Memoirs on Differential Equations and Mathematical Physics
Volume 56, 2012, 57–71
I. R. Lomidze and N. V. Makhaldiani
SOME PROPERTIES OF THE GENERALIZED
EULER BETA FUNCTION
Abstract. A generalization of the Euler beta function for the case of
multi-dimensional variable is proposed. In this context ordinary beta function is defined as a function of two-dimensional variable. An analogue of the
Euler formula for this new function is proved for arbitrary dimension. There
is found out the connection of defined function with multi-dimensional hypergeometric Laurichella’s function and the theorem on cancelation of multidimensional hypergeometric functions singularities is proved. Such generalizations (among others) may be helpful to construct corresponding physical
(string) models including different number of fields, as far the (bosonic)
string theory reproduces the Euler beta function (Veneziano amplitude)
and its multi-dimensional analogue.
2010 Mathematics Subject Classification. 33B15, 33C65, 81T30.
Key words and phrases. Generalized Euler integrals, multivariable
hypergeometric functions, strings theory.
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Some Properties of the Generalized Euler Beta Function
59
In the articles [1], [2] there was proposed and investigated the function
Bn (r0 , r1 , . . . , rn ) = Bn (r) =
(
1
£ i−1 ¤
=
det−1 [xi−1
j ]i,j=1,n det xj bij i,j=1,n
if n = 0,
if n ≥ 1
(1)
(0 = x0 < x1 , x2 < · · · < xn ),
where
Z1
ui−1 (1−u)rj −1
bij =
xj−1 /xj
n ³
Y
xj u−xk rk −1 ´
k=0
k6=j
xj −xk
du (i, j = 1, . . . , n).
The function (1) is a multidimensional generalization of the Euler beta
function
Z1
B(r0 , r1 ) = B1 (r) = ur0 −1 (1 − u)r1 −1 du.
0
A multidimensional analogues of Euler’s beta function had been studied
by mathematicians such as Selberg, Weil and Deligne among many others
(see e.g. [3]–[7]). In [1] we have shown that for any n ∈ N and for r0 > 0,
rj ∈ N (j = 1, . . . , n) an analogue of the Euler formula is valid:
n
Q
Bn (r) =
j=0
Γ
Γ(rj )
³P
n
j=0
where Γ(t) =
R∞
´,
(2)
rj
e−u ut−1 du is the Euler Gamma function.
0
In [2] there is investigated the case of the dimension n = 2. The analogue of the Euler formula has been proved for any complex parameters
r0 , r1 , r2 (Re rj > 0, j = 0, 1, 2) and the complications arising when n ≥ 3
are shown. The relations between Bn (r) and hypergeometric functions of
one and of many variables are shown too. Number of relations for the Gauss
hypergeometric function is obtained. The analytic formulae for some new
definite integrals of the special functions are obtained as well as for the
elementary ones.
In present article we prove (2) for any n ∈ N and for rj ∈ C (Re rj > 0,
j = 0, 1, . . . , n). The key of the proof is the known
Carlson theorem ([8]). If the function f (z) of a complex variable z is
regular in the semi-plane Re z > A, A ∈ R, and if the conditions
(a)
lim |f (z)| exp(−k|z|) ≤ const, 0 < k < π;
|z|→∞
(b) f (z) = 0 for z = 0, 1, 2, . . . ,
60
I. R. Lomidze and N. V. Makhaldiani
are valid, then f (z) = 0 for any z ∈ C.
For proving (2) we need the following
Lemma. On the complex plane of the variable rj ∈ C (j = 0, 1, . . . , n)
the function B0 (r0 , r1 , . . . , rn ) = Bn (r) is bounded when |rj | → ∞ if other
variables r0 , . . . , rj−1 , rj+1 , . . . , rn are fixed and Re rj ≥ 1 (j = 0, 1 . . . , n):
|Bn (r)| ≤ M < ∞.
(3)
The bound M does not depend on the variables r0 , r1 , . . . , rn (Re rj ≥ 1,
j = 0, 1 . . . , n).
