Generating Kummer Type Formulas forHypergeometric Functions
Nobuki Takayama
Kobe University, Rokko, Kobe 657-8501, Japan, takayama@math.kobe-u.ac.jp

Abstract. Kummer type formulas are identities of hypergeometric series. A symmetry by the permutations of n-letters yields these formulas. We will present analgorithmic method to derive known and new Kummer type formulas. The algorithm utilizes several algorithms in algebra and geometry for generating Kummer
type formulas.

1

Introduction

The Gauss hypergeometric function is defined by the series
F (a, b, c; x) =∞X(a)n (b)n(1)n (c)n
n=0

xn ,

(a)n = a(a+1) · · · (a+n−1) = Γ (a+n)/Γ (a)where a, b, c are complex parameters and we assume c 6∈ Z<0 . The series
in the right hand side converges in the unit disk and F (a, b, c; x) can beanalytically continued to C \ {0, 1} as a multi-valued analytic function. The
Gauss hypergeometric function satisfies several attractive formulas. Amongthem, we will consider Kummer’s formulax
F (a, b, c; x) = (1 − x)−a F a, c − b, c;
x−1
and its generalizations in this article.
There are several methods to prove it. Among them, we here refer toa method by the Chu-Vandermonde formula of binomial coefficients [19].
Although this proof presents an interesting interplay between the famous
identity in combinatorics and Kummer’s formula, it seems hard to generalize
this approach.
I.M.Gel’fand suggested an idea to derive and prove Kummer’s formula inhis series of lectures at Kyoto in 1989. The method is a natural generalization of the method to derive 24 = 4! solutions of the Gauss hypergeometricequation by Kummer. The point of his idea is that the method can be also
applied to hypergeometric functions associated to the product of simplices,which have the symmetry of permutations of n-letters.
Since his presentation of the idea, there have been only a few tries to study
systematically Kummer type formulas (see, e.g., [15]). The author thinks that

2

Nobuki Takayama

the reason for it is the lack of integrations of algebra and geometry softwaresystems. As readers will see in the body of this paper, the systematic studyof Kummer type formulas requires several kinds of explicit constructions inalgebra and geometry including Gröbner basis computation, triangulations,and constructions of fans, for which algorithms and systems have been intensively studied in the last 10 years.
In this article, we will present an algorithm to derive Kummer type formulas for hypergeometric functions associated to the product of simplices
developing the original idea by I.M.Gel’fand. We can experience an interplay
between algebra, geometry and software systems through deriving Kummer
type formulas of hypergeometric functions.

2Hypergeometric Function Associated to∆k−1 × ∆n−k−1

Let us recall the definition of hypergeometric functions associated to a product of linear forms [1], [8]. We fix P
two numbers k and n satisfying n ≥ 2k ≥ 4.
n
Let αj be parameters satisfying j=1 αj = −k. Hypergeometric function oftype Ek,n is defined by the integral
Φ(α; z) =

Z Y
k
n X
zij si )αj ds2 · · · dsk(
C j=1 i=1

where we put s1 = 1 and z ∈ Mk,n = the space of k × n matrices and C is aQn Pkbounded (k−1)-cell in the hyperplane arrangement defined by j=1 i=1 zij si =0 in the (s2 , . . . , sk )-space.
When k = 2, n = 4, it is written asZΦ(α; z) =(z11 + z21 s2 )α1 · · · (z14 + z24 s2 )α4 ds2 .
CThe integral Φ(z)agrees with the Gauss hypergeometric integral when z =
1 0 1
1
.
0 1 −1 −x
The hypergeometric function of type Ek,n is quasi-invariant under the action of complex torus (C∗ )n and the general linear group GL(k) = GL(k, C).


h1 0
0
0 
 0 h2
0 · · · 0  ∈ (C∗ )n and g ∈ GL(k),In fact, we have for h =  0·· ··· ·
00 · · · hnY αΦ(α; zh) =  hj j  Φ(α; z),
(1)
j

−1

Φ(α; gz) = |g|Φ(α; z).

