Harmonic Numbers and Cubed Binominal Coefficients

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This is the Published version of the following publication

Sofo, Anthony (2011) Harmonic Numbers and Cubed Binominal Coefficients.

International Journal of Combinatorics. ISSN 1678-9163 (print), 1687-9171 (online)

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Volume 2011, Article ID 208260,14pages doi:10.1155/2011/208260

Research Article

Harmonic Numbers and Cubed Binomial Coefficients

Anthony Sofo

Victoria University College, Victoria University, P.O. Box 14428, Melbourne City, VIC 8001, Australia

Correspondence should be addressed to Anthony Sofo,[email protected] Received 18 January 2011; Accepted 3 April 2011

Academic Editor: Toufik Mansour

Copyrightq2011 Anthony Sofo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Euler related results on the sum of the ratio of harmonic numbers and cubed binomial coeﬃcients are investigated in this paper. Integral and closed-form representation of sums are developed in terms of zeta and polygamma functions. The given representations are new.

1. Introduction

The well-known Riemann zeta function is defined as

ζz ^∞

r1

1

r^z, Rez>1. 1.1

The generalized harmonic numbers of orderαare given by

Hn^αⁿ

r1

1

r^α forα, n∈^Æ×^Æ, ^Æ :{1,2,3, . . .}, 1.2

and forα1,

Hn¹Hn ₁

0

1−tⁿ

1−t dtⁿ

r1

1

r γψn1, 1.3

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whereγdenotes the Euler-Mascheroni constant defined by

γ lim

n→ ∞

_n

r1

1

r −logn

−ψ1≈0.5772156649. . ., 1.4

and whereψzdenotes the Psi, or digamma function defined by

ψz d

dzlogΓz Γz Γz ^∞

n0

1

n1− 1 nz

−γ, 1.5

and the Gamma functionΓz _∞

0 u^z−1e^−udu, for^Êz>0.

Variant Euler sums of the form ∞

n1

H_2n¹−1/2Hn¹

x²ⁿ

n^p forp1 andp2 1.6

have been considered by Chen1 , and recently Boyadzhiev2 evaluated various binomial identities involving power sums with harmonic numbersHn¹. Other remarkable harmonic number identities known to Euler are

∞ n1

Hn¹

n1² ζ3, ^∞

n1

Hn¹

n³ 5

4ζ4, 1.7

there is also a recurrence formula

2n1ζ2n 2 n−1 r1

ζ2rζ2n−2r, 1.8

which shows that in particular, forn2, 5ζ4 2ζ2²and more generally thatζ2nis a rational multiple ofζn².Another elegant recursion known to Euler3 is

2 ∞ n1

Hn¹

n^q q2

ζ q1

−^q−2

r1

ζr1ζ q−r

. 1.9

Further work in the summation of harmonic numbers and binomial coeﬃcients has also been done by Flajolet and Salvy 4 and Basu 5 . In this paper it is intended to add, in a small way, some results related to 1.7 and to extend the result of Cloitre, as reported in 6 , _∞

n1H_n¹/_nk

k

k/k−1² for k > 1. Specifically, we investigate integral representations and closed form representations for sums of harmonic numbers and cubed binomial coeﬃcients. The works of7–13 also investigate various representations of binomial sums and zeta functions in simpler form by the use of the Beta function and other techniques. Some of the material in this paper was inspired by the work of Mansour, 8 , where he used, in part, the Beta function to obtain very general results for finite binomial sums.

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2. Integral Representations and Identities

The following Lemma, given by Sofo 11 , is stated without proof and deals with the derivative of a reciprocal binomial coeﬃcient.

Lemma 2.1. Letabe a positive real number,z≥0,nis a positive integer and letQan, z ^anzz ⁻¹ be an analytic function ofz. Then,

Qan, z dQ dz

⎧⎨

⎩

−Qan, z

ψz1an−ψz1

forz >0,

−Hn¹, forz0 anda1. 2.1

Theorem 2.2. Leta, b, c, d ≥0 be real positive numbers,|t| ≤1, p≥ 0 and letj, k, l, m ≥0 be real positive numbers. Then

n≥1

tⁿ ^np−1_n−1 ψ

j1an

−ψ j1 n ^anj_j _bnk

k

_cnl

l

_dnm

m

−aklmt ¹

0

1−x^j

1−y_k−1

1−z^l−11−w^m−1 1−tx^ay^bz^cw^d_p1

×ln1−x·x^a−1y^bz^cw^ddx dy dz dw.

