Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 7, Issue 5, Article 166, 2006
ESTIMATING THE SEQUENCE OF REAL BINOMIAL COEFFICIENTS
VITO LAMPRET
U NIVERSITY OF L JUBLJANA
S LOVENIA 386
vito.lampret@fgg.uni-lj.si
Received 02 March, 2006; accepted 17 March, 2006
Communicated by J. Sándor
A BSTRACT. The sequence n 7→ na of real binomial coefficients is studied in two main cases:
a n and n a. In the first case a uniform approximation with high accuracy is obtained, in
contrast to DeMoivre-Laplace approximation, which has essentially local character and is good
only for n ≈ a2 . In the second case, for every a ∈ R \ ( N ∪ {−1, 0} ), the functions A(a, m)
A(a,m)
= 1, and
and B(a, m) are determined, such that lim B(a,m)
m→∞
a −(a+1)
< B(a, m) · (n − a)−(a+1) ,
A(a, m) · (n − a)
< n for integers m and n, obeying n > m > |a|.
Key words and phrases: Approximation, Binomial, Coefficient, Convergence, Estimate, Error, DeMoivre-Laplace, EulerMaclaurin, Remainder, Sequence.
2000 Mathematics Subject Classification. 05A10, 26D07, 26D15, 33F05, 40A05, 40A25, 65B15, 65D20.
1. I NTRODUCTION
a
Binomial coefficients n , where a ∈ R and n ∈ N, occur frequently, for example, in analysis
[11, 13, 15], in combinatorics and discrete
mathematics and computer science [7, 8, 14], and
in probability [3, 4]. Computing na directly by computer is not problematic as long as a
and n stay within reasonable limits. However, for very large a or n the computation of binomial
coefficients becomes difficult, even in the case when a and n are positive integers. An interesting
discussion on computing the binomial coefficients
in this particular case can be found in [6]. In
a
this note we are interested in the estimate of n for a or n being very large, where a ∈ R and
n ∈ N.
ISSN (electronic): 1443-5756
c 2006 Victoria University. All rights reserved.
061-06
2
V ITO L AMPRET
We observe that the sequence n 7→ na converges in some cases but in the other cases it
diverges, depending on a. We want to determine exactly when the sequence of binomial coefficients does converge, i.e. we wish to show that
( ∞, if a < −1
a =
lim n∈N
n 0, if a > −1 .
n→∞
Moreover, we also want to estimate precisely the rate of divergence/convergence of the sequence
of binomial coefficients. For example, we are searching for estimates like
−π π−1
< 0.438 · (n + π)π−1
0.436 · (n + π)
< n and
−(π+1)
0.983 · (n − π)
π < 0.985 · (n − π)−(π+1) ,
< n valid for n ≥ 9000. In addition, we wish to estimate the binomial coefficients na for large
a and positive integers n < a. It turns out, in this case, that the construction of a binomial
coefficient approximation, based on Taylor’s formula, similar to the treatment in [4, pp. 174190], is less accurate than the construction based on the Euler-Maclaurin summation formula.
The latter produces two main results, Theorem 4.1 and Theorem 4.2, presented on pages 8 and
12, respectively.
2. P RELIMINARIES
For a ∈ R and n ∈ N we define the binomial coefficients C(a, n) as
a
a · (a − 1) · · · · · (a − n + 1)
C(a, n) :=
=
(2.1)
n
1 · 2 · ··· · n
n−1
1 Y
(2.2)
= ·
(a − i)
n ! i=0
n
Y
a
a−i
=
·
a − n i=1 i
(2.3)
for a 6= n.
Due to (2.2) we have recursion and addition relations, respectively,
(2.4)
(2.5)
C(a, n + 1) =
a−n
· C(a, n),
n+1
C(a, n) + C(a, n + 1) = C(a + 1, n + 1).
Substituting n + 1 = m in (2.4), we obtain the equality
(2.6)
m C(a, m) = (a − m + 1)C(a, m − 1) ,
which holds for every a ∈ R and m ∈ N, if, in addition, we define C(a, 0) := 1.
From (2.3) we obtain the relation
n
a−m+1 Y a−i
·
.
C(a, n) = C(a, m − 1) ·
a−n
i
i=m
J. Inequal. Pure and Appl. Math., 7(5) Art. 166, 2006
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E STIMATING THE S EQUENCE OF R EAL B INOMIAL C OEFFICIENTS
3
Hence, by (2.6),
C(a, n) = m C(a, m)
(2.7)
n
Y
a−i
1
·
a − n i=m i
!