Proof. The substitutions
u=u
e, j = 1;
³
xj−1 ´
u=1−u
e 1−
, j = 2, . . . , n (n ≥ 2),
xj
give to the formula (1) the form
Bn (r) =
e
det[xi−1
j bij ]i,j=1,n
det[xi−1
j ]i,j=1,n
,
(4)
where
Z1
n ³
Y
x1 ´1−rk
x1 ´rk −1
1−
u
er0 +i−2 (1 − u
e)r1 −1
1−
u
e
de
u,
xk
xk
k=2
k=2
0
³
´r j
ebij = 1 − xj−1
×
xj
¸r +i−2
Z1 ·
n ³
³
Y
xj−1 ´ 0
xj − xj−1 ´rk −1
rj −1
×
1− 1−
u
e
u
e
de
u,
1−
xj
xj − xk
ebi1 =
n ³
Y
k=0
k6=j
0
i = 1, . . . , n; j = 2 . . . , n.
All these integrals converge if the conditions
Re rj > 0, j = 0 . . . , n; 0 = x0 < x1 < x2 < · · · < xn
are fulfilled. Note that
Γ(c)
Γ(a)Γ(c − a)
Z1
ua−1 (1 − u)c−a−1
0
n
Y
(1 − zk u)bk du =
k=1
³a
´
= F ; b1 , z1 ; . . . ; bn , zn (Re c > Re a > 0),
c
³a
´
where F ; b1 , z1 ; . . . ; bn , zn denotes the multi-variable hypergeometric func
ction – one of the four Lauricella’s functions of the arguments z1 , . . . , zn (see
Some Properties of the Generalized Euler Beta Function
61
[9]), which can be expressed as absolutely convergent power-sum
F
³a
´
¡
¢
; b1 , z1 ; . . . ; bn , zn ≡ FD a; b1 , . . . , bn ; c; z1 , . . . , zn =
c
∞
X
=
k1 ,k2 ,...,kn =0
kj
n
(a)k1 +···+kn Y (bj )kj zj
,
(c)k1 +···+kn j=1
kj !
if |zj | < 1, j = 1, . . . , n. Here (a)k denotes the Pochhammer symbol:
(a)0 = 1, (a)k = a(a + 1) · · · (a + k − 1), k ∈ N.
Thus the formula (4) expresses the above-defined function (1) via the determinant of Lauricella’s multi-variable hypergeometric functions.
Let us rewrite the formula (4) as follows
e
Bn (r) det[zji−1 ]i,j=1,n = M1 det[zji−1ebij ]i,j=1,n ,
(5)
where
xk
, 0 < z0 < z1 < · · · < zn−1 < zn = 1,
xn
n ³
Y
z1 ´1−rk ³
zk−1 ´rk
M1 =
1−
1−
=
zk
zk
zk =
=
k=2
n ³
Y
k=2
z1 ´ Y ³ zk − zk−1 ´rk
1−
zk
zk − z1
(6)
n
(7)
k=3
and
e
ebi1 =
Z1
ur0 +i−2 (1 − u)r1 −1
n ³
Y
k=2
0
1−u
z1 ´rk −1
du,
zk
Z1 ³
n
zj − zj−1 ´r0 +i−2 rj −1 Y ³
zj − zj−1 ´rk −1
e
e
bij =
1−
u
du,
1−
zj
zj − zk
k=1
k6=j
0
j = 2 . . . , n.
Due to the inequalities (6), the expression (5) can be estimated as follows:
¯
¯
¯Bn (r)¯ det[z i−1 ]
i,j=1,n =
j
¯
¯
£
e
e ¤
¯
¯
= |M1 | ¯ det[zji−1ebij ]i,j=1,n ¯ ≤ |M1 | per zji−1 |ebij | i,j=1,n ,
(8)
where per[aij ]i,j=1,n stands for the permanent of a matrix [aij ]i,j=1,n (see
e.g. [10]):
X
a1σ(1) · · · anσ(n) , σ : i 7→ σ(i).