2.2

Proof. Expand

n≥1

tⁿ ^np−1_n−1 n ^anj_j _bnk

k

_cnl

l

_dnm

m

n≥1

tⁿ

np−1 n−1

anΓanΓ j1

Γbn1k·Γk nΓ

anj1

Γbnk1

×Γcn1l·ΓlΓdn1m·Γm Γcnl1Γdnm1

aklm ₁

0

1−x^j x

n≥1

x^antⁿ

np−1 n−1

×Bbn1, kBcn1, lBdn1, mdx, 2.3

where

B α, β

ΓαΓ β Γ

αβ ₁

0

1−y_α−1 y^β−1dy

₁

0

1−y_β−1

y^α−1dy, forα >0 andβ >0

2.4

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is the classical Beta function. Diﬀerentiating with respect to the parameterj, and utilizing Lemma 2.1implies the resulting equation is as follows:

n≥1

j1an

−ψ j1 n ^anj_j _bnk

k

_cnl

l

_dnm

m

−aklm

₁

0

1−x^j

x ln1−x

n≥1

x^antⁿ

np−1 n−1

×Bbn1, kBcn1, lBdn1, mdx,

−aklm ¹

0

1−x^jln1−x x

1−y_k−1

1−z^l−11−w^m−1

×

n≥1

np−1

n−1 tx^ay^bz^cw^d_n

dx dy dz dw

−aklmt ¹

0

1−x^jln1−x

1−y_k−1

×x^a−1y^bz^cw^ddx dy dz dw

2.5

for|tx^ay^bz^cw^d|<1.

In the following three corollaries we encounter harmonic numbers at possible rational values of the argument, of the formH_r/b−1^α wherer1,2,3, . . . , k, α1,2,3, . . ., andk∈^Æ. The polygamma functionψ^αzis defined as

ψ^αz d^α1 dz^α1

logΓz d^α

dz^α ψz

, z /{0,−1,−2,−3, . . .}. 2.6

To evaluateH_r/b−1^α we have available a relation in terms of the polygamma functionψ^αz, for rational argumentsz,

H_r/b−1^α1 ζα1 −1^α α! ψ^α r

b

, 2.7

whereζzis the Riemann zeta function. We also define

H_r/b−1¹ γψ r b

, H₀^α0. 2.8

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The evaluation of the polygamma functionψ^αr/bat rational values of the argument can be explicitly done via a formula as given by K ¨olbig14 see also15 or Choi and Cvijovi´c16 in terms of the polylogarithmic or other special functions. Some specific values are given as

ψⁿ 1

2

−1ⁿn! 2ⁿ¹−1

ζn1

H_−2/3⁵ ζ5− 1 24ψ⁴

1 3

−2π⁵√ 3

9 −120ζ5, H_1/4² 16−8G−5ζ2,

2.9

and can be confirmed on a mathematical computer package, such as Mathematica17 . Corollary 2.3. Leta1, dcb >0, t1, p0, j 0 and letl mk≥ 1 be a positive integer. Then

n≥1

Hn¹

n_bnk

k

₃

−k³ ¹

0

1−y

1−z1−w_k−1 1−x

yzw_b ln1−x· yzw_b

dx dy dz dw

2.10

^k

r1

−1^r1 k

r 3

×

− r² 4b²ζ4−

r²

b²H_r/b−1¹ r b r²

bXRk

ζ3

1−r

bH_r/b−1¹ −r²

b²H_r/b−1² r 1− r

bH_r/b−1¹

XRk r²Y Rk

ζ2 1

2 H_r/b−1¹ 2

H_r/b−1²

r b

H_r/b−1¹ H_r/b−1² H_r/b−1³

r²

2b² H_r/b−1² ₂

2H_r/b−1¹ H_r/b−1³ 3H_r/b−1⁴

r

2 H_r/b−1¹ ₂

H_r/b−1²

r² b

H_r/b−1¹ H_r/b−1² H_r/b−1³ XRk

r²

2 H_r/b−1¹ ₂

H_r/b−1²

Y Rk

,

2.11

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where

XRk:−3 H_k−r¹ −H_r−1¹

, 2.12

Y Rk: 3 2

XR²k

3 H_k−r² H_r−1²

. 2.13

Proof. Let

n≥1

Hn¹

n_bnk

k

3

n≥1

Hn¹k!³ n_k

r1bnr³

n≥1

Hn¹k!³ n

k r1

Ar

bnr Br

bnr² Cr

bnr³

,

2.14

where

Cr lim

n→−r/b

bnr³ _k

r1bnr³

−1^r1 r

k!

