A certain family of series associated with the Zeta and related functions

A certain family of series associated with the Zeta and related functions

Junesang Choi and H. M. Srivastava

(Received May 1, 2001) (Revised May 29, 2002)

Abstract. The history of problems of evaluation of series associated with the Riemann Zeta function can be traced back to Christian Goldbach (1690–1764) and Leonhard Euler (1707–1783). Many di¤erent techniques to evaluate various series involving the Zeta and related functions have since then been developed. The authors show how elegantly certain families of series involving the Zeta function can be evaluated by starting with a single known identity for the generalized (or Hurwitz) Zeta function. Some special cases and their connections with already developed series involving the Zeta and related functions are also considered.

1. Introduction, definitions, and the main result The Riemann Zeta function zðsÞ defined by

zðsÞ:¼ X^y

n¼1

1 n^s¼ 1

12^s X^y

n¼1

1

ð2n1Þ^s ðRðsÞ>1Þ ð12^1sÞ¹X^y

n¼1

ð1Þ^nþ1

n^s ðRðsÞ>0; s01Þ 8>

>>

><

>>

>>

: ð1:1Þ

satisfies the functional equation (see [24, p. 269]):

zðsÞ ¼2^sp^s1Gð1sÞzð1sÞsin ps ð1:2Þ 2

and takes on the following special or limit values (see [24, p. 271]):

zð1Þ ¼ 1

12; zð0Þ ¼ 1

2; and z⁰ð0Þ ¼ 1

2 logð2pÞ;

ð1:3Þ

2000 Mathematics Subject Classification. Primary 11M06, 11M35, 33B15; Secondary 11B68, 11B73, 33E20.

Key words and phrases. Multiple Gamma functions, Riemann Zeta function, generalized (or Hurwitz) Zeta function, Psi (or Digamma) function, determinants of Laplacians, generating functions, Bernoulli polynomials and numbers, Stirling numbers, harmonic numbers.

and

lims!1 zðsÞ 1 s1

¼g ð1:4Þ

in terms of the Euler-Mascheroni constant g given by g:¼ lim

n!y

Xⁿ

k¼1

1 klogn

!

G0:577 215 664 901 532 860 606 512. . .: The generalized (or Hurwitz) Zeta function zðs;aÞ is defined by

zðs;aÞ:¼X^y

n¼0

1

ðnþaÞ^s ðRðsÞ>1; a00;1;2;. . .Þ;

ð1:5Þ

which, just as zðsÞ, can be continued meromorphically everywhere in the complex s-plane except for a simple pole at s¼1 (with residue 1). It is not di‰cult to see from the definitions (1.1) and (1.5) that

zðs;aþnÞ ¼zðs;aÞ Xⁿ¹

k¼0

1

ðkþaÞ^s ðnAN:¼ f1;2;3;. . .gÞ ð1:6Þ

and

zðs;1Þ ¼zðsÞ ¼ ð2^s1Þ¹z s;1 2

ð1:7Þ :

The subject of evaluations of series involving the Zeta and related functions has a long history which can be traced back to Christian Goldbach (1690–1764) and Leonhard Euler (1707–1783) (see, for details, [19] and [20]).

Many di¤erent techniques to evaluate various families of series involving the Zeta and related functions have since then been developed (cf., e.g., [2], [5], [8]

to [12], [13], [19], and [20]). The main object of this paper is to show how nicely certain families of series involving the Zeta and related functions can be evaluated by starting with the following known identity for zðs;aÞ [19, p. 18, Eq. (6.13)]:

X^y

k¼0

ðsÞ_k

k! zðsþk;aÞt^k ¼zðs;atÞ ðjtj<jajÞ:

ð1:8Þ

where ðlÞ_n denotes the Pochhammer symbol defined, in terms of the familiar Gamma function, by

ðlÞ_n:¼GðlþnÞ

GðlÞ ¼ 1 ðn¼0; l00Þ

lðlþ1Þ. . .ðlþn1Þ ðnAN; lACÞ:

ð1:9Þ

The general series identity, which is to be proved in this paper, is con- tained in the following