per[aij ]i,j=1,n =
σ
62
I. R. Lomidze and N. V. Makhaldiani
Further estimations give:
e
|ebi1 | ≤
¯
Z1 ¯
n ³
Y
¯ r +i−2
z1 ´rk −1 ¯¯
r1 −1
¯u 0
1
−
u
(1
−
u)
¯
¯ du =
zk
k=2
0
Z1
Re r0 +i−2
=
u
Re r1 −1
(1 − u)
n ³
Y
k=2
0
¯
z1 ´Re rk −1 ¯¯
1−u
¯ du ≡
zk
Z1
≡
gi1 (u)f1 (u) du,
0
e
|ebij | ≤
Z1
≤
0
(9)
¯
¯
n
¯³
zj − zj−1 ´r0 +i−2 rj −1 Y ³
zj − zj−1 ´rk −1 ¯¯
¯
u
1−u
¯ 1−u
¯ du =
¯
¯
zj
zj − zk
k=1
k6=j
¯
Z1 ¯¯³
n
zj −zj−1 ´Re r0 +i−2 Re rj −1 Y ³
zj −zj−1 ´Re rk −1 ¯¯
¯
= ¯ 1−u
u
1−u
¯ du ≡
¯
¯
zj
zj −zk
k=1
k6=j
0
Z1
fij (u)gj (u) du, j = 2, . . . , n,
≡
0
where we have denoted
f1 (u) =
n ³
Y
1−u
k=1
z1 ´Re rk −1
,
zk
gi1 (u) = uRe r0 +i−2 ,
j−1
zj − zj−1 ´Re rk −1
zj − zj−1 ´Re r0 +i−2 Y ³
1−u
fij (u) = 1 − u
,
zj
zj − zk
³
k=1
gj (u) = uRe rj −1
n
³
Y
k=j+1
zj − zj−1 ´Re rk −1
1+u
,
zk − zj
j = 2, . . . , n; i = 1, . . . , n.
Let us use the mean value theorem in the integrals (9). As it is known,
if the function f (x) is monotonic and f (x) ≥ 0 when x ∈ [a, b], and if g(x)
is integrable, then the Bonnet formulae are valid (see e.g. [11, II, n◦ 306]):
Zb
Zη
f (u)g(u) du = f (a)
a
g(u) du, a ≤ η ≤ b if f (x) decreases,
a
Some Properties of the Generalized Euler Beta Function
Zb
63
Zb
f (u)g(u) du = f (b)
a
g(u) du, a ≤ ξ ≤ b if f (x) increases,
ξ
(x ∈ [a, b]).
It is obvious that if Re rj ≥ 1, j = 0, 1, . . . , n, then the factors f1 (u) and
fij (u) in (9) decrease for u ∈ [0, 1] and the factors gi1 (u) and gj (u) increase
(j = 2, . . . , n; i = 1, . . . , n). Hence, according to the Bonnet formulae, for
Re rj ≥ 1, j = 0, 1, . . . , n (due to this conditions all integrals converge) one
gets
e
|ebi1 | ≤ f1 (0)
Zηi1
Zηi1
gi1 (u) du = f1 (0)gi1 (ηi1 ) du ≤
0
ξi1
≤ f1 (0)gi1 (1)(ηi1 − ξi1 ) ≤ 1,
e
|ebij | ≤ gj (1)
Z1
Zηi1
fij (u) du = gj (1)fij (ξij ) du ≤
ξij
(10)
ξij
n
Y
≤ gj (1)fij (0)(ηij − ξij ) ≤
k=j+1
³z − z
´Re rk −1
k
j−1
,
zk − zj
0 = z0 < z1 < · · · < zn , 0 ≤ ξij ≤ ηij ≤ 1, i, j = 1 . . . , n.
