Lectures on String Theory

– 127 –
where 1 F 1 is a degenerate hypergeometric function:
C. Useful formulae
1F 1 (α, β; x) =
∞∑
k=0
(α) k
(β) k
x k
k! . (B.7)
In discussion of the central charge of the Virasoro algebra we encounter the sum
S n =
n∑
q 2 .
q=1
Here we present a general method of computing this and similar sums. The method
is based on considering the generating function
so that
For x < 1 we have
℘(x) =
∞∑
S n x n =
n=1
℘(x) =
S n = 1 n!
∞∑
n=1 q=1
∞∑
S n x n ,
n=1
( ∂ n ℘
∂x n )
| x=0 .
n∑
q 2 x n =
∞∑
q=1
q 2
∞
∑
n=q
x n =
∞∑
q=1
q 2 x q
1 − x .
We further notice that
∂ 2
∂x 2
∞
∑
q=1
x q =
∞∑
q(q − 1)x q−2 = 1 x 2
q=2
∞
∑
q=2
q 2 x q − 1 x 2
∞
∑
q=2
qx q = 1 x 2
∞
∑
q=1
q 2 x q − 1 x 2
∞
∑
q=1
qx q .
Thus,
∞∑ ( ∂
q 2 x q = x 2 2
x
∂x 2 1 − x + 1 ∂
x 2 ∂x
q=1
and, therefore, we obtain the generating function
Finally we compute
1
( ∂ n ℘
)
= 1 n! ∂x n n!
x
)
1 − x
= x2 + x
(1 − x) 3
℘(x) = x2 + x
(1 − x) 4 = 1
(1 − x) 2 − 3
(1 − x) 3 + 2
(1 − x) 4 .
( )
2 · 3 · · · (n + 1) 3 · 4 · · · (n + 2) 4 · 5 · · · (n + 3)
− 3 + 2 .
(1 − x) n+2 (1 − x) n+3 (1 − x) n+4
The last formula results into
S n = (n + 1)
(1 − 3 2 (n + 2) + 1 )
3 (n + 2)(n + 3)
= 1 n(n + 1)(2n + 1) .
6