ, Γλ,i + λ 0, ν + λ,2 uλ 0,ν 0 x.

14 4 TOSHIO OSHIMA Some combinations of 9.0 and 9. give similar identities of the Riemann schemes corresponding to the transformations x x, x and x x. In general, for the Riemann scheme x c 0 c c 2 P λ 0, λ, λ 2, ; x with {c 0, c, c 2 } {0,, } λ 0,2 λ,2 λ 2,2 satisfying the Fuchs relation λ j,ν, we fix local functions u j,ν x belonging to the Riemann scheme corresponding to the characteristic exponents λ j,ν of the points x c j, respectively. We normalize u 0,, u, and u,2 as follows. Putting I 0,, φ 0 x x, φ x x if c 0, c 0,, 9.3 we have I 0,, φ 0 x x, φ x x if c 0, c, 0, I,, φ 0 x x, φ x x if c 0, c,, I,, φ 0 x x, φ x x if c 0, c,, I, 0, φ 0 x x, φ x x if c 0, c 0,, I, 0, φ 0 x x, φ x x if c 0, c, 0, 9.4 lim I x c 0 φ 0 x λ0, u 0, x, lim φ x λ,ν u,ν x I x c and φ 0 x λ 0, u 0, x is holomorphic at x c 0 and φ x λ,ν u,ν x are holomorphic at x c for ν and 2. Note that φ 0 x > 0 and φ x > 0 when x I. Then we have the connection formula 2 λ0, +λ u 0, x C, ν +λ, λ 0, +λ, ν +λ,2 λ 0, λ 0,2 + λ, ν λ,ν u,ν x 9.5 ν 2 ν Γλ 0, λ 0,2 + Γλ, ν λ,ν Γλ 0, + λ, ν + λ, Γλ 0, + λ, ν + λ,2 u,νx for x I. Here we note that α + α 2 β + β 2 in the definition of C α α 2 β β 2. In particular, we have F α, β, γ; x ΓγΓγ α β F α, β, α + β γ + ; x Γγ αγγ β γ α β ΓγΓα + β γ + x F γ α, γ β, γ α β + ; x ΓαΓβ α ΓγΓβ α x α, ΓβΓγ α F α γ +, α β + ; x β ΓγΓα β + x β, ΓαΓγ β F β γ +, β α + ; x Here F a, b, c; z is considered to be a holomorphic function on C \ [,. In general, if the fractional linear transformation x y ax+b cx+d with ad bc 0 transforms 0,, to c 0, c and c 2, respectively, then by this coordinate transformation we have the Riemann scheme y c 0 c c P λ 0, λ, λ 2, ; y λ j,ν λ 0,2 λ,2 λ 2,2 j0 ν.

15 AN ELEMENTARY APPROACH TO THE GAUSS HYPERGEOMETRIC FUNCTION 5 and the corresponding differential equation and its solutions. For simplicity, 9.6 is simply expressed by c 0 c c 2 { } P λ 0, λ, λ 2, or P λ0, λ, λ 2,. λ λ 0,2 λ,2 λ 0,2 λ,2 λ 2,2 2,2 If c j < for j 0,, 2, the equation ˆP û 0 of this Riemann scheme is given by 9.7 ˆP d2 dx 2 2 j0 λ j, + λ j,2 d 2 x c j dx + j0 λ j, λ j,2 ν {0,,2}\{j} c j c ν x c j x c 0 x c x c 2, which is obtained by a direct calculation or by characteristic exponents at every singular point and the fact that is not a singular point. This equation was first given by Papperitz. Applying Ad x λ 0, x, λ to 6.2, we have linear maps x P λ 0, λ, λ, ; x λ 0,2 λ,2 λ,2 x d dx + λ, x λ 0, λ 0,2 λ, x d dx + λ 0, x λ0, λ, λ,2+ P which are bijective if and only if x 0 λ 0, λ, λ, ; x, λ 0,2 + λ,2 λ,2 9.9 λ 0,2 + λ, + λ, λ 0,2 + λ, + λ,2 0. Hence in general, there are non-zero linear maps x c 0 c c 2 P x c 0 c c P λ 0, λ, λ 2, ; x P λ 0, λ, λ 2, ; x P λ 0,2 λ,2 λ 2 2,2 λ 0,2 + λ,2 λ 2,2 given by differential operators P and P 2 and they are bijective if and only if 9.2 λ 0,2 + λ, + λ 2, λ 0,2 + λ, + λ 2, Monodromy and irreducibility The equation P ũ 0 has singularities at 0, and but any local solution ux in a small neighborhood of a point x 0 X : C { } \ {0, } is analytically continued along any path in X starting from x 0. It defines a single-valued holomorphic function on the simply connected domain X : C \, 0] [,. In particular, F α, β, γ; x defines a holomorphic function on C \ [,. Let u, u 2 be a base of local solutions of P ũ 0 at x 0. Let γ p be closed paths starting from x 0 and circling around the point x p once in a counterclockwise direction for p 0, and, respectively, as follows. γ 0 x 0 0 γ Let γ p u j be the local solutions in a neighborhood of x 0 obtained by the analytic continuation of u j along γ p, respectively. Then there exist M p GL2, C satisfying γ