,
and, consequently,
(2.8)
C(a, n) = (−1)n−m · m C(a, m) ·
n
Y
i−a
1
·
n − a i=m i
!
,
valid for integers m and n such that a 6= n ≥ m ≥ 1.
For any n ∈ N and a ∈ R, using (2.2), we read the equivalence
(2.9)
C(a, n) = 0 ⇐⇒ a ∈ {0, 1, . . . , n − 1} .
We find, using induction and (2.5), that C(a, n) are positive integers, provided that a and n are
positive integers as well and, 1 ≤ n ≤ a. Further, C(a, n) = 0 for a ∈ N and n > a.
3. M ONOTONY
If C(a, n) 6= 0, we have, according to (2.6), the following equivalences
(3.1)
|C(a, n + 1)| < |C(a, n)| ⇔ |a − n| < n + 1 ⇔ −1 < a < 2n + 1
and
(3.2)
|C(a, n + 1)| = |C(a, n)| ⇔ |a − n| = n + 1 ⇔ a = −1 or a = 2n + 1 .
3.1. Referring to (2.9), we have C(0, n) = 0 for every positive integer n.
3.2. Considering (2.2), the equality C(−k, n) = (−1)n · C(k + n − 1, n) holds for any k, n ∈ N.
3.3. Let n and a be integers and 1 ≤ n ≤ a. Then, according to (2.9), (3.1), and (3.2), the
sequence n 7→ |C(a, n)| ≡ C(a, n) strictly increases for n < a−1
and strictly decreases for
2
a−1
n > a−1
,
while
for
n
=
the
equality
C(a,
n
+
1)
=
C(a,
n)
holds.
This means:
2
2
(i) If ais an even
integer, then the sequence
n 7→ C(a,n) strictly increases
on the
positive
a+1
a+1
set 1, . . . , a+1
and
strictly
decreases
on
,
.
.
.
,
a
,
where
denotes
the
2
2
2
a+1
integer part of 2 .
(ii) If a is an odd positive
integerand a ≥ 3, then the sequence
in-
a+1 n 7→ C(a,
n) strictlya−1
a−1
creases on the set 1, . . . , 2 and strictly decreases on 2 , . . . , a , where C a, 2 =
C a, a+1
.
2
From the considerations above we conclude that
(3.3)
max C(a, n) = C a, a+1
,
2
1≤n≤a
if the conditions, quoted in this subsection, are satisfied.
3.4. Let a ∈
/ {−1, 0} ∪ N. Then, by (2.9), all C(a, n) are different from zero. Consequently,
considering (2.9), (3.1), and (3.2), we find for the sequence n 7→ |C(a, n)| the following result:
(i) a < −1 ⇒ The sequence strictly increases on the entire set N.
(ii) a ∈ (−1, 0) ∪ (0, 3) ⇒ The sequence strictly decreaseson the entire
set N.
a+1 (iii) a > 3⇒ The sequence
and strictly de strictly increases on the set 1, . . . , 2
.
(Here
bxc
means
the
integer
part
of
x.)
Consequently,
creases for n ≥ a+1
2
.
(3.4)
max |C(a, n)| = C a, a+1
2
n∈N
Figures 3.1 – 3.5 illustrate the sequences n 7→ |C(a, n)| for several values of a.
J. Inequal. Pure and Appl. Math., 7(5) Art. 166, 2006
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4
V ITO L AMPRET
2000
1500
1000
500
aΠ
10 20 30 40 50
2
1.8
1.6
1.4
1.2
8
a 7
20
40
60
80
Figure 3.1: The sequences n 7→ |C(a, n)|.
0.2
0.15
0.1
0.05
0.8
0.75
0.7
0.65
0.6
1
a Π
10
20
30
7
a 8
10 20 30 40 50
40
Figure 3.2: The sequences n 7→ |C(a, n)|.
0.004
0.003
0.002
0.001
1
a Π
0.02
0.015
0.01
0.005
a 2
10 20 30 40 50 60
10
20
30
40
Figure 3.3: The sequences n 7→ |C(a, n)|.
2.5
2
1.5
1
0.5
3
2.5
2
1.5
1
0.5
a
1
2
3
4
5
6
aΠ
1
2
3
4
5
Figure 3.4: The sequences n 7→ |C(a, n)|.