(2)

Kummer type formulas

3

It follows from the quasi-invariant property and the integral representationthat the function Φ(α; z) satisfies a system of first order equations and asystem of second order equations respectively.
Theorem 1. (Gel’fand [8]) The function Φ(α; z) satisfieskX!

f = 0, p = 1, . . . , nnX!

f = 0, i, j = 1, . . . , k

∂
zip
− αp
∂zip
i=1

∂
+ δij
zip
∂z
jp
p=1



∂2
∂2
−
∂zip ∂zjq∂ziq ∂zjp

f = 0, i, j = 1, . . . , k, p, q = 1, . . . , n

We call this system of equations Ek,n .
Let A = (aij ) be an integer d × N -matrix of rank d and β a vector
over complex numbers (β1 , . . . , βd )T . The A-hypergeometric (GKZ hypergeometric) system HA (β) is the following system of linear partial differential
equations for the indeterminate function f (x1 , . . . , xN ):


N
X

aij xj ∂j − βi  f = 0,

for i = 1, . . . , d

j=1(∂ u − ∂ v )f = 0 for all u, v ∈ NN
0 with Au = Av.
Q
N
Here, we use the multi-index notation ∂ u = i=1 ∂iui .
The A-hypergeometric system was first introduced and studied by Gel’fand,Kapranov and Zelevinsky [9].
If we restrict the system of differential equations in Theorem 1 to theaffine chart
1 · · · 0 x11 · · · x1`
 0 · · · 0 x21 · · · x2` , ` = m − k
0 ··· 0···0 · · · 1 xk1 · · · xk`of GL(k)\Mk,n , we obtain the A-hypergeometric system associated to the
product of simplices ∆k−1 × ∆n−k−1 . More precisely, put
 
 
1
0
0
0
 
 
a1 =  ·  , . . . , an−k =  ·  ∈ Zn−k ,
 
 
·
·
0
1

4

Nobuki Takayama

1
 0

 0

 ·
Ak,n = 
 ·

 ·

0
a1


···
1
···
0
···
0
···
·
···
·
···
·
···
0
· · · an−k

0
1
0
·
·
·
0
a1

···
0
···
1
···
0
···
·
···
·
···
·
···
0
· · · an−k000·
· · · ·
·
1
a1


···
0
···
0 

···
0 

···
· 
,
···
· 

···
· 

···
1
· · · an−kand β = (−α1 − 1, . . . , −αk − 1, αk+1 , . . . , αn )T . Then, the restricted system of Ek,n is the A-hypergeometric system HAk,n (β) with x1 = x11 , x2 =x12 , . . . , x` = x1` , x`+1 = x21 , . . . , xN = xk` . Note that, by using the GL(k)quasi-invariant property (2), solutions of Ek,n can be expressed in terms ofsolutions of HAk,n (β). We will present later some explicit constructions ofsolutions of E2,4 and E2,5 .

3Configuration SpaceWe denote by Xk,n the quotient space∗
GL(k)\Mk,n/(C∗ )n .
∗Here, Mk,ndenotes the set of the k×n matrices of which any k×k minor does
not vanish. The space Xk,n is called the configuration space of n-points inthe (k − 1)-dimensional project space. It is a (k − 1) × (n − k − 1) dimensional
affine variety. The group of the permutations of n-columns of Mk,n acts on
Xk,n . We denote the group by Sn .
∗Regular functions on Mk,nwhich are invariant under the action of GL(k)∗ nand (C ) are regular functions on Xk,n . Let us denote by

yi1 ,···,ikthe determinant of (i1 , . . . , ik )-th columns of the k×n matrix z = (zij ), which∗is called the Plücker coordinate. Then, the affine chart xij of GL(k)\Mk,nisexpressed as
y1,2,···,i−1,k+j,i+1,···,k
xij =
,
(3)
y1,2,···,kwhich is invariant under the action of GL(k).
Let us take a vector
Ak,np = (pij ) ∈ Ker Zk×(n−k) −→ Zn .
Q pijThen, it is easy to see that xp =xij is invariant as a function of y =(yi1 ,...,ik ) by the actions of GL(k) and (C∗ )n . Hence, xp is a regular function
on the configuration space Xk,n .