16 6 TOSHIO OSHIMA γ p u, γ p u 2 u, u 2 M p. Here GL2, C is the group of invertible matrices of size 2 with entries in C. The matrices M p are called the local generator matrices of monodromy of the solution space of P ũ and the subgroup of GLn, C generated by M 0, M and M is called the monodromy group. Here we note 0. M M M 0 I 2 the identity matrix and if we differently choose x 0 and u, u 2, the set of local generator matrices of monodromy M 0, M, M p changes into gm 0 g, gm g, gm g with a certain g GL2, C. If there exists a subspace V of C 2 such that {0} V C 2 and M p V V for p 0,,, then we say that the monodromy of P ũ 0 is reducible. If it is not reducible, it is called irreducible. Remark 5. Suppose P ũ 0 is reducible. Then there exists a non-zero local solution vx in a neighborhood of x 0 which satisfies γ p v C p v with C p C \ {0}. Then the function bx v x vx satisfies γ pb b and therefore bx Cx and vx is a solution of the differential equation v bxv. Here Cx is the ring of rational functions with the variable x. Since P a x bx a 0 x + cx with a x x 2 x 2, a 0 x, cx Cx, we have a division P a x + a 0 x bx + rx with rx Cx. The condition P vx 0 implies rx 0 and therefore we have P a x + a 0 x bx. In general, let W x denote the ring of ordinary differential operators with coefficients in rational functions. Then P W x and the equation P u 0 are called reducible if there exist Q, R W x such that P QR and the order of Q and that of R are both positive. If they are not reducible, they are called irreducible. We note that P ũ 0 is irreducible if and only if its monodromy is irreducible. λ0, a Lemma 6. Let A 0 0 λ, 0 and A 0 λ in GL2, C. Then 0,2 a λ,2 there exists a non-trivial proper simultaneous invariant subspace under the linear transformations of C 2 defined by A 0 and A if and only if 0.2 a 0 a a0 a + λ 0, λ 0,2 λ, λ,2 0. Proof. The lemma is clear when a 0 a 0 and therefore we may assume a 0 a 0. In this case if there exists a -dimensional invariant subspace, it is of the form C c with c 0 and A0 c λ0,2 c and A c λ, c, which is equivalent to λ 0, + a 0 c λ 0,2 and a + λ,2 c λ, c. This means c λ0,2 λ0, a a 0 λ, λ,2. Theorem 7. i When Re γ, the local monodromy of the Riemann Scheme P 0 0 α ; x at the origin is not semisimple if and only if γ γ α β β γ {0,, 2,...} and α, β / {0,,..., γ}. When γ {0,, 2,...}, the condition α, β / {0,,..., γ} is equal to the non-existence of a polynomial solution of degree γ with a non-zero value at the origin. ii The Riemann Scheme P λ 0, λ, λ, ; x with the Fuchs relation λ 0,2 λ,2 λ,2 p,ν λ p,ν has a non-semisimple local monodromy at the origin if and only if 0.3 λ 0, λ 0,2 Z and λ 0,k λ, λ,ν / {0,,..., λ 0, λ 0,2 } for ν, 2.