4. A PPROXIMATING THE B INOMIAL C OEFFICIENTS
Concerning the method of approximating binomial coefficients, we consider two main cases:
the case when a is very large and 1 ≤ n < a, and the case when a is any real number and
integer n a. The corresponding results for the first and for the second case, respectively, are
presented in Theorem 4.1 and Theorem 4.2, pp. 8 and 12.
J. Inequal. Pure and Appl. Math., 7(5) Art. 166, 2006
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E STIMATING THE S EQUENCE OF R EAL B INOMIAL C OEFFICIENTS
200
150
100
50
0.002
aΠ
0.001
5
10
15
a3Π
2
20
5
4
6
10
8
Figure 3.5: The sequences n 7→ |C(a, n)|.
40
30
20
10
80
60
40
20
a7
2
4
6
n
8
a8
n
2
4
6
8
Figure 3.6: The sequences n 7→ C(a, n).
In the case a 1 and n < a we use the expression (2.7), and in the case n a, we use
the equation (2.8). In both cases we can obtain for the last factor on the right of (2.7) and (2.8),
respectively, some product with all factors positive. Surely, this can be achieved in the first case
by choosing for m any positive integer and, in the second case, by selecting any positive integer
m > a. In the sequel we shall see that parameter m plays an important role concerning the
accuracy of the obtained approximation; the larger m is, the more accurate the approximation
is. Therefore, we demand that at least, m ≥ 2 and, m − a ≥ 2, in the first and in the second
case, respectively. However, to compute C(a, m) directly, using (2.7) and (2.8), both m and
m − a should not be large.
Since all factors in the above mentioned product are positive, we can use the real logarithmic function to transform it into a sum, which could be approximated easily using the EulerMaclaurin summation formula [2, 7, 9, 11, 12]. For example, from [12, p. 117 - items (21a) and
(21b)], setting p = 3, we obtain the Euler-Maclaurin formula1 of the third order
Z n
n
X
f (m) + f (n) f 0 (n) − f 0 (m)
(4.1)
f (j) =
f (x) dx +
+
+ R(m, n),
2
12
m
j=m
having the remainder
(4.2)
1
R(m, n) := ρ3 (m, n) = −
3!
Z
n
P3 (−x)f (3) (x) dx,
m
3
where m and n are integers, m < n, and f ∈ C [m, n]. P3 (x) stands for the third Bernoulli
1-periodic function [12, p. 114 - items (13) and (14)]
(i) P3 (x) = x x − 12 (x − 1) for x ∈ [0, 1]
(4.3)
(ii) P3 (x + 1) = P3 (x)
for x ∈ R .
1
In 2007 we shall be celebrating the third-centenary of Euler’s birth: April 15, 1707.
J. Inequal. Pure and Appl. Math., 7(5) Art. 166, 2006
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6
V ITO L AMPRET
Therefore, due to
P3 (−x) ≡ P3 (1 − x) ≡ −P3 (x),
(4.4)
we have, referring to (4.2),
1
R(m, n) =
6
(4.5)
Z
n
P3 (x)f (3) (x) dx.
m
We wish to obtain a better estimate of the remainder R(m, n). Indeed, due to (4.3), we have
2 1 1
= 1√ .
= max t −
P3
t
+
t
max |P3 (x)| = max
0≤x≤1
4 12 3
2
− 12 ≤t≤ 12
− 21 ≤t≤ 12
Hence, using (4.5), we estimate
1
|R(m, n)| ≤ √
72 3
(4.6)
Z
n
(3) f (x) dx,
m
for integers n > m. Moreover, if f (3) (x) is a monotonic function, not necessarily keeping its
sign, then, for integers n > m:
(i)
(4.7)
R(m, n) ≤ 0 if f (3) (x) grows
(ii) R(m, n) ≥ 0 if f (3) (x) decreases .
Indeed, substituting x = i + t, i being an integer, and considering the periodicity, P3 (i + t) ≡
P3 (t), we have
Z i+1
Z 1/2
Z 1
(3)
(3)
(4.8)
P3 (x)f (x) dx =
P3 (t)f (i + t) dt +
P3 (t)f (3) (i + t) dt .
i
0
1/2
Additionally, substituting t = 1 − τ and referring to the identity (4.4), we obtain
Z 1
Z 1/2
(3)
P3 (t)f (i + t) dt = −
P3 (τ )f (3) (i + 1 − τ ) dτ .