Theorem. For every nonnegative integer n, X^y

k¼2

zðk;aÞ

ðkÞ_nþ1t^nþk¼ð1Þⁿ

n! ½z⁰ðn;atÞ z⁰ðn;aÞ ð1:10Þ

þXⁿ

k¼1

ð1Þ^nþk n!

n

k ½ðHnHnkÞzðkn;aÞ z⁰ðkn;aÞt^k þ ½HnþcðaÞ t^nþ1

ðnþ1Þ! ðjtj<jaj; nAN₀Þ;

where Hn denotes the harmonic numbers defined by

H_n:¼Xⁿ

j¼1

1 ð1:11Þ j

and it is understood (as elsewhere in this paper) that an empty sum is nil.

For the sake of ready reference, we recall here the following identities and relationships which will be required in our proof of the above Theorem.

First of all, there exists a relationship between the generalized Zeta functionzðs;aÞand the Bernoulli polynomialsBnðaÞin the form (see [3, p. 264, Theorem 12.13]):

zðn;aÞ ¼ Bnþ1ðaÞ

nþ1 ðnAN₀:¼NUf0gÞ;

ð1:12Þ

which can be applied in order to evaluate zðn;aÞfor special values of nAN₀. The following identity involving the Bernoulli polynomials:

Bnþ1ðaþtÞ nþ1 ¼Xⁿ

k¼0

n k

Bkþ1ðaÞ

kþ1 t^nkþ t^nþ1

nþ1 ðnAN₀Þ ð1:13Þ

results from the known formula (see [3, p. 275]):

B_nðaþtÞ ¼Xⁿ

k¼0

n

k B_kðaÞt^nk ðnAN₀Þ ð1:14Þ

when we replace n in (1.14) by nþ1 and divide both sides of the resulting equation by nþ1.

We also recall here the following Laurent series expansion of zðs;aÞ at s¼1 (see [24, p. 271]):

zðs;aÞ ¼ 1

s1cðaÞ þX^y

n¼1

c_nðs1Þⁿ; ð1:15Þ

where the coe‰cients c_n are constants to be determined and the Psi (or Digamma) function cðzÞ defined by

cðzÞ:¼G⁰ðzÞ

GðzÞ or logGðzÞ ¼ ðz

1

cðtÞdt ð1:16Þ

is meromorphic in the complex z-plane with simple poles at z¼0;1;2;. . . (with residue 1). In fact, we have (see, e.g., [21, pp. 24–25]):

cðzþnÞ ¼cðzÞ þXⁿ

j¼1

1

zþj1 ðnANÞ ð1:17Þ

and

d

dzfðzÞ_ng ¼ ðzÞ_n½cðzþnÞ cðzÞ;

ð1:18Þ

which follows easily from the definitions (1.9) and (1.16).

2. Proof of the Theorem

Upon transposing the first nþ2 terms from k¼0 to k¼nþ1 in (1.8) to the right-hand side, if we divide both sides of the resulting equation by sþn, we get

X^y

k¼nþ2

ðsÞ_nðsþnþ1Þ_kn1zðsþk;aÞt^k

k!¼gnðs;t;aÞ

sþn ðjtj<jaj; nAN₀Þ;

ð2:1Þ

where, for convenience,

g_nðs;t;aÞ:¼zðs;atÞ X^nþ1

k¼0

ðsÞ_k

k! zðsþk;aÞt^k: ð2:2Þ

Now we shall show that

s!nlim g_nðs;t;aÞ ¼0:

ð2:3Þ

Since zðsþnþ1;aÞhas a simple pole at s¼ n with its residue 1, we find that

s!nlimðsþnÞzðsþnþ1;aÞ ¼1:

ð2:4Þ

By rewriting (2.2) in the form:

gnðs;t;aÞ ¼zðs;atÞ Xⁿ

k¼0

ðsÞ_k

k! zðsþk;aÞt^k ðsÞ_nðsþnÞzðsþnþ1;aÞ t^nþ1 ðnþ1Þ!; if we take the limit as s! n with the aid of (2.4) and make use of the elementary identity:

l

n ¼ð1ÞⁿðlÞ_n

n! ðnAN₀; lACÞ;

we obtain

s!nlim g_nðs;t;aÞ ¼zðn;atÞ Xⁿ

k¼0

ð1Þ^nk n

k zðk;aÞt^nkþð1Þ^nþ1 nþ1 t^nþ1; which, in view of the relationship (1.12), can be put in its equivalent form:

s!nlim gnðs;t;aÞ ¼ Bnþ1ðatÞ nþ1 þXⁿ

k¼0

ð1Þ^nk n k

Bkþ1ðaÞ

kþ1 t^nkþð1Þ^nþ1 nþ1 t^nþ1; from which our assertion (2.3) follows easily by applying (1.13). Thus, by l’Hoˆpital’s rule, we have

s!nlim

gnðs;t;aÞ sþn ¼ lim

s!n

q

qsfgnðs;t;aÞg:

ð2:5Þ

Next, by appealing to (2.2) and (1.18), we observe that q

qsfgnðs;t;aÞg ¼z⁰ðs;atÞ z⁰ðs;aÞ ð2:6Þ

Xⁿ

k¼1

hðs;a;kÞt^k

k!hðs;a;nþ1Þ t^nþ1 ðnþ1Þ!; where, for convenience,

hðs;a;kÞ:¼ ðsÞ_k½fcðsþkÞ cðsÞgzðsþk;aÞ þz⁰ðsþk;aÞ:

ð2:7Þ

In view of (1.17), (2.7) yields

s!nlim hðs;a;kÞ ¼ ðnÞ_k½ðHnHnkÞzðkn;aÞ z⁰ðkn;aÞ ð2:8Þ

ðk¼1;. . .;nÞ

and

s!nlim hðs;a;nþ1Þ ð2:9Þ

¼ lim

s!n

Xⁿ

j¼0

ðsÞ_n sþj

!

ðsþnÞzðsþnþ1;aÞ þ ðsÞ_nþ1z⁰ðsþnþ1;aÞ

" #

;

which, upon writing Xⁿ

j¼0

ðsÞ_n sþj¼Xⁿ¹

j¼0

ðsÞ_n sþjþ ðsÞ_n

sþn; reduces to the form:

s!nlim hðs;a;nþ1Þ ¼ Xⁿ¹

j¼0

ðnÞ_n nj þ lim

s!n fðs;a;nÞ;

ð2:10Þ

where, for convenience,

fðs;a;nÞ:¼ ðsÞ_nzðsþnþ1;aÞ þ ðsÞ_nþ1z⁰ðsþnþ1;aÞ: ð2:11Þ

Now, by virtue of (1.15), we readily have

s!nlim fðs;a;nÞ ¼ lim

s!nðsÞ_n cðaÞ þX^y

j¼1

ðjþ1ÞcjðsþnÞ^j

" #

ð2:12Þ

¼ ðnÞ_ncðaÞ

¼ ð1Þ^nþ1n!cðaÞ: It follows from (2.10) and (2.12) that

s!nlim hðs;a;nþ1Þ ¼ ð1Þ^nþ1n!½HnþcðaÞ;

ð2:13Þ

where Hn denotes the harmonic numbers defined by (1.11).

Making use of (2.8) and (2.13), we find from (2.6) that

s!nlim q

qsfgnðs;t;aÞg ¼z⁰ðn;atÞ z⁰ðn;aÞ ð2:14Þ

þXⁿ

k¼1

ð1Þ^k n

k ½ðHnHnkÞzðkn;aÞ z⁰ðkn;aÞt^k þð1Þⁿ

nþ1 ½HnþcðaÞt^nþ1:

Finally, since

s!nlim X^y

k¼nþ2

ðsÞ_nðsþnþ1Þ_kn1zðsþk;aÞt^k

k!¼ ð1Þⁿn!X^y

k¼2

zðk;aÞ ðkÞ_nþ1t^kþn; ð2:15Þ

by equating the second members of (2.14) and (2.15), we are led immediately to the desired series identity (1.10). This evidently completes our proof of the Theorem.

Infinite sums of the type occurring in (1.10) can also be evaluated, in a markedly di¤erent manner, in terms of such higher transcendental functions as the multiple Gamma functions (see, for details, [16, p. 10, Theorem 1]).