Kummer type formulas

Let σ be an element of Sn . The action of σ on xp is defined asY  yσ(1),σ(2),···,σ(i−1),σ(k+j),σ(i+1),···,σ(k) pijσ • xp =yσ(1),σ(2),···,σ(k)
i,j

5

(4)

The action induces a biholomorphic transformation on Xk,n .
Example 1. k = 2, n = 4
The configuration space X2,4 is isomorphic to P1 \ {0, 1, ∞} = C \ {0, 1}. An
isomorphism is given by
y4,2 y1,3= x ∈ C \ {0, 1}.
X2,4 3 (zij ) 7→
y3,2 y1,4
We denote the Plücker coordinate yij by [ij]. Then, the action of S4 on
C1 \ {0, 1} is summarized as follows.
[42][13][32][14][41][23]1(12) • x ==[31][24]x[43][12][43][13](23) • x ==−1=x−1[23][14][32][14]([42][13] − [41][23] + [21][43] = 0, Plücker’s relation)1(34) • x =x

1 2 3 4Here, for example, (23) means∈ S4 .
1 3 2 4x :=Example 2. k = 2, n = 4
The configuration space X2,5 is isomorphic to
X = C2 \ {(x, y) | xy(x − 1)(y − 1)(x − y) = 0}
by the isomorphism
X2,5 3 (zij ) 7→

[42][13] [14][52],[32][14] [42][15]= (x, y) ∈ X

where [ij] = yi,j . The action of S5 is as follows.
(12) • x =1,x(12) • y =(23) • x = 1 − x,(34) • x =1,x(45) • x = xy,1y(23) • y =(34) • y = xy(45) • y =1y

1 − xy1−x6Nobuki Takayama

where Plücker relations
2X(−1)k yi1 ,jk yj0 ,···,ĵk ,···,j2 = 0

k=0

are used to reduce expressions in yij into expressions in x and y. The index
sets {i1 } and {j0 , j1 , j2 } run over the subsets of {1, 2, 3, 4, 5}. For instance,
when i1 = 1 and j0 = 3, j1 = 4, j2 = 5, the relation is y1,3 y4,5 − y1,4 y3,5 +
y1,5 y3,4 = 0.
We are interested in deriving explicit expressions of the action of Sn on
the configuration space Xk,n as in the two examples. It can be done by anelimination. Put m = (k − 1)(n − k − 1), which is the dimension of theconfiguration space Xk,n .

Data

p(1)

p(m)

: x1 = yyq(1) , . . . , xm = yyq(m) : a system of local coordinates ofa compactification of Xk,n , and σ ∈ Sn .
f (x ,...,x )Result : Polynomials fj and gj such that σ • xj = gjj (x11 ,...,xmm)
1. PutP = {the Plücker relations, yi1 ,···,ik yi∗1 ,···,ik − 1 |i1 , . . . , ik ⊂ {1, . . . , n}}.
DefineI = hP, y q(i) xi − y p(i) (i = 1, . . . , m), (σ • y q(j) )wj − (σ • y p(j) )i
as an ideal in∗∗Q[y1,2,...,k , . . . , yn−k+1,...,n , y1,2,...,k, . . . , yn−k+1,...,n, x1 , . . . , xm , wj ].

2. Find the ideal intersection J=
Q(x1 , . . . , xm )I ∩Q(x1 , . . . , xm )[wj ].
f (x)It is generated by an element of the form wj − gjj (x) .
f

3. Output gjj .

Algorithm 1: Action of Sn on the configuration space
The correctness follows from that the ideal generated by the Plücker relations is prime and hence radical. Here is a sketch of a proof of the primenessKummer type formulas