1/2
0
Therefore, using (4.8), we find
Z n
n−1 Z
X
(3)
(4.9)
P3 (x)f (x) dx =
m
i=m
1/2
P3 (t) f (3) (i + t) − f (3) (i + 1 − t) dt .
0
Because i + t ≤ i + 1 − t for t ∈ 0, 12 , the difference f (3) (i + t) − f (3) (i + 1 − t), occurring in
(4.9), is non-positive
or non-negative if f (3) (x) grows or decreases, respectively. But, P3 (t) ≥ 0
for t ∈ 0, 12 , due to (4.3). Hence,
in the right hand side of (4.9) keep their
1 all the integrands
(3)
sign over the entire interval 0, 2 , provided that f (x) is monotonous. According to (4.5),
this confirms the assertion (4.7).
4.1. Case a 1, 2 ≤ m ≤ n ≤ a − m. In this section we are supposing that the real number
a and the integers m and n satisfy the following conditions:
(4.10)
a 1,
2 ≤ m ≤ n ≤ a − m.
Using the logarithmic function we can transform the last product in (2.7) into the sum
n
n
Y
a−i X
ln
=
f (i),
i
i=m
i=m
J. Inequal. Pure and Appl. Math., 7(5) Art. 166, 2006
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E STIMATING THE S EQUENCE OF R EAL B INOMIAL C OEFFICIENTS
7
where f (x) ≡ ln a−x
. Unfortunately the second and the fourth derivatives of the function f do
x
not keep their respective sign for x ∈ [m, a − m], which has some disadvantage for estimating
the remainder in the summation formula. However,
n
n
Y
a−i Y a−i
=
.
i
m
+
n
−
i
i=m
i=m
Thus,
n
n
Y
a−i X
ln
=
fa,b (i),
i
i=m
i=m
(4.11)
where
(4.12)
fa,b (x) := ln
a−x
≡ ln(a − x) − ln(b − x),
b−x
b = m + n, and 0 < x < b ≤ a.
We have the derivatives
(4.13)
(1)
fa,b (x) ≡ −
(2)
1
1
+
≥ 0,
2
(a − x)
(b − x)2
(4.14)
fa,b (x) ≡ −
(4.15)
fa,b (x) ≡ −
(3)
1
1
+
≥ 0,
a−x b−x
2
2
+
≥0
3
(a − x)
(b − x)3
and
(4.16)
(4)
fa,b (x) ≡ −
6
6
+
≥ 0,
(a − x)4 (b − x)4
for 0 < x < a.
Formula (4.1), applied to the function fa,b , determines the remainder, denoted as Ra (m, n).
Referring to (4.6), (4.14) and (4.15), we have
Z n
1
(3)
|Ra (m, n)| ≤ √ ·
fa,b (x)dx
72 3 m
1
(2)
(2)
= √ · fa,b (n) − fa,b (m)
72 3
1
(2)
≤ √ · fa,b (n)
72 3
−1
1
1
= √
+
72 3 (a − n)2 (b − n)2
−1
1
1
= √
+
,
72 3 (a − n)2 m2
that is
(4.17)
|Ra (m, n)| ≤
1
√ .
72 m2 3
Moreover, due to (4.7) and (4.16), we have
Ra (m, n) ≤ 0 .
J. Inequal. Pure and Appl. Math., 7(5) Art. 166, 2006
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8
V ITO L AMPRET
Therefore, considering (4.17), we conclude that for a, m and n, as determined by (4.10), there
exists some ϑ ∈ [0, 1], depending on a, m and n, such that
(4.18)
Ra (m, n) = −
ϑ
√ ,
72 m2 3
for n ∈ [m, a − m].
After the remainder has been uniformly estimated, the summation formula (4.1) can be applied. According to (4.12), we find
m
n
Z n
(b − x)b−x
m (a − m)a−m
(4.19)
= ln
fa,b (x) dx = ln
(a − x)a−x m
nn (a − n)a−n
m
and
r
1
(a − m)(a − n)
(4.20)
[fa,b (m) + fa,b (n)] = ln
.
2
m·n
Referring to (4.13), and recalling that b = m + n, defined below (4.12), we have
i
1 h (1)
a
1
1
(1)
(4.21)
f (n) − fa,b (m) =
−
.
12 a,b
12 m(a − m) n(a − n)
From (4.11) and (4.1), using (4.19)–(4.21), we obtain the expression
!
r
n
Y
a−i
(a − m)(a − n) mm (a − m)a−m
ln
= ln
· n
i
m
·
n
n (a − n)a−n
i=m
a
1
1
+
−
+ Ra (m, n) .