17 AN ELEMENTARY APPROACH TO THE GAUSS HYPERGEOMETRIC FUNCTION 7 Here k if λ 0,2 λ 0, 0 and k 2 otherwise. Proof. i Recall the description of local functions belonging to the Riemann scheme in 3. Then the local monodromy around the origin is not semisimple if γ and it is semisimple if γ / Z. Putting m γ, we may assume m {0,, 2,...} to prove the claim. Then the semisimplicity of the monodromy is equivalent to the condition that w m+ [α,β, m] in 3 has no logarithmic term, which equals the condition α m+ β m+ 0. Then Remark implies that it also equals the existence of a polynomial ux with u0 0 in the Riemann scheme. The claim in ii follows from i by the transformation of functions ux x λ0, x λ, ux or x λ0,2 x λ, ux according to Re λ 0,2 λ 0, is non-negative or negative, respectively. Theorem 8. For the Riemann scheme 0.4 P λ 0, λ, λ, ; x with the Fuchs relation λ p,ν λ 0,2 λ,2 λ,2 p, ν let M 0, M and M be the monodromy matrices around the points 0,,, respectively, under a suitable base. i M 0, M, M is irreducible if and only if 0.5 λ 0, + λ,ν + λ,ν / Z ν, ν {, 2}. ii Suppose 0.6 λ 0,2 + λ,2 + λ,ν / {0,, 2,...} ν, 2. We may assume 0.7 λ p, λ p,2 / {, 2, 3,...} p 0, by some of the permutations λ 0, λ 0,2 and λ, λ,2 if necessary. Under these conditions the functions u λ 0,2 0 x and u λ,2 x given by 9.5 are well-defined and linearly independent functions on 0,. When 0.8 λ 0,2 + λ, + λ,ν / Z ν, 2, there exists g GL2, C such that the monodromy matrices satisfy 0.9 gm 0 g, gm g e 2πiλ 0,2 a 0 e 2πiλ, 0, 0 e 2πiλ0, with 0.0 a 0 2e πiλ,2 sin πλ 0,2 + λ, + λ,2, a 2e πiλ, sin πλ 0,2 + λ, + λ,. a e 2πiλ,2 When 0.8 is not valid, we have 0.9 with a certain g GL2, C and { if λ 0, + λ,2 + λ,ν / {0,, 2,...} ν, 2, a 0 0 otherwise, 0. { if λ 0,2 + λ, + λ,ν / {0,, 2,...} ν, 2, a 0 otherwise. Note that a 0 a 0 in this case. iii Under a change of indices λ p,ν λ σp,σpν with suitable permutations σ, σ 0, σ, σ S 3 S 3 2 we have 0.6 and 0.7. Here S 3 and S 2 are identified with the permutation groups of {0,, } and {, 2}, respectively.