k r

3

,

Br lim

n→−r/b

d dn

bnr³ _k

r1bnr³

3−1^r r

k!

k r

3

H_k−r¹ −H_r−1¹ ,

Ar 1 2 lim

n→−r/b

d² dn²

bnr³ _k

r1bnr³

3 2−1^r1

r k!

k r

3

3 H_k−r¹ −H_r−1¹₂

H_k−r² H_r−1²

.

2.15

Now, by interchanging sums, we have

n≥1

Hn¹

n_bnk

k

3 ^k

r1

Ark!³

n≥1

Hn¹

nbnrBrk!³

n≥1

Hn¹

nbnr² Crk!³

n≥1

Hn¹

nbnr³

. 2.16

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We can evaluate

n≥1

Hn¹

nbnr

n≥1

ψn 1/n

nbnr γ

nbnr

n≥1

ψn

nbnr 1

n²bnr γ nbnr

n≥1

ψn nbnr 1

rn²

γ−b/r r

n≥0

1

n1 − 1 n r/b

1

n r/b− 1 n1 r/b

−bγ r² −γ²

2r 3ζ2

2r

ψr/b₂

2r −ψr/b

2r γ

r ψ r b

γb ,

2.17

here we have used the result from18

n≥1

ψn nbnr 1

2r ψ r b1

₂

−γ²ζ2−ψ r b1

. 2.18

Now using2.7and2.8, we may write

n≥1

Hn¹

nbnr ζ2 r 1

2r H_r/b−1¹ ₂

H_r/b−1²

. 2.19

Similarly

n≥1

Hn¹

nbnr² 1 r

n≥1

Hn¹

nbnr−b r

n≥1

Hn¹

bnr²

−ζ3 br ζ2

r² −ζ2H_r/b−1¹

br 1

2r² H_r/b−1¹ 2

H_r/b−1²

1 br

H_r/b−1¹ H_r/b−1² H_r/b−1³ ,

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n≥1

Hn¹

nbnr³

n≥1

Hn¹

1

r²nbnr− b

r²bnr² − b rbnr³

−ζ4 4b²r −ζ3

br² − ζ3H_r/b−1¹ b²r ζ2

r³ −ζ2H_r/b−1¹ br²

−ζ2H_r/b−1² b²r 1

2r³ H_r/b−1¹ ₂

H_r/b−1²

1 br²

H_r/b−1¹ H_r/b−1² H_r/b−1³ 1

2b²r H_r/b−1² ₂

2H_r/b−1¹ H_r/b−1³ 3H_r/b−1⁴

.

2.20

Substituting2.19,2.20into2.16whereXRkandY Rkare given by2.12and2.13, respectively, on simplifying the identity2.11is realized.

Fork1 andb1 the following identity is valid:

n≥1

Hn¹

nn1³ ζ2−ζ3−ζ4

4 . 2.21

Theorem 2.4.

n≥1

j1an

−ψ j1 n² ^anj_j _bnk

k

_cnl

l

_dnm

m

−ablmt ¹

0

1−x^j 1−y_k

×ln1−x·x^a−1y^b−1z^cw^ddx dy dz dw.

2.22

Proof. The proof of this theorem is very similar to that ofTheorem 2.2and will not be given here.

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Corollary 2.5. Leta1, dc b >0, t 1, p0, j 0, and letlm k≥ 1 be a positive integer. Then

n≥1

Hn¹

n² ^bnk_k ₃

−bk² ¹

0

1−y_k

1−z1−w ^k−1 1−x

yzwb ln1−x·y^b−1zw^bdx dy dz dw

2.23

^k

r1

−1^r1 k

r 3

× r

4bζ4 4r

bH_r/b−1¹ 3rXRk 2r²Y Rk ζ3

−3b

r 2H_r/b−1¹ r

bH_r/b−1²

rH_r/b−1¹ −2b

XRk−brY Rk

ζ2

− 3b

2r H_r/b−1¹ ₂

H_r/b−1²

−2

H_r/b−1¹ H_r/b−1² H_r/b−1³

− r

2b H_r/b−1² 2

2H_r/b−1¹ H_r/b−1³ 3H_r/b−1⁴

−

b H_r/b−1¹ ₂

H_r/b−1²

r

H_r/b−1¹ H_r/b−1² H_r/b−1³ XRk

−br

2 H_r/b−1¹ ₂

H_r/b−1²

Y Rk

,

2.24

whereXRkis given by2.12andY Rkis given by2.13.