12 m(a − m) n(a − n)
Consequently, we conclude, according to (2.7) and (4.18), that the following theorem holds.
Theorem 4.1. For any integers m and n, obeying condition (4.10), there exists some ϑ =
ϑ(a, m, n) ∈ [0, 1], depending on a, m and n, such that
ϑ
√ ,
(4.22)
C(a, n) = B(a, m, n) · exp −
72 m2 3
where
r
mm (a − m)a−m
a−m
B(a, m, n) = m C(a, m) · n
·
a−n
n (a − n)
m n(a − n)
a
1
1
· exp
−
,
12 m(a − m) n(a − n)
i.e.
(4.23)
s
mm (a − m)a−m
m (a − m)
B(a, m, n) = C(a, m) · n
·
a−n
n (a − n)
n (a − n)
a
1
1
· exp
−
.
12 m(a − m) n(a − n)
For every a and m, satisfying (4.10), the function x 7→ B(a, m, x) has the symmetric property
a
a
B a, m, − x ≡ B a, m, + x
2
2
J. Inequal. Pure and Appl. Math., 7(5) Art. 166, 2006
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E STIMATING THE S EQUENCE OF R EAL B INOMIAL C OEFFICIENTS
9
and, moreover, strictly increases/decreases on the intervals 0, a2 and a2 , a respectively. Indeed, due to (4.23) we have
d
B (a, m, x) ≡ B(a, m, x) · ϕ(x),
dx
where ϕ(x) ≡ ψ(a − x) − ψ(x) and ψ(x) ≡ 1/(2x) − 1/(12x2 ) + ln x. Because
6x2 − 3x + 1
>0
6x3
for x > 0 and consequently ϕ0 (x) ≡ −ψ 0 (a − x) − ψ 0 (x) < 0, function ϕ(x) strictly decreases.
a
Thus, ϕ(x) > ϕ( a2 ) = 0 for x ∈ 0, a2 and ϕ(x) < ϕ( a2 ) = 0 for
x
∈
,
a
. Hence
2
d
d
a
a
B (a, m, x) > 0 for x ∈ 0, 2 and dx B (a, m, x) < 0 for x ∈ 2 , a .
dx
The expression for B(a, m, n) can be further simplified. Namely, thanks to the estimate
0 < n < a, the relative deviation
t − a2
t
(4.24)
d(a, t) := a = 2 − 1
a
2
ψ 0 (x) ≡
lies within the open interval (−1, 1) and generates the equalities
a 2 (4.25)
t(a − t) =
1 − d2 (a, t)
2
and
a a a
t
a−t
(4.26)
t (a − t)
=
(1 + x)1+x (1 − x)1−x 2 , x = d(a, t).
2
Using (4.25) and (4.26), the equation (4.23) can be written in a more compact form as
D(a, µ)
(4.27) B(a, m, n) =C(a, m) ·
D(a, ν)
a/2 r
F (µ)F (−µ)
1 − µ2
3
1
1
(4.28)
=C(a, m)
exp
−
,
F (ν)F (−ν)
1 − ν2
a 1 − µ2 1 − ν 2
where µ = d(a, m), ν = d(a, n), F (t) ≡ (1 + t)1+t and
√
a/2
2
D(a, t) = [F (t) F (−t)]
1 − t exp
3
a(1 − t2 )
.
The graphs of the functions x 7→ 1/D(a, x), for a = 3π and a = 33π, are shown in Figure 4.1,
and the graphs of the sequences n 7→ C(a, n) and the functions x 7→ B(a, 2, x) are illustrated
in Figure 4.2.
1
1
0.5
a3Π
0.5
0
a33Π
0
1
0
1
1
0
1
Figure 4.1: Graphs of functions x 7→ 1/D(a, x).
Figure 4.2 indicates that the approximation B(a, m, n) is very close to C(a, n), even for
small m, for example, m = 2. Here, "to be close" has its meaning in the absolute sense, i.e.
J. Inequal. Pure and Appl. Math., 7(5) Art. 166, 2006
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10
V ITO L AMPRET
150
121029
100
81029
a3Π
50
1
a33Π
41029
5
10
20
40
60
80
100
Figure 4.2: Approximations B(a, 2, x) to sequences n 7→ C(a, n).
proportionally to max{C(a, n) : 1 ≤ n ≤ a}. The curves in these figures are reminiscent of
the Gauss (normal) bell-shaped curves arising from the function x 7→ exp(−x2 ). Indeed, in
probability theory we have the well known DeMoivre-Laplace local approximation [4, pp. 174190]

!2 
n a−n
a
1
1
1
1 n− 2 
pa
C(a, n) ·
·
≈p
· exp −
,
a
2
2
2
2π · 4
4
from which we obtain a DeMoivre-Laplace approximation to the sequence of binomial coefficients
!