3. Applications of the Theorem

Upon setting n¼0;1;2;3, and 4 in (1.10), if we make use of the appropriate identities which are readily available in the mathematical litera- ture, we shall obtain the following known or new formulas for closed-form evaluations of several families of series involving the generalized (or Hurwitz) Zeta function zðs;aÞ:

X^y

k¼2

zðk;aÞ

k t^k¼logGðatÞ logGðaÞ þtcðaÞ ðjtj<jajÞ;

ð3:1Þ

which is given (for example) in [24, p. 276], [15, p. 358, Entry (54.11.1)], and [8, p. 107, Eq. (2.11)];

X^y

k¼2

zðk;aÞ

kðkþ1Þt^kþ1 ¼z⁰ð1;aÞ z⁰ð1;atÞ ð3:2Þ

þ 1

2alogGðaÞ þ1

2 logð2pÞ

t

þ ½1þcðaÞt²

2 ðjtj<jajÞ; X^y

k¼2

zðk;aÞ

kðkþ1Þðkþ2Þt^kþ2 ð3:3Þ

¼1

2½z⁰ð2;atÞ z⁰ð2;aÞ þ 1

4 a²aþ1 6

þz⁰ð1;aÞ

t

þ 3

23aþlogð2pÞ 2 logGðaÞ

t²

2 þ ½3þ2cðaÞt³

12 ðjtj<jajÞ;

X^y

k¼2

zðk;aÞ

kðkþ1Þðkþ2Þðkþ3Þt^kþ3 ð3:4Þ

¼1

6½z⁰ð3;aÞ z⁰ð3;atÞ 1

3 a³3 2a²þ1

2a

þ3z⁰ð2;aÞ

t

6 þ 5

2 a²aþ1 6

þ6z⁰ð1;aÞ

t² 12 þ 11

2 11aþ3 logð2pÞ 6 logGðaÞ

t³ 36 þ ½11þ6cðaÞ t⁴

144 ðjtj<jajÞ;

X^y

k¼2

zðk;aÞ

kðkþ1Þðkþ2Þðkþ3Þðkþ4Þt^kþ4 ð3:5Þ

¼ 1

24½z⁰ð4;atÞ z⁰ð4;aÞ þ 1

4 a⁴2a³þa² 1 30

þ4z⁰ð3;aÞ

t 24 7

3 a³3 2a²þ1

2a

þ12z⁰ð2;aÞ

t² 48 þ 13 a²aþ1

6

þ24z⁰ð1;aÞ

t³

144 þ½2550aþ12 logð2pÞ 24 logGðaÞ t⁴

576 þ ½25þ12cðaÞ t⁵

1440 ðjtj<jajÞ:

Setting a¼1 in (3.5), we readily obtain X^y

k¼2

zðkÞ

kðkþ1Þðkþ2Þðkþ3Þðkþ4Þt^kþ4 ð3:6Þ

¼ 1

24 z⁰ð4;1tÞ 3zð5Þ 4p⁴

5

72þ4 logC

t

24þzð3Þ 16p²t² þ 25

6 24 logA t³

144þ ½12 logð2pÞ 25 t⁴ 576 þ ð2512gÞ t⁵

1440 ðjtj<1Þ;

which, for t¼¹₂, yields X^y

k¼2

zðkÞ

kðkþ1Þðkþ2Þðkþ3Þðkþ4Þ2^k ð3:7Þ

¼ 1

48 logð2pÞ g 2401

3 logA4

3 logCþzð3Þ

4p² 31zð5Þ 32p⁴ ;

where we have made use of such results as (for example) the relationship (1.7) and the derivative formula (cf., e.g., [20, p. 387, Eq. (1.15)]):

z⁰ð2nÞ ¼ ð1Þⁿ ð2nÞ!