7suggested by B.Sturmfels. “Establish the SAGBI basis property of the k × kminors and then to infer that the Plücker ideal is precisely the ideal of all
algebraic relations on the k × k-minors. This implies primeness, since thekernel of a map from a domain to a domain is always prime.” Gröbner basis
of Plücker relations and the SAGBI argument are described in the chapter
three of the book by B.Sturmfels on invariant theory [17].
Elimination can be done by the characteristic set method or the Gröbner
basis method. Readers are invited to articles by M.Noro[10], H.Schönemann[14],and D.M.Wang[20] in this volume for algorithms, implementations and related topics in this area.
We close this section with a short remark on efficiency. Although themethod above is simple, it seems that it is not the most efficient method to
explicitly derive the action of Sn on a coordinate system of the configuration
space. It seems more efficient to use a method based on transforming a given
1 0 ··· 0 1 1 ··· 1
0 1 ··· 0 1 ∗ ··· ∗k × n matrix to the normal form ···
1 ∗ · · · ∗  by the left
···
1 ∗ ··· ∗0 0 ··· 1 1 ∗ ··· ∗GL(k) and the right (C∗ )n action.

4

Series solutions

Several methods to construct series solutions of A-hypergeometric systemhave been known. Among them, we would like to mention here the following
two constructions.
Theorem 2. [9]1. There is a correspondence between a regular triangulation T of A and abasis of series solutions of HA (β) for generic β.
2. A translation of the secondary cone of a regular triangulation T of A isthe domain of convergence of the basis of series solutions associated to
T.
Theorem 3. [12] There is a construction algorithm of series solution basis
of HA (β) for any β. The series solutions are constructed by extending thesolutions of the initial ideal in(−w,w) (HA (β)) with respect to a given weight(−w, w).
Details on these methods are explained in [9] and section 3.4 and chapter
two of the book [12]. Triangulations play an important role to construct series
solutions. Readers are invited to consult the article by J.Pfeifle and J.Rambau[11] on algorithms and implementations for triangulations.
We will add one more note here. Series solutions can be analytically continued to the complement of the singular locus of HAk,n (β), of which defining

8

Nobuki Takayama

equation is expressed in terms of sparse resultants. In case of A2,4 , the defining equation isx11 x12 x21 x22 (x11 x22 − x12 x21 ).
Readers are invited to the article by I.Emiris[6] on sparse resultants in thisbook.
Let φ(α; x) be a series solution of Ak,n -hypergeometric system. Then, by
the quasi-invariant relation (2), the function
Φ(α; y) =1φ(α; x(y))
y1,2,...,k(5)

is a solution of Ek,n where x(y) is a coordinate system defined in (3).
The invariant property of the system of equations Ek,n under the actionof Sn yields the following theorem.
Theorem 4. If the function Φ(α; y) is a solution of Ek,n , then the functionσ • Φ(α; y) := Φ(σ • α; σ • y) is also a solution of Ek,n for any σ ∈ Sn .
Example 3. k = 2, n = 4.
x
 11
1
 0
The matrix A2,4 is 
 1
0
HA2,4 (β) is generated by

x12
1
0
0
1

x21
0
1
1
0

x22

0
1 
. The hypergeometric system
0 
1x11 ∂11 + x12 ∂12 − (−α1 − 1), x21 ∂21 + x22 ∂22 − (−α2 − 1),x11 ∂11 + x21 ∂21 − α3 , x12 ∂12 + x22 ∂22 − α3 , ∂11 ∂22 − ∂12 ∂21
where ∂ij = ∂x∂ij and α1 + α2 + α3 + α4 = −2. Take the triangulation
T = {124, 134} or 
the Gröbner
deformation with respect to the weight
0 1
w = (0, 1, 1, 0) =. Then the starting terms of series solutions
1 0



p11 p12α3 −α1 − α3 − 1
are xp and xq where p ==and q =
p21 p220−α2 − 1−α1 − 10
. Note that−α2 − α4 − 1 α4
!

−1
[42]
1 0 [32]z11 z12[12][12].
(zij ) =[14]
z21 z220 1 [13]
[12]
[12]
[42]
[13]
[14]
Then, we put x11 = [32][12] , x12 = [12] , x21 = [12] , x22 = [12] . By extending
the starting terms to series solutions, we obtain solutions of HA2,4 (β) andconsequently those of E25 :η1 F (−α3 , α2 + 1, −α1 − α3 ; s)
η2 F (−α4 , α1 + 1, −α2 − α4 ; s)Kummer type formulas9x21= [42][13]where s = xx12[32][14] and11 x22α1 −α1 −α2 x11 3
1
x12 x22[12]x12x12 x22

α1
1
x21
x21
3 α4
η2 =
xα
,
21 x22[12] x11
x11
η1 =

P4

i=1 αi = −2. F is the Gauss hypergeometric function. We obtain |S4 | = 24
fundamental sets of solutions by applying the Theorem 4.