According to (7), (8) and (10) one has
¯
¯
¯Bn (r)¯ det[z i−1 ]
i,j=1,n ≤
j
£
e ¤
≤ |M1 | per zji−1 |ebij | i,j=1,n ≤ M2 per[zji−1 ]i,j=1,n ,
M2 =
n ³
Y
k=2
n
n−1
n
z1 ´ Y ³ zk − zk−1 ´Re rk Y Y ³ zk − zj−1 ´Re rk −1
1−
.
zk
zk − z1
zk − zj
j=2
k=3
k=j+1
Because of obvious equalities
n−1
Y
n
³z − z
´Re rk −1
Y
k
j−1
=
zk − zj
j=2 k=j+1
=
n · k−1
Y
Y ³ zk − zj−1 ´¸Re rk −1
k=3
j=2
zk − zj
=
n ³
Y
zk − z1 ´Re rk −1
zk − zk−1
k=3
one obtains
M2 =
n ³
Y
k=2
1−
n
n
zk−1 ´
z1 ´ Y ³ zk − zk−1 ´ Y ³
1−
=
.
zk
zk − z1
zk
k=3
k=2
64
I. R. Lomidze and N. V. Makhaldiani
Inserting these results into the inequality (8), one obtains the estimation:
n ³
¯
¯ per[zji−1 ]i,j=1,n Y
zk−1 ´
¯Bn (r)¯ ≤
, Re rj ≥ 1, j = 0, 1, . . . , n.
1−
i−1
zk
det[zj ]i,j=1,n
k=2
Hence, we have got the estimation (3) with M to be expressed as
n ³
per[zji−1 ]i,j=1,n Y
zk−1 ´
M=
1−
,
i−1
zk
det[zj ]i,j=1,n
(11)
k=1
which, obviously, does not depend on the variables r0 , r1 , . . . , rn (Re rj ≥ 1,
j = 0, 1, . . . , n).
¤
Note. The restriction Re rj ≥ 1, j = 0, 1, . . . , n, is essential. Let, e.g.,
n = 1. In this case (11) gives M = 1 and one gets
¯ Z1
¯ ¯
¯
¯
¯ ¯
¯
¯B1 (r)¯ = ¯ ur0 −1 (1 − u)r1 −1 du¯ = ¯¯ Γ(r0 )Γ(r1 ) ¯¯ ≤ 1 if Re r0 , r1 ≥ 1,
¯
¯
Γ(r0 + r1 )
0
while in the opposite case when Re r0 , r1 < 1, e.g. for r0 = r1 = 1/2 one
has B1 (1/2, 1/2) = π > 1, and the estimation (3) is not fulfilled.
Now we get the following
Statement. The function
n
Q
j=0
Γ(rj )
f (r0 , r1 , . . . , rn ) = Bn (r0 , r1 , . . . , rn ) − ¡ P
n
¢
Γ
rj
(12)
j=0
satisfies all conditions of Carlson theorem on the complex plane of each variable rj if other variables r0 , r1 , . . . , rj−1 , rj+1 , . . . , rn are fixed and Re rj ≥ 1,
j = 0, 1, . . . , n.
Proof of the Statement. Due to the fact that the function Γ(z) is analytic
everywhere on the (open) complex plane except the points z = 0, −1, −2, . . .,
the function (12) is analytic if Re rj > 0, j = 0, 1, . . . , n (rj = +∞ is a
regular point of both summands of (12), j = 0, 1, . . . , n). In [1] we have
proved that f0 (r0 ) = f (r0 , r1 , . . . , rn ) = 0 if the variable r0 is real and
r0 > 0, rj ∈ N, j = 1, . . . , n. Hence, according to the analytic function
uniqueness theorem, if rj ∈ N, j = 1, . . . , n, then f0 (r0 ) = 0 everywhere
on the complex plane of the variable r0 except, may be, the points z =
0, −1, −2, . . . . So, the function f0 (r0 ) satisfies the Statement.
Let us fix the numbers rj ∈ N, j = 1, . . . , n and r0 ∈ C, Re r0 > 0.
Under these conditions the function f1 (r1 ) = f (r0 , r1 , . . . , rn ) is analytic on
the complex semi-plane Re r1 > 0 and f1 (r1 ) = 0 for r1 = 1, 2, . . . .
Let us show that if Re r0 ≥ 1, rj ∈ N, j = 2, . . . , n, then
lim |f1 (r1 )| exp(−k|r1 |) ≤ const, Re r1 ≥ 1, 0 < k < π.
|r1 |→∞
Some Properties of the Generalized Euler Beta Function
65
Indeed, if |z| → ∞ and Re z > 0, in accordance with the asymptotic
behavior of the Euler’s Gamma function (see e.g. [12, Eq. 1.18(5)]) for any
fixed number ρ ∈ C we have
¡
¢
Γ(z)
Γ(z)
lim
exp ρ ln |z| = lim
|z|ρ = 1.
