A CURIOUS IDENTITY PROVED BY CAUCHY’S INTEGRAL
FORMULA
HELMUT PRODINGER
†
Simons [2] has proved the identity
q
X
r=0
(−1)
q+r
(q + r)!(1 + x)
r
(q − r )!r!
2
=
q
X
r=0
(q + r)!x
r
(q − r)!r!
2
; (1)
Chapman [1] gave a nice and short proof of it. In this note, I want to give another
attractive proof. It uses Cauchy’s integral formula to pull out coefficients of generating
functions.
We divide (1) by q! and prove the equivalent version
S =
q
X
r=0
µ
q
r
¶µ
q + r
r
¶
(−1)
q+r
(1 + x)
r
=
q
X
r=0
µ
q
r
¶µ
q + r
r
¶
x
r
.
We start with the righthand-side:
S = [t
q
]
X
i≥0
µ
q
i
¶
t
i
·
X
i≥0
µ
q + i
i
¶
(tx)
i
= [t
q
](1 + t)
q
· (1 − tx)
−q−1
=
1
2πi
I
dt
t
q+1
(1 + t)
q
· (1 − tx)
−q−1
.
Now we substitute t = u/(1 − u), so that dt = du/(1 − u)
2
and obtain
S =
1
2πi
I
du
(1 − u)
2
(1 − u)
q+1
u
q+1
(1 − u)
−q
·
µ
1 − u(1 + x)
1 − u
¶
−q−1
= [u
q
]
¡
1 − u)
q
(1 − u(1 + x)
¢
−q−1
=
q
X
r=0
µ
−q − 1
r
¶
(−1)
r
(1 + x)
r
µ
q
q − r
¶
(−1)
q−r
=
q
X
r=0
µ
q + r
r
¶µ
q
r
¶
(1 + x)
r
(−1)
q−r
,
which is the lefthand-side.
Date: May 8, 2003.
†
Supported by NRF Grant 2053748.
1
2 H. PRODINGER
References
[1] R. Chapman. A curious identity revisited. The Mathematical Gazette, 87:139–141, 2003.
[2] S. Simons. A curious identity. The Mathematical Gazette, 85:296–298, 2001.
H. Prodinger, The John Knopfmacher Centre for Applicable Analysis and Number
Theory, School of Mathematics, University of the Witwatersrand, P. O. Wits, 2050
Johannesburg, South Africa
E-mail address: helmut@maths.wits.ac.za