5Deriving Kummer Type Formulas

Let η1 (y)φ1 (α; x(y)), η2 (y)φ2 (α; x(y)), . . . , ηr (y)φr (α; x(y)) be a basis of solutions of Ek,n . It follows from the quasi-invariant property (1) and (2) thatthe quotient function(σ • ηi (y))φi (σ • α; σ • x(y))/η1 (y)φ1 (α; x(y))
is a multi-valued analytic function on Xk,n for σ ∈ Sn . Since, the function
φ1 (α; x(y)) is a multi-valued analytic function in Xk,n , the quotient function(σ • ηi (y))φi (σ • α; σ • x(y))/η1 (y)is also a multi-valued analytic function on the configuration space Xk,n .
We take a coordinate system
s1 = x(y)p(1) , s2 = x(y)p(2) , . . . , sm = x(y)p(m)(6)

of a compactification of the configuration space Xk,n which satisfies thecondition: all φi (α; x(y)) can be expressed in a convergent power series ofs1 , . . . , sm .
If we take a compatible basis {p(1), . . . , p(m)} [9], it satisfies the condition
above. And, we need a “small” basis in a suitable sense. However, the author
does not know an efficient method to find a compatible basis and does notknow what compatible basis is “small” for our purpose. They are questionsof geometry.
Now, we are ready to state our method to derive Kummer type formulasof hypergeometric series associated to Ek,n .

10Nobuki Takayama

Data : ηi (y), φi (α; x(y)), sj = x(y)p(j)Result : Kummer Type Formulas
Suppose that ηi (y)/η1 (y), i ≥ 2 are not holomorphic at s1 = · · · = sm .
1. Let G be the isotropy group in Sn that fixes the point s1 = · · · =sm .
2. for σ ∈ G doσ•ηii
consider functions { σ•ηη1 | i = 1, . . . , r}. If the function η1 is holoσ•ηmorphic at s1 = · · · = sm = 0 and other functions η1 j , j 6= i are
not holomorphic, then outputcσ • ηiφi (σ • α; σ • x(y)) = φ1 (α; x(y)).
η1(7)

Here, c is a suitable constant. Note that the both sides are functionsin s1 , . . . , sm .
end

Algorithm 2: Deriving Kummer type formulas
Since the formula (7) contains complex power functions, we have to becareful about choices of branches. As to this topics, readers are invited to thearticle of J.Davenport[4] in this book. The problem of choices of branches
in formulas on hypergeometric functions has a long history. For example,
É.Goursat gave connection formulas of the Gauss hypergeometric function
on the upper half plane and the lower half plane in his long paper in 1881 [7].
This approach of giving connection formulas on simply connected domains
is nice, however, most editors of formula books do not take his original nicerigorous approach and list connection formulas without the side conditions
on branches. In the several variable case, the problem of finding a nice decomposition into simply connected domains is more interesting and difficult.
See, for example, [22, Chapter V] and [13].
Example 4. k = 2, n = 4.
By a calculation, we have η2 /η1 = sα1 +α3 +1 , which is not holomorphic at
s = 0. Put σ = (13) = [3, 2, 1, 4, 5]. Then, we have
α +1
σ • η1[32][14] 2=
η1[34][12]andσ • F (−α3 , α2 + 1, −α1 − α3 ; s) = F

−α1 , α2 + 1, −α1 − α3 ;[42][31][12][34]

.