Proof. Following similar steps toCorollary 2.3, we may write

n≥1

Hn¹

n²_bnk

k

3

^k

r1

Ark!³

n≥1

Hn¹

n²bnrBrk!³

n≥1

Hn¹

n²bnr² Crk!³

n≥1

Hn¹

n²bnr³

,

2.25

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and evaluate

n≥1

Hn¹

n²bnr

n≥1

Hn¹

1

rn² − b rnbnr

2ζ3

r −bζ2 r² − b

2r² H_r/b−1¹ ₂

H_r/b−1²

,

n≥1

Hn¹

n²bnr² 3ζ3

r² −2bζ2

r³ ζ2H_r/b−1¹ r²

− b

r³ H_r/b−1¹ ₂

H_r/b−1²

− 1 r²

H_r/b−1¹ H_r/b−1² H_r/b−1³ ,

n≥1

Hn¹

n²bnr³ ζ4

4br² − 3bζ2

r⁴ 2ζ2H_r/b−1¹

r³ ζ2H_r/b−1²

br² 4ζ3

r³ ζ3H_r/b−1¹ br²

− 3b

2r⁴ H_r/b−1¹ 2

H_r/b−1²

− 2 r³

H_r/b−1¹ H_r/b−1² H_r/b−1³

− 1

2br² H_r/b−1² ₂

2H_r/b−1¹ H_r/b−1³ 3H_r/b−1⁴

.

2.26

By substituting2.26into2.25and collecting zeta functions, the identity2.24is obtained.

Fork1 andb1 the following identity is valid:

n≥1

Hn¹

n²n1³ ζ4

4 4ζ3−3ζ2. 2.27

Theorem 2.6.

n≥1

j1an

−ψ j1 n³ ^anj_j _bnk

k

_cnl

l

_dnm

m

−abcmt ¹

0

1−x^j 1−y_k

1−z^l1−w^m−1 1−tx^ay^bz^cw^d_p1

×ln1−x·x^a−1y^b−1z^c−1w^ddx dy dz dw.

2.28

Proof. The proof of this theorem is very similar to that ofTheorem 2.2and will not be given here.

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Corollary 2.7. Leta1, dcb > 0, t1, j 0, and letl mk≥1 be a positive integer.

Then

n≥1

Hn¹

n³_bnk

k

₃ −b²k

1 0

1−y

1−z_k

1−w^k−1 1−x

yzw_b ln1−x· yz_b−1

w^bdx dy dz dw

2.29

^k

r1

−1^r1 k

r 3

× 15

4XRk 5

4r²Y Rk

ζ4

− 9b

r H_r/b−1¹ 5bXRk 2brY Rk

ζ3

⎛

⎝6b²

r² − 3bH_r/b−1¹

r −H_r/b−1² − 3b²

r bH_r/b−1¹

XRk b²Y Rk

⎞

⎠ζ2

3b²

r² H_r/b−1¹ ₂

H_r/b−1²

3b r

H_r/b−1¹ H_r/b−1² H_r/b−1³ 1

2 H_r/b−1² ₂

2H_r/b−1¹ H_r/b−1³ 3H_r/b−1⁴

3b²

2r H_r/b−1¹ 2

H_r/b−1²

b H_r/b−1¹ H_r/b−1² H_r/b−1³ XRk

b²

2 H_r/b−1¹ 2

H_r/b−1²

Y Rk

,

2.30

whereXRkis given by2.12andY Rkis given by2.13.

Proof. We follow similar steps as the previous corollary so that

n≥1

Hn¹

n³ ^bnk_k ₃

^k

r1

Ark!³

n≥1

Hn¹

n³bnrBrk!³

n≥1

Hn¹

n³bnr² Crk!³

n≥1

Hn¹

n³bnr³

.