2ð2pÞ²ⁿzð2nþ1Þ ðnANÞ;

which follows easily from

zð2nÞ ¼0 ðnANÞ

and Riemann’s functional equation (1.2). Here, and elsewhere in this paper, A denotes the Glaisher-Kinkelin constant defined by

logA¼ lim

n!y

Xⁿ

k¼1

klogk 1 2n²þ1

2nþ 1 12

lognþ1 4n²

( )

;

the numerical value of A being given by

AG1:282427130. . .; and C is a mathematical constant defined by

logC¼ lim

n!y

Xⁿ

k¼1

k³logk 1 4n⁴þ1

2n³þ1 4n² 1

120

lognþ 1 16n⁴ 1

12n²

" #

;

the numerical value of C being given by

CG301 393 392 412 467 8410^2714341:

Some of these and other mathematical constants have already occurred in the recent revival of the multiple Gamma functions [4] in the study of the determinants of the Laplacians on the n-dimensional unit sphere Sⁿ (see [6], [18], [22], and [23]).

Similarly, we obtain X^y

k¼2

zðkÞ

kðkþ1Þðkþ2Þðkþ3Þ2^k ð3:8Þ

¼ 59

720 log 2þ 1

12 logp g

48logA5

2 logCþzð3Þ 2p²; X^y

k¼2

zðkÞ kðkþ1Þðkþ2Þ2^k ð3:9Þ

¼ 3 8þ1

2 logð2pÞ g

122 logAþ7zð3Þ 8p² ; X^y

k¼2

zðkÞ

kðkþ1Þ2^k ¼ g 4þ 7

12 log 2þ1

2 logp3 logA;

ð3:10Þ

which is recorded in [8, p. 109, Eq. (2.23)];

X^y

k¼2

zðkÞ k2^k ¼ g

2þ1 2 logp;

ð3:11Þ

which is also recorded in [8, p. 109, Eq. (2.21)].

We remark in passing that the various series identities presented here are potentially useful in deriving further identities for series involving the Zeta and related functions. For example, if we replace t in (3.2) by t, we get

X^y

k¼2

ð1Þ^k zðk;aÞ kðkþ1Þt^kþ1 ð3:12Þ

¼z⁰ð1;atÞ z⁰ð1;aÞ þ 1

2alogGðaÞ þ1

2 logð2pÞ

t

½1þcðaÞt²

2 ðjtj<jajÞ:

And since [7, p. 164, Eq. (2.8)]

G2ðaÞ ¼A ð2pÞð1=2Þð1=2Þa

exp 1

12þz⁰ð1;aÞ þ ð1aÞz⁰ð0;aÞ

ð3:13Þ ;

by making use of the following consequence of Hermite’s formula for zðs;aÞ (see [24, pp. 270–271]):

z⁰ð0;aÞ ¼ q

qsfzðs;aÞgj_s¼0¼logGðaÞ 1

2 logð2pÞ;

the series identity (3.12) can be written as follows in an equivalent form involving the double Gamma function G₂:

X^y

k¼2

ð1Þ^k zðk;aÞ

kðkþ1Þt^kþ1¼ ð1aÞlogGðaÞ logG2ðaÞ ð3:14Þ

þ ðtþa1ÞlogGðaþtÞ þlogG2ðaþtÞ þ 1

2aþ1

2 logð2pÞ logGðaÞ

t

½cðaÞ þ1t²

2 ðjtj<jajÞ;

which was proven, in a markedly di¤erent way, by Choi and Srivastava [8, p. 108, Eq. (2.14)].

Finally, we deduce yet another interesting identity by suitably combining the special cases of (1.10) whena¼1 anda¼2. By applying (1.7), (1.8), and the familiar relationship:

cðnÞ ¼ gþXⁿ¹

k¼1

1

k ðnANÞ;

we thus find that X^y

k¼1

t^nþk

ðkÞ_nþ1¼ð1Þ^nþ1

n! ð1tÞⁿlogð1tÞ ð3:15Þ

þXⁿ

k¼1

ð1Þ^nþk n!

n

k ðHnHnkÞt^k ðjtj<1; nAN₀Þ;

which, in the special case when n¼2, is a known result recorded (for example) by Hansen [15, p. 37, Entry (5.7.40); p. 74, Entry (5.16.26)].

Acknowledgements

The present investigation was carried out during the first-named author’s visit to the University of Victoria from August 1999 to August 2000 while he was on Study Leave from Dongguk University at Kyongju. For the first- named author, this work was supported by the Research Funds of Dongguk

University. The second-named author was supported, in part, by the Nat- ural Sciences and Engineering Research Council of Canada under Grant OGP0007353.