Kummer type formulas

11sSince [12][34] − [13][24] + [23][14] = 0 we have [42][31][12][34] = s−1 . Similarly, weshave [32][14][12][34] = 1 − s−1 . Therefore, we obtain

F (−α3 , α2 + 1, −α1 − α3 ; s) = (1−s)−α2 −1 F

−α1 , α2 + 1, −α1 − α3 ;

s
s−1



,

which is well-known Kummer’s formula for the Gauss hypergeometric function.
[42]
[52]
Example 5. k = 2, n = 5. Put x11 = [32][12] , x12 = [12] , x13 = [12] , x21 =
[13]
[14]
[15]
[12] , x22 = [12] , x23 = [12] ,





 

x21 x12x21 x13
φ1 (x(y)) = F1 α1 + 1, −α4 , −α5 , α3 + α1 + 2;,,
x11 x22x11 x23 
x21 x12x22 x13
φ2 (x(y)) = G2 −α3 , −α5 , α3 + α1 + 1, α5 + α2 + 1; −, −,x11 x22x12 x23 
x21 x13x22 x13
φ3 (x(y)) = F1 α2 + 1, −α3 , −α4 , α5 + α2 + 2;,,
x11 x23x12 x23
and1 −1 α3 +α1 +1 α4 α5η1 = [12]−1 x−αx21x22 x2311
3 −α1 −1−α3 −α2 −1−α5 α5η2 = [12]−1 xα
x
x
x23
11 12
22
−1 α3 α4 α5 +α2 +1 −α2 −1
η3 = [12] x11 x12 x13
x23
.

Here
F1 (a, b, b0 , c; x1 , x2 ) :=∞X(a)m+n (b)m (b0 )n m nx x(c)m+n (1)m (1)n 1 2m,n=0

is the Appell hypergeometric function and G2 is the hypergeometric function
G2 in Horn’s list. Then, the functions ηi φi are solutions of Ek,n . Put
s1 =[42][13]x22 x13[52][14]x21 x12=, s2 ==.
x11 x22[32][14]x12 x23[42][15]

The functions ηη1i φi (x(y)) are expressed in terms of s1 and s2 as follows.
F1 (s1 , s1 s2 )−α1 −α3s−1ssG2 (−s1 , −s2 )
1
1
1
−1
−α1
−α3 −α4(s1 s2 ) (s1 s2 )(s1 s2 )s2 F1 (s1 s2 , s2 )These functions converge in |s1 |, |s2 | < 1 and the first function φ1 (s1 , s1 s2 )is holomorphic at s1 = s2 = 0.
The isotropy group G at s1 = s2 = 0 consists of eight elements. The group
G is the dihedral group.

12

Nobuki Takayama(σ • s1 , σ • s2 )σ
2 −s1 s2 +s1( (−ss12+1)s
,
)
[5,
3,
2, 4, 1]
−1
s1 −1
2
1
( s2s−1, ss11ss22−s)[5,
1,2, 4, 3]−1
−s1 s2 +s1 (−s1 +1)s2( s1 −1 , s2 −1 ) [3, 5, 1, 4, 2]
2
1( s1s−1, (ss11 −1)s[3, 2, 1, 4, 5]
s2 −1 )
(s1 −1)s2
s1( s1 s2 −1 , s1 −1 ) [2, 3, 5, 4, 1](s2 , s1 )[2, 1, 5, 4, 3]
s2
1( ss11ss22−s,)[1, 5, 3, 4, 2]−1 s2 −1(s1 , s2 )[1, 2, 3, 4, 5]
1 2Here, [i1 , i2 , i3 , i4 , i5 ] denotes the permutation
i1 i2Here is a list ofσ • η1 σ • η2 σ • η2
,
,
,σ
η1
η1
η1

3
i3

4
i4

5
i5



∈ S5 .