(13)
|z|→∞ Γ(z + ρ)
|z|→∞ Γ(z + ρ)
Hence, due to the estimation (3) we obtain:
0 ≤ lim |f1 (r1 )| exp(−k|r1 |) =
|r1 |→∞
¯
¯
n
Y
¯
¯
Γ(r1 )
¯
= lim exp(−k|r1 |)¯Bn (r) −
Γ(r0 )
Γ(rj )¯¯ ≤
Γ(r1 + ρ1 )
|r1 |→∞
j=2
(
¯
¯)
n
Y
¯
¯ ¯ Γ(r1 )
¯
≤ lim exp(−k|r1 |) ¯Bn (r)¯ + ¯¯
Γ(r0 )
Γ(rj )¯¯ ≤
Γ(r1 + ρ1 )
|r1 |→∞
j=2
≤ M lim exp(−k|r1 |) + Γ(r0 )
|r1 |→∞
½
× lim
|r1 |→∞
¡
¢
Γ(r1 )
exp ρ1 ln |r1 |
Γ(r1 + ρ1 )
n
Y
Γ(rj )×
j=2
¾
lim
|r1 |→∞
n
o
|r1 |−ρ1 exp(−k|r1 |) = 0,
where we denote ρ1 = r0 + r2 + · · · + rn . Therefore we get
lim |f1 (r1 )|e−k|r1 | = 0
|r1 |→∞
for any fixed values of k > 0 and ρ1 . In particular, one can choose 0 < k < π,
Re r0 ≥ 1, rj ∈ N, j = 2, . . . , n.
Thus, if Re r0 ≥ 1, rj ∈ N, j = 2, . . . , n, then the function f1 (r1 ) =
f (r0 , r1 , . . . , rn ) satisfy all conditions of the Carlson Theorem and therefore
f1 (r1 ) = 0, Re r1 ≥ 1.
Now let us suppose that the Statement is valid for the variables rj with
the indices j = 0, 1, . . . , m, where 1 ≤ m ≤ n − 1, i.e. let the function
f (r0 , r1 , . . . , rm−1 , rm , . . . , rn ) ≡ fm (rm )
satisfies the conditions:
fm (rm ) = f (r0 , r1 , . . . , rm−1 , rm , . . . , rn ) = 0 if rm = 1, 2, . . . ,
lim
|fm (rm )| exp(−k|rm |) ≤ const, 0 < k < π if Re rm ≥ 1
¢
Re rj ≥ 1, j = 0, 1, . . . , m − 1, rm+k ∈ N, k = 1, . . . , n − m .
|rm |→∞
¡
Therefore, according to the Carlson Theorem one gets
¡
fm (rm ) = f (r0 , r1 , . . . , rm−1 , rm , . . . , rn ) = 0 if rm ∈ C
¢
Re rj ≥ 1, j = 0, 1, . . . , m − 1, rm+k ∈ N, k = 1, . . . , n − m .
Thus we have shown that
fm+1 (rm+1 ) = f (r0 , r1 , . . . , rm , rm+1 , . . . , rn ) = 0
(14)
66
I. R. Lomidze and N. V. Makhaldiani
if Re rj ≥ 1, j = 0, 1, . . . , m, rm+k ∈ N, k = 1, . . . , n − m. Besides, due to
the estimates (3) and (13)
¯
¯
lim ¯fm+1 (rm+1 )¯ exp(−k|rm+1 |) ≤
|rm+1 |→∞
≤
lim
|rm+1 |→∞
(
¯
¯ exp(−k|rm+1 |) M +
Γ(rm+1 )
Γ(rm+1 + ρm+1 )
n
Y
)
Γ(rj )
=0
j=0
j6=m+1
if Re rm+1 ≥ 1, where k and ρm+1 are fixed numbers such that
¡
0 < k < π, ρm+1 = r0 + r1 + · · · + rm + rm+2 + · · · + rn
¢
Re rj ≥ 1, j = 0, 1, . . . , m, rm+k ∈ N, k = 2, . . . , n − m .