References

[ 1 ] M. Abramowitz and I. A. Stegun (Editors), Handbook of Mathematical Functions, Dover Publications, New York, 1965.

[ 2 ] T. M. Apostol, Some series involving the Riemann Zeta function, Proc. Amer. Math. Soc.

5 (1954), 239–243.

[ 3 ] T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, New York, Heidelberg, and Berlin, 1976.

[ 4 ] E. W. Barnes, The theory of the G-function, Quart. J. Math. 31 (1899), 264–314.

[ 5 ] M.-P. Chen and H. M. Srivastava, Some families of series representations for the Riemann zð3Þ, Resultate Math. 33 (1998), 179–197.

[ 6 ] J. Choi, Determinant of Laplacian on S³, Math. Japon. 40(1994), 155–166.

[ 7 ] J. Choi and T. Y. Seo, The double Gamma function, East Asian Math. J. 13 (1997), 159–174.

[ 8 ] J. Choi and H. M. Srivastava, Sums associated with the Zeta function, J. Math. Anal.

Appl. 206 (1997), 103–120.

[ 9 ] J. Choi and H. M. Srivastava, Certain classes of series involving the Zeta function, J.

Math. Anal. Appl. 231 (1999), 91–117.

[10] J. Choi and H. M. Srivastava, An application of the theory of the double Gamma function, Kyushu J. Math. 53 (1999), 209–222.

[11] J. Choi and H. M. Srivastava, Certain classes of infinite series, Monatsh. Math. 127 (1999), 15–25.

[12] J. Choi and H. M. Srivastava, Certain classes of series associated with the Zeta function and multiple Gamma functions, J. Comput. Appl. Math. 118 (2000), 87–109.

[13] J. Choi, H. M. Srivastava, and J. R. Quine, Some series involving the Zeta function, Bull.

Austral. Math. Soc. 51 (1995), 383–393.

[14] L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions (Translated from the French by J. W. Nienhuys), Reidel, Dordrecht and Boston, 1974.

[15] E. R. Hansen, A Table of Series and Products, Prentice-Hall, Englewood Cli¤s, New Jersey, 1975.

[16] S. Kanemitsu, H. Kumagai, and M. Yoshimoto, Sums involving the Hurwitz Zeta func- tion, Ramanujan J.5 (2001), 5–19.

[17] G. Po´lya and G. Szego¨, Problems and Theorems in Analysis (Translated from the German by D. Aeppli), Vol. I, Springer-Verlag, New York, Heidelberg, and Berlin, 1972.

[18] J. R. Quine and J. Choi, Zeta regularized products and functional determinants on spheres, Rocky Mountain J. Math. 26 (1996), 719–729.

[19] H. M. Srivastava, A unified presentation of certain classes of series of the Riemann Zeta function, Riv. Mat. Univ. Parma (Ser. 4) 14 (1988), 1–23.

[20] H. M. Srivastava, Some rapidly converging series for zð2nþ1Þ, Proc. Amer. Math. Soc.

127 (1999), 385–396.

[21] H. M. Srivastava and H. L. Manocha, A Treatise on Generating Functions, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane, and Toronto, 1984.

[22] I. Vardi, Determinants of Laplacians and multiple Gamma functions, SIAM J. Math.

Anal. 19 (1988), 493–507.

[23] A. Voros, Special functions, spectral functions and the Selberg Zeta function, Comm.

Math. Phys. 110 (1987), 439–465.

[24] E. T. Whittaker and G. N. Watson, A Course of Modern Analysis: An Introduction to the General Thoery of Infinite Processes and of Analytic Functions; With an Account of the Principal Transcendental Functions, Fourth Ed., Cambridge University Press, Cambridge, London, and New York, 1963.

Junesang Choi Department of Mathematics College of Natural Sciences

Dongguk University Kyongju 780-714

Korea

E-Mail: junesang@mail.dongguk.ac.kr H. M. Srivastava

Department of Mathematics and Statistics University of Victoria

Victoria, British Columbia V8W 3P4 Canada

E-Mail: harimsri@math.uvic.ca