generated by Algorithm 1 in this article. For example, the first line means
that

α 
α1

α
s1 s2 − 1 3−s1 s2 + 1 s2 − 1 4 −1(−s1 s2 + 1)α5(σ • η1 )/η1 =
s1 s2
s2
s1 s2
s1 s2
for σ = [5, 3, 2, 4, 1].
[[0,(-s1*s2+1)/(s1*s2)],[4,(s2-1)/(s2)],[1,(-1)/(s1*s2)],[5,-s1*s2+1],[3,(s1*s2-1)/(s1*s2)]][[0,((s1^2-s1)*s2-s1+1)/(s1*s2-s1)],[4,-s1+1],[1,(s1-1)/(s1*s2-s1)],[5,-s1*s2+1],[3,((-s1^2+s1)*s2+s1-1)/(s1*s2-s1)]][[0,-s1*s2+1],[4,-s1+1],[1,-1],[5,-s1*s2+1],[3,s1*s2-1]][5,3,2,4,1][[0,(1)/(s1*s2)],[4,(s2-1)/(s2)],[3,(s1*s2-1)/(s1*s2)],[1,(-1)/(s1*s2)]][[0,(1)/(s1*s2-s1)],[3,(s1*s2-1)/(s1*s2-s1)],[1,(-1)/(s1*s2-s1)]][[0,(1)/(s1*s2-1)],[1,(-1)/(s1*s2-1)]][5,1,2,4,3][[0,s1*s2-1],[1,-1],[4,-s1+1],[3,s1*s2-1],[5,-s1*s2+1]][[0,((-s1^2+s1)*s2+s1-1)/(s1*s2-s1)],[1,(s1-1)/(s1*s2-s1)],[4,-s1+1],[3,((-s1^2+s1)*s2+s1-1)/(s1*s2-s1)],[5,-s1*s2+1]][[0,(s1*s2-1)/(s1*s2)],[1,(-1)/(s1*s2)],[4,(s2-1)/(s2)],[3,(s1*s2-1)/(s1*s2)],[5,-s1*s2+1]][3,5,1,4,2][[0,-1],[1,-1],[4,-s1+1],[5,-s1*s2+1],[3,-1]][[0,(-s1+1)/(s1)],[1,(-s1+1)/(s1)],[4,-s1+1],[5,-s1*s2+1],[3,(-s1+1)/(s1)]][[0,(-s1*s2+1)/(s1*s2)],[1,(-s1*s2+1)/(s1*s2)],[4,(-s1*s2+1)/(s2)],[5,-s1*s2+1],[3,(-s1*s2+1)/(s1*s2)]]

Kummer type formulas

13[3,2,1,4,5][[0,(s1*s2-1)/(s1*s2)],[5,-s1*s2+1],[4,(-s1*s2+1)/(s2)],[1,(-s1*s2+1)/(s1*s2)],[3,(-s1*s2+1)/(s1*s2)]][[0,(s1-1)/(s1)],[5,-s1*s2+1],[4,-s1+1],[1,(-s1+1)/(s1)],[3,(-s1+1)/(s1)]][[5,-s1*s2+1],[4,-s1+1],[1,-1],[3,-1]][2,3,5,4,1][[0,(-1)/(s1*s2)],[4,(1)/(s2)],[3,(1)/(s1*s2)],[1,(1)/(s1*s2)]][[0,(-1)/(s1)],[3,(1)/(s1)],[1,(1)/(s1)]][[0,-1]][2,1,5,4,3][[0,(-1)/(s1*s2-1)],[1,(-1)/(s1*s2-1)]][[0,(-1)/(s1*s2-s1)],[3,(s1*s2-1)/(s1*s2-s1)],[1,(-1)/(s1*s2-s1)]][[0,(-1)/(s1*s2)],[3,(s1*s2-1)/(s1*s2)],[4,(s2-1)/(s2)],[1,(-1)/(s1*s2)]][1,5,3,4,2][][[0,(1)/(s1)],[3,(1)/(s1)],[1,(1)/(s1)]][[0,(1)/(s1*s2)],[3,(1)/(s1*s2)],[4,(1)/(s2)],[1,(1)/(s1*s2)]][1,2,3,4,5]