So, the proposition of the Statement is fulfilled according to Full Mathematical Induction Principle.
¤
The Statement enables one to prove the following
Theorem 1. For any number n ∈ N the function Bn (r) satisfies the
formula (2) – n-dimensional analogue of the Euler formula – if Re rj > 0,
j = 0, 1, . . . , n.
Proof. According to the Statement the formula (2) is fulfilled if Re rj ≥
1, j = 0, 1, . . . , n. Therefore, due to analyticity of the function (12) if
Re rj > 0, j = 0, 1, . . . , n, (2) is fulfilled on the open semi-plane Re rj > 0
of each variable rj ∈ C, j = 0, 1, . . . , n.
¤
We have shown in [2] that the limits of the function B2 (r) = B2 (r0 , r1 , r2 )
when x1 → x0 = 0 (x1 /x2 = z → 0) and x2 → x1 (x1 /x2 = z → 1) (see the
definition (1)) exist and satisfy the relation
¯
¯
det[xi−1
det[xi−1
j bij ]i,j=1,2 ¯¯
j bij ]i,j=1,2 ¯¯
=
=
¯
¯
det[xi−1
det[xi−1
x1 →x0 =0
x2 →x1
j ]i,j=1,2
j ]i,j=1,2
= B1 (r0 , r1 )B1 (r0 + r1 , r2 ).
Therefore for n = 2 the formula (2) remains valid even if the only restrictions
on the variables x0 , x1 , . . . , xn are xj ≥ x0 = 0, j = 1, . . . , n (instead of the
restrictions 0 = x0 < x1 < · · · < xn which we have in (1)). It is easy to
show that the same is valid for any n ≥ 3. Namely, one has the following
Theorem 2. For any n, l, k ∈ N, l ≤ k ≤ n − l and xj ≥ x0 = 0,
Re rj > 0, j = 1, · · · , n, the function Bn (r) satisfies the formula
Bn (r0 , r1 , . . . , rn ) =
= Bn−k (r0 , r1 , . . . , rl−1 , rl + · · · + rl+k , rl+k+1 , . . . , rn )Bk (rl , . . . , rl+k ) =
n
Q
Γ(rj )
=
j=0
Γ
n
¡P
j=0
rj
¢.
(15)
Some Properties of the Generalized Euler Beta Function
67
Proof. The theorem is trivial for n = 0 and n = 1; for n = 2, in fact,
the formula (15) is proved in [2]. In the case n ≥ 3 one has to consider
separately the cases x1 → x0 = 0 and xl → xl−1 , l ≥ 2.
In the case x1 → x0 = 0 from the definition (1) one gets:
Z1
i−1
xi−1
1 bi1 = x1
ur0 +i−2 (1 − u)r1 −1
0
(
−→
x1 →0
n ³
Y
x1 u − xk ´rk −1
x1 − xk
k=2
du −→
x1 →0
B1 (r0 , r1 ), i = 1,
0
i ≥ 2,
xi−1
j bij =
Z1
= xi−1
j
ur0 +i−2 (1−u)rj −1
n ³
Y
xj u−xk ´rk −1 ³ xj u − x1 ´r1 −1
k=2
k6=j
xj−1 /xj
xj −xk
Z1
−→ xi−1
x1 →0 j
xj−1 /xj
ur0 +r1 +i−3 (1 − u)rj −1
xj −x1
n ³
Y
xj u − xk ´rk −1
xj − xk
k=2
k6=j
du −→
x1 →0
du, j ≥ 2,
i = 1, . . . , n.