1
2 a+ 2
(2n − a)2
C(a, n) ≈ M (a, n) := √
· exp −
.
2a
aπ
A figure representing the graphs of the sequences n 7→ C(a, n) and the function x 7→ M (a, x)
for a = 3π and a = 33π, is not shown because it is indistinguishable from Figure 4.2. Consequently, it seems that the approximation C(a, n) ≈ M (a, n) should be very good for a and n
obeying (4.10). Unfortunately, this approximation is good relatively to max{C(a, n) : 1 ≤ n ≤
a}. It is true that the relative error
C(a, n) − M (a, n)
M (a, n)
is small for n ≈ a2 . However, it can approach even to the number −1 for n ≈ 1 or n ≈ a, as is
evident from Figure 4.3, which shows the sequences of errors ρ(a, n) for a = 3π and a = 33π.
ρ(a, n) :=
0.05
0.05
3
6
9
40
50
a3Π
6103
60
60
90
0.3
0.6
3103
0.1
0.15
30
3103
a33Π
0.9
a33Π
Figure 4.3: The graphs of sequences n 7→ ρ(a, n).
Fortunately, the situation is quite different concerning the approximation C(a, n) ≈ B(a, m, n).
Indeed, due to (4.22), the absolute value of the relative error
C(a, n) − B(a, m, n)
ϑ
√
(4.29)
r(a, m, n) :=
= exp −
−1
B(a, m, n)
72 m2 3
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E STIMATING THE S EQUENCE OF R EAL B INOMIAL C OEFFICIENTS
11
becomes small uniformly for large m. Because the function ϕ(x) := e−x − 1 + x strictly
increases on [0, ∞) and ϕ(0) = 0, we have |exp(−x) − 1| < x for x > 0. Thus, considering
(4.29), we find the uniform estimate
1
(4.30)
−
< r(a, m, n) ≤ 0,
124 m2
for a, m and n, which obey condition (4.10). For example |r(a, 2, n)| < 2.1×10−3 , |r(a, 10, n)| <
8.4 × 10−5 , |r(a, 100, n)| < 8.4 × 10−7 , and |r(a, 1000, n)| < 8.4 × 10−9 . Direct computations,
using [19], give −3.3 × 10−4 < r(33π, 2, 50) < −3.2 × 10−4 , −2.8 × 10−6 < r(33π, 10, 50) <
−2.7 × 10−6 , and −2.8 × 10−9 < r(333π, 100, 500) < −2.7 × 10−9 . Hence, a priori estimate
(4.30) appears as rather rough.
4.2. Case n a. Let us suppose that the real number a and the integers m and n satisfy the
following conditions:
(4.31)
a ∈ R \ ( N ∪ {−1, 0} )
and n > m > |a| .
We relate the last product in (2.8) with a sum
ln
(4.32)
n
n
Y
i−a X
=
fa (i) ,
i
i=m
i=m
where
(4.33)
fa (x) ≡ ln
x−a
x
≡ ln(x − a) − ln x .
We shall use formula (4.1) for the function fa , which determines the remainder, denoted as
Ra (m, n). To this effect we need the derivatives
1
1
(4.34)
fa0 (x) ≡
− ,
x−a x
1
−1
+
,
(x − a)2 x2
(4.35)
fa00 (x) ≡
(4.36)
fa(3) (x) ≡
2
2
− 3
3
(x − a)
x
fa(4) (x) ≡
−6
6
+ 4,
4
(x − a)
x
and
(4.37)
for x > a. It is evident from these expressions that, for a > 0, all derivatives of odd/even orders
are positive/negative and, for a < 0, all derivatives of odd/even orders are negative/positive.
Thus, using the function signum, sgn(a) := −1 for a < 0 and sgn(a) = 1 for a > 0, we have
(i)
(4.38)
(ii)
sgn(a) · fa00 (x) ≤ 0
(3) (3)
f
(x)
a
≡ sgn(a) · fa (x)
(4)
(iii) sgn(a) · fa (x) ≤ 0 .