18 8 TOSHIO OSHIMA Proof. i, ii Assume 0.7 in the proof. Then the functions u λ0,2 0 and u λ,2 are well-defined and they satisfy γ 0 u λ 0,2 0 e 2πiλ 0,2 u λ 0,2 0 and γ u λ,2 e 2πiλ,2 u λ,2. Suppose that u λ 0,2 0 and u λ,2 are linearly dependent. Then v 0 : x λ0,2 x λ,2 u λ 0,2 0 P x 0 λ 0, λ 0,2 λ, λ,2 λ 0,2 + λ,2 + λ, 0 0 λ 0,2 + λ,2 + λ,2 is an entire function, therefore a polynomial cf. the last part of and 0.2 λ 0,2 + λ,2 + λ,k {0,, 2,...} for k or 2. Suppose 0.2. Since λ 0, λ 0,2 / {, 2,...} and v 0 x F λ 0,2 + λ,2 + λ,, λ 0,2 + λ,2 + λ,2, λ 0, + λ 0,2 ; x, Remark shows that v 0 is a polynomial and hence 0.4 is reducible. Suppose u λ 0,2 0 and u λ,2 are linearly independent. Under the base u λ 0,2 0, u λ,2 e 2πiλ 0,2 a M 0 0 e 2πiλ,, M e 2πiλ0,, M e 2πiλ,2 M M 0 I 2 and therefore trace M M 0 e 2πiλ0,2+λ, + a 0 a + e 2πiλ0,+λ,2 e 2πiλ, + e 2πiλ,2, a 0 a e 2πiλ, + e 2πiλ,2 e 2πiλ 0,2+λ, e 2πiλ 0,+λ,2 e 2πiλ,2 e 2πiλ 0,2+λ, +λ,2 e 2πiλ 0,+λ,2 +λ,2 e πiλ0,+λ0,2+λ,+λ,2 2i sin πλ 0,2 + λ, + λ,2 2i sin πλ 0, + λ,2 + λ,2 a 4e πiλ,+λ,2 sin πλ 0,2 + λ, + λ,2 sin πλ 0,2 + λ, + λ,. Hence the condition 0.8 implies a 0 a 0 and then we have 0.9 with 0.0 by multiplying u λ 0,2 0 by a suitable non-zero number. Since a 0 a + e 2πiλ 0,2 e 2πiλ 0, e 2πiλ, e 2πiλ,2 e 2πiλ, + e 2πiλ,2 e 2πiλ 0,+λ, e 2πiλ 0,2+λ,2 e 2πiλ,2 e 2πiλ0,+λ,+λ,2 e 2πiλ0,2+λ,2+λ,2, we have i in view of Lemma 6. Note that a 0 if and only if γ u λ 0,2 0 Cu λ 0,2 0. Since u λ 0,2 0 / Cu λ,2 and λ,2 λ, / {, 2,...}, the condition γ u λ 0,2 0 Cu λ 0,2 0 is valid if and only if x 0 ṽ 0 : x λ 0,2 x λ, u λ0,2 0 P λ 0, λ 0,2 λ,2 λ, λ 0,2 + λ, + λ, 0 0 λ 0,2 + λ, + λ,2 is a polynomial. Remark shows that this is also equivalent to λ 0,2 + λ, + λ,k {0,, 2,...} for k or 2 because λ 0, λ 0,2 / {, 2,...}. The condition a 0 0 is similarly examined. iii Suppose the claim iii is not valid. If λ 0,2 + λ,2 + λ, {0,, 2,...} and λ,2 λ, Z, λ 0, + λ, + λ,ν / {0,, 2,...} for ν, 2. Hence we may assume λ p,2 λ p, {0,, 2,...} for p 0,, and that the Riemann scheme is P { } λ0, λ, λ, P λ 0,2 λ,2 λ,2 { c0 c c 0 c n 0 n m c 0 + n 0 c + n c 0 c + m + with n 0, n, m {0,, 2,...}. Then λ 0,ν +λ,2 +λ,2 {0,,...} for ν, 2. }

19 AN ELEMENTARY APPROACH TO THE GAUSS HYPERGEOMETRIC FUNCTION 9 Remark 9. We calculate the monodromy matrices. We may assume 0.6 and 0.7. Then the monodromy matrices with respect to the base u λ0,2 0, u λ,2 depend holomorphically on the parameters λ p,ν. The function corresponding to the Riemann scheme P λ 0, λ, λ, has the connection formula λ 0,2 λ,2 λ,2 λ0,2 +λ a C,2 +λ, λ 0,2 +λ,2 +λ,2 u λ0,2 0 a u λ, + b u λ,2 λ with 0,2 λ 0, + λ,2 λ, λ0,2+λ b C,+λ, λ 0,2+λ,+λ,2 λ 0,2 λ 0,+ λ, λ,2 as is given in 9.7 and therefore 6 γ u λ0,2 0, γ u λ,2 γ u λ,, γ u λ,2 a 0 b u λ,, u λ,2 e 2πiλ, u λ 0,2 0, u λ,2 u