Hence, according to (1) and (2),
¯
Bn (r)¯x1 →0 = B1 (r0 , r1 )
Y
× det
Z1
xi−1
j
xj−1 /xj
r0 +r1 +i−3
u
x−1
j ×
2≤j≤n
2≤k<j≤n
"
Y
(xj − xk )−1
rj −1
(1 − u)
n
Y
k=2
k6=j
#n
³ x u − x ´rk −1
j
k
du
xj − xk
=
i,j=2
(0 = x1 < · · · < xn )
= B1 (r0 , r1 )Bn−1 (r0 + r1 , r2 , . . . , rn ) =
=
Γ(r0 )Γ(r1 ) · · · Γ(rn )
Γ(r0 )Γ(r1 )Γ(r0 + r1 )Γ(r2 ) · · · Γ(rn )
=
. (16)
Γ(r0 + r1 )Γ(r0 + r1 + r2 + · · · + rn )
Γ(r0 + r1 + · · · + rn )
Similarly, in the case when xl → xl−1 , l ≥ 2, one obtains:
³
xl−1 ´1−rl
i−1
×
xl−1
bi l−1 = xi−1
l−1 1 −
xl
Z1
×
n
³
xl−1 ´rl −1 Y ³ xl−1 u−xk ´rk −1
ui−1 (1−u)rl−1−1 1−
u
du,
xl
xl−1 −xk
xl−2 /xl−1
k=0
k6=l−1,l
68
I. R. Lomidze and N. V. Makhaldiani
Z1
n ³
³ x u−x ´rl−1 −1
Y
xl u−xk ´rk −1
l
l−1
ui−1
(1−u)rl −1
du,
xl −xl−1
xl −xk
xi−1
bil = xi−1
l
l
k=0
k6=l−1,l
xl−1 /xl
Z1
xi−1
j bij
xji−1
=
ui−1 (1 − u)rj −1
n ³
Y
xj u − xk ´rk −1
xj − xk
k=0
k6=j
xj−1 /xj
du,
j = 1, . . . , n, j 6= l − 1, l
(i = 1, . . . , n).
The substitution
³
xl u − xl−1
xl−1 ´
=1−u
e, u = 1 − 1 −
u
e,
xl − xl−1
xl
xl u − xk = xl − xk − u
e(xl − xl−1 )
gives to the second of these integrals the form
³
xl−1 ´rl
i−1
xi−1
×
b
=
x
1
−
il
l
l
xl
Z1³
n ³
Y
xl −xl−1 ´i−1
xl −xl−1 ´rk −1
rl−1 −1 rl −1
× 1−u
(1−u)
u
du.
1−u
xl
xl −xk
k=0
k6=l−1,l
0
Inserting these results in (1), after obvious simplifications we find:
¯
Bn (r)¯x →x =
l
=
xl−1 det[xi−1
j ]i=1,n−2
l−1
B1 (rl−1 , rl ) det B (l)
Q
(xl−1 − xk )2
Q
(xm − xl−1 )2
, (17)
l+1≤m≤n
j=1,n 1≤k≤l−2
j6=l−1,l
(l)
where B (l) = [Bik ] is the matrix whose i-th row, i = 1, . . . , n, has the form
h
i
ebi1 , . . . , xi−1ebi l−2 , xi−1ebi l−1 , xi−1ebi l+1 , . . . , xi−1ebin
xi−1
(18)
n
1
l−2
l−1
l+1
and
Z1
ebi l−1 =
ui−1 (1 − u)rl−1 +rl −2
xl−2 /xl−1
Z1
ebij =
ui−1 (1−u)rj −1
xj−1 /xj
n
³ x u − x ´rk −1
Y
l−1
k
du,
xl−1 − xk
k=0
k6=l−1,l
n ³
³ x u−x ´rl−1 +rl −2 Y
xj u−xk ´rk −1
j
l−1
du,
xj −xl−1
xj −xk
k=0
k6=j,l−1,l
i, j = 1, . . . , n, j 6= l − 1, l.