From this table, we obtain one of Kummer type formulas for the Appellfunction F1 firstly studied by Vavasseure [2](1 − s1 s2 )(1 − s1 s2 )α3 (1 − s1 )α4 (1 − s1 s2 )α5s1 (1 − s2 )×F1 α3 + 1, −α2 , −α4 , α1 + α3 + 2; s1 s2 ,s1 − 1= F1 (α1 + 1, −α4 , −α5 , α3 + α1 + 2; s1 , s1 s2 )standing for σ = [5, 3, 2, 4, 1] and (σ • η3 )/η1 .
We have sketched a method to generate Kummer type formulas. However,
it is not the end of the story and the following problems are arising.
1. Improve efficiency. How far can we derive Kummer type formulas of Ek,nfor large k and n? Can we find an interesting structure?
2. We need an efficient method to find a “small” compatible basis.
3. We will obtain a lot of Kummer type identities. However, we have no
method to classify them by a suitable mathematical meaning.
4. Building a system to study and generate formulas of hypergeometric functions by integrating algebra and geometry systems (http://www.openxm.org).
5. How to store generated formulas and proofs in an electronic or digitalformula book on hypergeometric functions[5], [21], [18]?
As to the last problem, OpenMath approaches seem to be promissing. Readers
are invited to articles by A.Cohen, H.Cuypers, E.R.Barreiro, H.Sterk[3], and
A.Solomon[16] in this book on OpenMath activities.

14

Nobuki Takayama

References
1. K.Aomoto and M.Kita, Theory of Hypergeometric Functions, (in Japanese)
Springer-Tokyo, 1994.
2. P.Appell, J.Kampé de Fériet, Fonctions Hypergéometrique et Hypersphériques– Polynomes d’Hermite. Gauthier-Villars, 1926, Paris.
3. A.Cohen, H.Cuypers, E.R.Barreiro, H.Sterk, Interactive Mathematical Documents on Web, in this volume.
4. J.Davenport, The Geometry of Cn is Important for the Algebra of Elementary
Functions, in this volume.
5. Digital Library of Mathematical Functions (Abramowitz and Stegun Mk II)
http://dlmf.nist.gov/
6. I.Emiris, “Discrete Geometry for Algebraic Elimination” and “Sparse Resultant
Perturbations”, in this volume.
7. É.Goursat, Sur L’Équation Différentielle Linéaire qui Admet pour Intégrale laSérie Hypergéométrique, Annales Scientifique l’École Normal Supérieure, Series
II, Vol. 10, (1881) 3–142.
8. I.M.Gel’fand, General theory of hypergeometric functions. Soviet Mathematics
Doklady 33, (1986), 573–577.
9. I.M.Gel’fand, A.V.Zelevinsky, M.M.Kapranov, Hypergeometric functions andtoral manifolds. Functional Analysis and its Applications 23, (1989), 94–106.
10. M.Noro, A Computer Algebra System Risa/Asir, in this volume.
11. J.Pfeifle, J.Rambau, Computing Triangulations Using Oriented Matroids, inthis volume.
12. M.Saito, B.Sturmfels and N.Takayama, Gröbner Deformations of Hypergeometric Differential Equations, Algorithms and Computation in Mathematics 6,
Springer, 2000.
13. M.Saito and N.Takayama, Restrictions of A-hypergeometric systems and connection formulas of the ∆1 ×∆n−1 -hypergeometric function, International Journal of Mathematics 5 (1994), 537-560.
14. H.Schönemann, Singular in a Framework for Polynomial Computations, inthis volume.
15. J.Sekiguchi and N.Takayama, Compactifications of the configuration space of 6
points of the projective plane and fundamental solutions of the hypergeometricsystem of type (3, 6), Tohoku Mathematical Journal 49 (1997), 379 – 413.
16. A.Solomon, Distributed Computing for Conglomerate Mathematical Systems,in this volume.
17. B.Sturmfels, Algorithms in Invariant Theory, Texts and Monographs in Symbolic Computation. Springer-Verlag, Vienna, 1993.
18. Y.Tamura, The design and implementation of an interactive formula book forgeneralized hypergeometric functions, Thesis, Kobe University, 2002, in preparation.
19. K.Ueno, Hypergeometric series formulas generated by the Chu-Vandermondeconvolution. Memoirs of the Faculty of Science. Kyushu University. Series A.
Mathematics 44 (1990), 11–26.
20. D.M.Wang, Automated Generation of Diagrams with Maple and Java, in thisvolume.
21. Wolfram Research’s Mathematical Functions, http://functions.wolfram.com
22. M.Yoshida, Hypergeometric functions, my love. Modular interpretations of configuration spaces. Aspects of Mathematics, Vieweg, 1997