λ 0,2 0, u λ,2 u λ0,2 0, u λ,2 e 2πiλ,2 a 0 b a 0 e 2πiλ, a 0 b e 2πiλ,2 b a 0 ae 2πiλ, 0 a b be 2πiλ,2 e 2πiλ,2 e 2πiλ, 0 be 2πiλ,2 e 2πiλ, e 2πiλ,2 e 2πiλ, e 2πiλ,2 2ie πiλ,+λ,2 sin πλ, λ,2 2πie πiλ,+λ,2 λ, λ,2 λ, λ,2 2 n 2 n, 2πie πiλ,+λ,2 Γλ, λ,2 Γ λ, + λ,2, λ0,2 +λ b C, +λ, λ 0,2 +λ, +λ,2 λ 0,2 λ 0, + λ, λ,2 Γλ 0,2 λ 0, + Γλ, λ,2 Γλ 0,2 + λ, + λ, Γλ 0,2 + λ, + λ,2. Hence putting a be 2πiλ, e 2πiλ,2, we have 7 γ u λ 0,2 0, γ u λ,2 u λ 0,2 0, u λ,2 e 2πiλ, a e 2πiλ,,2 a 2πie πiλ,+λ,2 Γλ 0,2 λ 0, + Γλ,2 λ, Γλ 0,2 + λ, + λ, Γλ 0,2 + λ, + λ,2. Note that under the conditions 0.6 and 0.7, the right hand side of the first line of 0.3 is holomorphic and never vanishes and the equality 0.3 is valid. In the same way we have γ 0 u λ 0,2 0, γ 0 u λ,2 u λ 0,2 0, u λ,2 e 2πiλ 0, a 0, e 2πiλ0,2 6 λ The Wronskian of u 0,2 0, u λ,2 equals λ,2 λ, ax λ 0,+λ 0,2 x λ,+λ,2. 7 Under the base ũ λ 0,2 0, ũ λ,2 with the functions ũ λ p,2 p matrices M 0 and M are given by a p Note that the functions ũ λ p,2 p u λ p,2 p, the monodromy Γλ p,2 λ p, + 2πie πiλ p, +λ p,2 for p 0,. Γλ p, +λ p,2 +λ, Γλ p, +λ p,2 +λ,2 holomorphically depend on the parameters λ j,ν C without a pole.

20 20 TOSHIO OSHIMA 0.4 a 0 2πie πiλ 0,+λ 0,2 Γλ,2 λ, + Γλ 0,2 λ 0, + Γλ 0, + λ,2 + λ, Γλ 0, + λ,2 + λ,2. Here we use Gamma functions for the convenience to the reader but we may use functions of infinite products of linear functions in place of Gamma functions. Remark 0. The differential operators P and P 2 in 9.20 induce the identity maps of the monodromy groups of the Riemann schemes under the condition 9.2. In particular, let P {α, β, γ} denote the Riemann scheme 4.3 which contains F α, β, γ; x. Then we have the isomorphisms: 0.5 P {α, β, γ} P {α +, β, γ} if αγ α 0, P {α, β, γ} P {α, β +, γ} if βγ β 0, P {α, β, γ} P {α, β, γ } if γ α γ β 0. The isomorphisms are given by the differential operators appeared in 6. and 6.2. The above conditions are equivalent to say that under the shift of the parameters, any integer contained in {α, β, γ α, γ β} does not change its property that it is positive or not. Put {α,..., α N } {α, β, γ α, γ β} Z. Then N 0 or or 2 or 4. When N or 2, we have α j 0 or for j N by suitable successive applications of the above isomorphisms. Let α, β, γ Z. Then it is easy to see that suitable successive applications of the above isomorphisms and the maps Adx ±, Ad x ±, T 0 and T 0 transform the equation P α, β, γu 0 into 2 u 0 with α, β, γ 0,, 0 or ϑ + u 0 with α, β, γ 0, 0, if #{α ν > 0 ν,..., 4} is odd or even, respectively. Note that their solution spaces are C +Cx or C +C log x, respectively.. Integral representation Lastly we give an integral representation of Gauss hypergeometric function:. z2 t z a z 2 t b z 3 t c Γa + Γb + dt z Γa + b + 2 z 2 z a+b+ z 3 z c F a +, c, a + b + 2; z 2 z z 3 z a >, b >, z 2 z < z 3 z. Putting z, z 2, z 3 0,, x and a, b, c α, γ α, β, we have an integral representation 8 of Gauss hypergeometric series, namely,.2 0 t α t γ α xt β dt ΓαΓγ α F α, β, γ; x. Γγ By the complex integral through the Pochhammer contour +, 0+, 0, along a double loop circuit, we have +,0+,,0 z α z γ α xz β dz.3 4π 2 e πiγ 0 starting point F α, β, γ; x. +, 0+, 0, Γ αγ + α γγγ Here we have no restriction on the values of the parameters α, β and γ. Put γ γ α. If α or γ is a positive integer, the both sides in the above vanish. 8 Putting x in.2, we get the Gauss summation formula.

21 AN ELEMENTARY APPROACH TO THE GAUSS HYPERGEOMETRIC FUNCTION 2 But replacing the integrand Φα, β, γ ; x, z : z α z γ xz β by the function Φ m α, β, γ ; x, z : Φα, β, γ ; x, z Φα, β, m; x, z γ, m we get the integral representation of F α, β, γ; x even if γ α m is a positive integer because the function Φ m α, β, γ, x, z holomorphically depends on γ. For example, when m, we have +,0+,,0 z α log z xz β dz 4π 2 e πiα F α, β, α + ; x. Γ αγ + α A similar replacement of the integrand is valid when α is a positive integer. Namely, the function Γ αγ + α γ +,0+,,0 z α z γ α xz β dz holomorphically depends on the parameters. Putting t z + z 2 z s and w z2 z z 3 z, the integral. equals z 2 z a+b+ z 3 z c s a s b ws c ds z 2 z a+b+ z 3 z c s a s b z 2 z a+b+ z 3 z c 0 0 n0 z 2 z a+b+ z 3 z c n0 w n n! 0 n0 c n ws n ds n! s a+n s b c n ds Γa + + nγb + c n w n, Γa + b + n + 2 which implies.. This function belongs to z 3 z z 2 P 0 0 c ; z 3 a + c + b + c + a b c as a function of z 3. Remark. An analysis of the monodromy group associated with the Gauss hypergeometric function using integrals is given in [MS]. References [EMO] A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, 3 volumes, McGraw-Hill Book Co., New York, 953. [MS] K. Mimachi and T. Sasaki, Monodromy representations associated with the Gauss hypergeometric function using integrals of a multivalued function, Kyushu J. Math , [O] T. Oshima, Special functions and algebraic linear ordinary differential equations, Lecture Notes in Mathematical Sciences, the University of Tokyo, 20, in Japanese, [O2] T. Oshima, Fractional calculus of Weyl algebra and Fuchsian differential equations, MSJ Memoirs 28, Mathematical Society of Japan, Tokyo, 202. [SW] S. Yu. Slavyanov and W. Lay, Spherical Functions, A Unified Theory Based on Singularities, Oxford Univ. Press, New York, [WG] Z. X. Wang and D. R. Guo, Special Functions, World Scientific, Singapore, 989. [WW] E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th edition, Cambridge University Press, London, 955. Graduate School of Mathematical Sciences, University of Tokyo, 7-3-, Komaba, Meguro-ku, Tokyo , Japan address: oshima@ms.u-tokyo.ac.jp