According to (4.33) we get
n
m
Z n
(x − a)x−a
m (n − a)n−a
(4.39)
fa (x) dx = ln
= ln
xx
nn (m − a)m−a
m
m
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12
V ITO L AMPRET
and
(4.40)
1
[fa (m) + fa (n)] = ln
2
r
(m − a)(n − a)
.
m·n
Due to (4.34), we have
1 0
a
1
1
0
[f (n) − fa (m)] =
−
.
(4.41)
12 a
12 n(n − a) m(m − a)
Considering (4.6), (4.38)(i) – (ii) and (4.35) we estimate the remainder Ra (m, n) as
Z n
1
fa(3) (x)dx
|Ra (, m, n)| ≤ √ · sgn(a)
72 3
m
1
00
00
= √ · sgn(a) · fa (n) − sgn(a) · fa (m)
72 3
− sgn(a) 00
√ · fa (m)
≤
72 3
sgn(a)
1
1
√
−
=
72 3 (m − a)2 m2
sgn(a) 2ma − a2
√
=
,
72 3 (m − a)2 m2
i.e.
|a|
(4.42)
|Ra (m, n)| ≤ √
.
36 3 m(m − a)2
However, from (4.7) and (4.38)(iii) we conclude that
sgn(a) · Ra (m, n) ≥ 0 .
Consequently, according to (4.42), there exists some ϑ ∈ [0, 1] such that
sgn(a) · Ra (m, n) =
ϑ |a|
√
.
36 3 m(m − a)2
Hence, for integers n > m > |a|, we have
(4.43)
Ra (m, n) =
ϑ·a
√
36 3 m(m − a)2
for some ϑ ∈ [0, 1], depending on a, m and n.
Inserting expressions (4.39) – (4.41), and (4.43) into the summation formula (4.1), we obtain
the expression
!
r
n
Y
i−a
(m − a)(n − a) mm (n − a)n−a
ln
= ln
· n
i
mn
n (m − a)m−a
i=m
a
1
1
ϑ·a
+
−
+ √
.
12 n(n − a) m(m − a)
36 3 m(m − a)2
Hence, considering (2.8), we conclude that the following theorem was proved true.
Theorem 4.2. For any integers m and n, which obey (4.31), there exists some ϑ ∈ [0, 1],
depending on a, m and n, such that
ϑ·a
∗
√
(4.44)
C(a, n) = B (a, m, n) · exp
,
36 3 m(m − a)2
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E STIMATING THE S EQUENCE OF R EAL B INOMIAL C OEFFICIENTS
13
where
1
B ∗ (a, m, n) = (−1)n−m · m C(a, m) ·
n−a
r
(m − a)(n − a) mm (n − a)n−a
·
· n
mn
n (m − a)m−a
a
1
1
· exp
−
12 n(n − a) m(m − a)
i.e.
(4.45)
mm+1/2
B ∗ (a, m, n) = (−1)n−m · C(a, m) ·
(m − a)m−a−1/2
1
1
a
· exp
−
12 n(n − a) m(m − a)
r
a n
a
· 1−
· 1 − · (n − a)−(a+1) .
n
n
The rate of convergence
a n
= e−a , a ∈ R,
n→∞
n
of the sequence occurring in (4.45), can be estimated using the following lemma:
lim
1−
Lemma 4.3. For any positive real x and t ≥ 2x there hold the estimates
t
2
2
(i) exp −x − xt < 1 − xt < exp −x − x2t
t
2
2
(ii) exp x − x2t < 1 + xt < exp x − x3t .
Proof. Indeed, integrating the inequalities
1+τ <
1
< 1 + 2τ
1−τ
and
1
2
<1− τ,
1+τ
3
valid for τ ∈ 0, 21 , we obtain the relations
Z y
y2
dτ
y+
<
= − ln(1 − y) < y + y 2
2
0 1−τ
and
Z y
y2
dt
y2
y−
<
= ln(1 + y) < y − ,
2
3
0 1+t
1
true for
y ∈ 0, 2 . Moreover, for x > 0 and t ≥ 2x the number y := xt lies in the interval
0, 12 and the relations above could be applied for this y and the lemma is thus verified.
1−τ <
Remark 4.4. From the above lemma we obtain
x2
x t
x2
(4.46)
exp x −
< 1+
< exp x −
t
t
3t
for any real x 6= 0 and t ≥ 2|x|.