The last step of our transformations is to multiply the row with number
i − 1 in the determinant of the matrix (18) on xl−1 and to extract it from
Some Properties of the Generalized Euler Beta Function
69
the row with number i. Then in the l-th column of the determinant we
obtain the Kroneker symbol δ1i ,
(
1, i = 1
δ1i =
0, i 6= 1
and in the other ones we get:
i−1e0
e
xi−1
l−1 bi l−1 −→ xl−1 bi l−1 =
Z1
=
−xi−1
l−1
ui−2 (1 − u)rl−1 +rl −1
n
³ x u − x ´rk −1
Y
l−1
k
du,
xl−1 − xk
k=0
k6=l−1,l
xl−2 /xl−1
i−1e0
i−2
e
xi−1
j bij −→ xj bij = (xj − xl−1 )xj ×
Z1
×
ui−2 (1−u)rj −1
(19)
n ³
³ x u−x ´rl−1 +rl −1 Y
xj u−xk ´rk −1
j
l−1
du,
xj −xl−1
xj −xk
k=0
k6=j,l−1,l
xj−1 /xj
i, j = 1, . . . , n, j 6= l − 1, l.
Expanding the determinant obtained with respect to the l-th column
elements and inserting the result in (17), one gets
¯
Bn (r)¯x →x
=
l
=
l−1
det[xi−1
j ]i=1,n−2
B1 (rl−1 , rl )(−1)l−1
Q
Q
(xl−1 − xk )
1≤k≤l−2
j=1,n
j6=l−1,l
(xm − xl−1 )
(−1)l+1 ×
l+1≤m≤n
i
h
i−1e0
i−1e0
i−1e0
i−1e0
e0
=
× det xi−1
1 bi1 , . . . , xl−2 bi l−2 , xl−1 bi l−1 , xl+1 bi l+1 , . . . , xn bin
i=1,n−1
£ i−1 0
¤
i−1e0
i−1e0
e0
det x1 ebi1 , . . . , xi−1
l−1 bi l−1 , xl+1 bi l+1 , . . . , xn bin i=1,n−1
£ i−1
¤
= B1 (rl−1 , rl )
,
i−1
i−1
det x1 , . . . , xl−1
, xl+1
, . . . , xi−1
n
i=1,n−1
where the entries ebij , i, j = 1, . . . , n, are defined in (19). So, we obtain
the formula (16) in the case under consideration, too. Now the statement
of Theorem 2 follows from (16) according to Full Mathematical Induction
Principle.
¤
Theorem 3. The integrals’ singularities of the formula (1) determinant’s
entries, i.e. the singularities of Lauricella’s hypergeometric functions on the
complex plane of each variable xj ∈ C, j = 0, . . . , n, cancel each other in
the formula (1).
Proof. According to Theorem 2, the function (1) does not depend on the
variables xj ≥ 0 if Re rj > 0, j = 0, . . . , n. Hence, according to the analytic
function uniqueness theorem, the function (1) does not depend on the variables xj ∈ C if Re rj > 0, j = 0, . . . , n, while the integrals in the formula
70
I. R. Lomidze and N. V. Makhaldiani
(1) – Lauricella’s hypergeometric functions – have singularities with respect
to variables xj ∈ C.
¤
As far as the (bosonic) string theory [13] reproduces the Euler beta function (Veneziano amplitude) and its multidimensional analogue, it seems to
be helpful to take an advantage of the proposed generalization of the Euler
beta function when one attempts to construct physical (string) models [14]
including a number of fermionic fields, as far as the expression
"
#
Z1
Y ³ xj u − xk ´rk −1
i−1
i−1
rj −1
det xj
u (1 − u)
du
=
xj − xk
k=0
k6=j
xj−1 /xj
i,j=1,n
n
Q
=
Γ(rj )
j=0
det[xji−1 ]i,j=1,n ¡ P
n
Γ
j=0
rj
¢
is skew-symmetryc with respect to the variables xj ∈ C.
Acknowledgement
The authors thank gratefully to prof. V. Paatashvili for helpful discussions.
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Some Properties of the Generalized Euler Beta Function
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(Received 17.11.2009)
Authors’ addresses:
I. R. Lomidze
Physics Departments School of Natural Sciences, University of Georgia,
77a Kostava St., Tbilisi 0175, Georgia.
E-mail: lomiltsu@gmail.com
N. V. Makhaldiani
Laboratory of Information Technologies, Joint Institute for Nuclear Research, Dubna, Moscow Region, Russia.
E-mail: mnv@jinr.ru