Sums of Involving the Harmonic Numbers and the Binomial Coefficients ()
1. Introduction and Preliminaries
Let
be the exponential complete Bell polynomials and
In [1] , Zave established the following series expansion:
![](//html.scirp.org/file/5-1100402x12.png)
(1)
where
for
,
and
.
Spiess [2] introduced the numbers
and
,
for
; then Equation (1.1) is equivalent to
![]()
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where
,
,
,
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The paper is organized as follows. In Section 2, we obtain some for
and binomial coefficients by means of the Riordan arrays. In Section 3, we establish some identities involving the numbers
and inverse of binomial coefficients. Finally, in Section 4, we give the asymptotic expansions of some summations
involving the numbers
by Darboux’s method. Due to [3] [4] , a Riordan array is a pair
of formal power series with
. It defines an infinite lower triangular array
according to the rule
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Hence we write
. If
is an Riordan array and
is the generating function of the sequence
, i.e.,
. Then we have
(2)
Based on the generating function (1), we obtain the next Riordan arrays, to which we pay particular attention in the present paper:
(3)
Lemma 1 (see [5] ) Let
be a real number and
. When
,
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2. Identities Involving the Numbers
and Binomial Coefficients
Theorem 1. Let
,
,
, then
(4)
Proof. By (1), we have
(5)
Comparing the coefficients of
on both sides of (5), we completes the proof of Theorem 1.
Recall that
Thus, setting
in Theorem 1 gives the next three identities, respectively.
Corollary 1. Let
,
, the following relations hold
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Theorem 2. Let
,
, then
(6)
Proof. To obtain the result, make use of the Theorem 1.
Theorem 3. Let
,
, then
(7)
Proof. Applying the summation property (2) to the Riordan arrays (3), we have
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which is just the desired result.
Setting
in Theorem 3 gives the next Corollary.
Corollary 2 Let
, then
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Corollary 3 Let
,
, then
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Proof. Setting
in Theorem 3 gives Corollary 3.
Corollary 4. Let
, then
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Proof. Setting
in Corollary 2 yields Corollary 4.
Theorem 4. Let
,
, then
(8)
Proof. which is just the desired result.
Setting
in Theorem 4 gives the next Corollary.
Corollary 5. Let
, then
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Corollary 6. The substitutions
in Theorem 4 gives the next four identities, respectively.
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Setting
in Corollary 5 gives the next four identities, respectively.
Corollary 7. Let
, then
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Theorem 5. Let
,
, then
(9)
where
are the Stirling numbers of the first kind.
Proof. By (1) and (2), we have
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which is just the desired result.
Setting
in Theorem 5 gives the next Corollary.
Corollary 8. Let
, then
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Setting
in Theorem 6 gives the next Corollary.
Corollary 9. Let
,
, then
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We give four applications of Corollary 9:
Corollary 10. Let
, then
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3. Identities Involving
and Inverse of Binomial Coefficients
For identities involving Harmonic numbers and inverse of binomial coefficients
in given in [6] .
In Section, we obtain some for
and binomial coefficients by means of the Riordan arrays. From these identities, we deduce some identities involving binomial coefficients, Harmonic numbers and identities related to
, ![]()
In [7] , the inverse of a binomial coefficient is related to an integral, as follows
(10)
From the generating function of
and (10), we have
Theorem 6. For
be any integer, then
(11)
Proof. From (1) and (10), we obtain
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This gives (11).
Corollary 11 Setting
in Theorem 6, The following relation holds:
(12)
(13)
(14)
(15)
Setting
in Corollary 11, gives the next identities.
Corollary 12 The following relation holds
(16)
(17)
(18)
(19)
(20)
(21)
(22)
(23)
Corollary 13. The following relation holds
(24)
(25)
(26)
(27)
Proof. (16) minus(20) give (24); (17) minus (21), (18) minus (22) and (19) minus (23), yields (25), (26) and (27), respectively.
Leonhard Euler (1707-1783) had already stated the equation
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Recall the Euler sum identities [8] [9] .
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The next, we gives identities related to
, ![]()
For completeness we supply proofs:
(28)
(29)
Similarly, we obtain summation formulas related
, ![]()
(30)
(31)
By (18) and (28), (19) and (31), we have
(32)
(33)
Similarly, for completeness we supply a proof:
(34)
By (28) minus (30), we get
(35)
Applying (25) and (34), (26) and (32), we have
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4. Asymptotics
Theorem 7 For
be any integer, as
, we have
(36)
Proof. By Lemma 1, we have
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and this complete the proof.
Similarly, we can obtain the next Theorem.
Theorem 8. Let
be any integer, as
, we have
(37)
Theorem 9. For
be any integer, as
, we have
(38)
Proof. By Lemma 1, we have
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this give (38).
Theorem 10. For
be any integer, as
, we have
(39)
Proof. By Corollary 3 of [10] , immediately complete the proof of Theorem 10.
Acknowledgements
The author would like to thank an anonymous referee whose helpful suggestions and comments have led to much improvement of the paper. The research is supported by the Natural Science Foundation of China under Grant 11461050 and Natural Science Foundation of Inner Mongolia under Grant 2012MS0118.