J. Inequal. Pure and Appl. Math., 7(5) Art. 166, 2006
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14
V ITO L AMPRET
√
From (4.44) and (4.45),
√ using our Lemma, and considering the inequalities 1 < 1 1 + h 1<
1 + h/2 and 1 − h < 1 − h < 1, true for h ∈ (0, 1), together with the estimate − n > − m ,
valid for positive integers n > m, we find that the next proposition is valid.
Proposition 4.5. For a, m and n, which follow (4.31), the following estimates hold
(i)
|C(a, n)| > C ∗ (a, m) · I(a, m) · (n − a)−(a+1)
(4.47)
(ii) |C(a, n)| < C ∗ (a, m) · J(a, m) · (n − a)−(a+1) ,
where
e−a mm+1/2
C (a, m) = |C(a, m)| ·
(m − a)m−a−1/2

2
|a|
a

 exp − 2m
− 36√3 m(m+|a|)
if a < 0
2 ,
I(a, m) =

a
 1 − a exp − a2 −
, if a > 0
m
m
12 m(m−a)
∗
(4.48)
and
 |a|
|a|

 1 + 2m
exp 12 m(m+|a|)
, if a < 0
J(a, m) =

 exp
√ a
,
if a > 0 .
36 3 m(m−a)2
(4.49)
We emphasize that |C(a, n)| ≈ C ∗ (a, m) · (n − a)−(a+1) ) represents a good approximation
for large n. From relations (4.47)–(4.49) we conclude that the following proposition also holds.
Proposition 4.6. For every1 real number a we have

∞, if a < −1


1, if a = −1
(4.50)
lim |C(a, n)| =
n∈N


n→∞
0, if a > −1 .
Computing C ∗ (a, m), |I(a, m)| and |J(a, m)| directly, using e.g. [19], we obtain from relations (4.47)–(4.49) good estimates of binomial coefficients. For example, because
C ∗ (−π, 2099) · I(−π, 2099) = 0.436029 . . . > 0.436
and
C ∗ (−π, 2099) · J(−π, 2099) = 0.437382 . . . < 0.438,
we have
0.436 · (n + π)π−1 < |C(−π, n)| < 0.438 · (n + π)π−1 ,
true for n ≥ 2100.
Similarly, since
C ∗ (π, 8999) · I(π, 8999) = 0.983122 . . . > 0.983
and
C ∗ (π, 8999) · J(π, 8999) = 0.984545 . . . < 0.985,
we have
0.983 · (n − π)−(π+1) < |C(π, n)| < 0.985 · (n − π)−(π+1) ,
valid for n ≥ 9000.
1
Obviously
−1
n
≡ (−1)n .
J. Inequal. Pure and Appl. Math., 7(5) Art. 166, 2006
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E STIMATING THE S EQUENCE OF R EAL B INOMIAL C OEFFICIENTS
15
According to (4.47), the quotient
Q(a, n) :=
|C(a, n)|
· (n − a)−(a+1)
C ∗ (a, m)
lies close to 1 and is bounded as
I(a, m) < Q(a, n) < J(a, m),
(4.51)
for a, m and n, which obey (4.31). Figure 4.4 illustrates the estimate (4.51) for a ∈ {−π, π}
and n = m + 1 ∈ {5, 6, . . . , 101}.
1.4
1.2
1
0.8
0.6
0.4
0.2
1
0.8
0.6
0.4
0.2
aΠ
20
40
60
80 100
aΠ
20
40
60
80 100
Figure 4.4: Estimating the sequence of binomial coefficients.
Remark 4.7. Using the Gamma function, the definition of binomial coefficient C(a, b) could
be extended as
Γ(a + 1)
C(a, b) :=
Γ(b + 1) · Γ(a − b + 1)
for a and b being arbitrary complex numbers, different from any negative integer. From this
expression the symmetric property, C(a, b) = C(a, a − b), is evident.
Using Stirling’s approximation to the Gamma function, see e.g. [1],
x x ϑ
√
· e 12x
Γ(x + 1) = x Γ(x) = 2π x ·
e
for x ∈ R+ and some ϑ ∈ (0, 1), which depends on x, it is possible to obtain approximations,
which are close to ours. For example, for positive integer n and real a > n we obtain the
formula
r
a
aa
1 ϑ ϑ0
ϑ00
·
· exp
− −
,
C(a, n) =
2π n(a − n) nn (a − n)a−n
12 a
n
a−n
true for some ϑ, ϑ0 , ϑ00 ∈ (0, 1) which depend on a and n.
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16
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