2.31

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After much algebraic simplification, the following identity is obtained:

n≥1

Hn¹

n³bnr³

n≥1

Hn¹

1

r³n³ − 3b

r⁴n²− b³

r³bnr³ − 3b³

r⁴bnr² 6b² r⁴nbnr

ζ4

r³ − 9bζ3

r⁴ −ζ3H_r/b−1¹

r³ 6b²ζ2

r⁵ −3bζ2H_r/b−1¹

r⁴ −ζ2H_r/b−1²

r³

3b²

r⁵ H_r/b−1¹ ₂

H_r/b−1²

3b r⁴

H_r/b−1¹ H_r/b−1² H_r/b−1³ 1

2r³ H_r/b−1² ₂

2H_r/b−1¹ H_r/b−1³ 3H_r/b−1⁴

,

n≥1

Hn¹

n³bnr² 5ζ4

4r² − 5bζ3

r³ 3b²ζ2

r⁴ −bζ2H_r/b−1¹ r³

3b²

2r⁴ H_r/b−1¹ ₂

H_r/b−1²

b r³

H_r/b−1¹ H_r/b−1² H_r/b−1³ ,

n≥1

Hn¹

n³bnr 5ζ4

4r b²ζ2

r³ −2bζ3 r² b²

2r³ H_r/b−1¹ ₂

H_r/b−1²

.

2.32 Now we can substitute2.32into2.31, collecting zeta functions and using2.12and2.13 forXRkandY Rk, respectively, the identity2.30is obtained.

Some specific examples ofCorollary 2.7are as follows.

Fork1 andb1 the following identity is valid,

n≥1

Hn¹

n³n1³ ζ4−9ζ3 1 6ζ2,

n≥1

Hn¹

n³_4n8

8

₃ 12729073000

27 − 2641017400832

11025 ln 2440380018688 3675 ln 2²

−16604790784

315 G11272192G²1576747008 35 Gln 2

−1390679293952π

33075 −172233728π³

105 − 203992775989

2450 ζ2

−4069970159

210 ζ3 28974848 ln 2ζ3−62246737

2 ζ4,

2.33

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whereGis Catalan’s constant, defined by

G 1 2

₁

0

Ksds^∞

r1

−1^r

2r1² ≈0.915965. . . , 2.34

andKsis the complete elliptic integral of the first kind. The degenerate casek 0, gives the well-known result

n≥1

Hn¹

n³ 5

4ζ4. 2.35

Remark 2.8. Corollaries2.3,2.5, and2.7are important and can be evaluated as demonstrated independently of their integral representations. Similarly the proofs of Corollaries 2.3,2.5, and2.7are not obvious therefore their explicit representations is desired.

Remark 2.9. Theoretically it should be possible to obtain an integral representation for the general sum

n≥1

j1an

−ψ j1 n^q ^anj_j _bnk

k

_cnl

l

_dnm

m

, forq1,2,3, . . . , 2.36

with its associated corollaries. This work will be investigated in a forthcoming paper.

References

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2 K. N. Boyadzhiev, “Harmonic number identities via Euler’s transform,” Journal of Integer Sequences, vol. 12, no. 6, article 09.6.1, p. 8, 2009.

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8 T. Mansour, “Combinatorial identities and inverse binomial coeﬃcients,” Advances in Applied Mathematics, vol. 28, no. 2, pp. 196–202, 2002.

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11 A. Sofo, “Sums of derivatives of binomial coeﬃcients,” Advances in Applied Mathematics, vol. 42, no. 1, pp. 123–134, 2009.

12 A. Sofo, “Harmonic numbers and double binomial coeﬃcients,” Integral Transforms and Special Functions, vol. 20, no. 11-12, pp. 847–857, 2009.

13 A. Sofo, “Harmonic sums and integral representations,” Journal of Applied Analysis, vol. 16, no. 2, pp.

265–277, 2010.

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14 K. S. K ¨olbig, “The polygamma function and the derivatives of the cotangent function for rational arguments,” CERN-IT-Reports CERN-CN 96-005, 1996.

15 K. S. K ¨olbig, “The polygamma functionψxforx1/4 andx3/4,” Journal of Computational and Applied Mathematics, vol. 75, no. 1, pp. 43–46, 1996.

16 J. Choi and D. Cvijovi´c, “Values of the polygamma functions at rational arguments,” Journal of Physics.

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http://www.hindawi.com Volume 2014 Hindawi Publishing Corporation

Harmonic Numbers and Cubed Binominal Coefficients | VU Research Repository | Victoria University | Melbourne Australia