Integral Transforms and Special Functions

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A certain class of series associated with the zeta function

J. Choi a; H. M. Srivastava b a Department of Mathematics, College of Natural Sciences, Dongguk University, Kyongju, Korea b Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia, Canada Online Publication Date: 01 November 2001 To cite this Article: Choi, J. and Srivastava, H. M. (2001) 'A certain class of series associated with the zeta function', Integral Transforms and Special Functions, 12:3, 237 - 250 To link to this article: DOI: 10.1080/10652460108819348 URL: http://dx.doi.org/10.1080/10652460108819348

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A CERTAIN CLASS OF SERIES ASSOCIATED WITH THE ZETA FUNCTION J. CHOP and H. M. SRIVASTAVA2

'Department of Mathematics, College of Natural Sciences, Dongguk University, Kyongju 780-714, Korea 2 ~ e p a r t m etnof Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3P4, Canada

(Receaved January 20, 2001 ) The authors apply the theory of the double Gamma function, which was recently revived in the study of the determinants of the Laplacians, in order to give closed-form evaluations of some series involving the Zeta function. Each of these series is evaluated at different arguments from those considered in several earlier works. The possibility of numerical computations of the sums of the series considered here is also indicated.

KEY WORDS: multiple Gamma functions, Riemann's C-function, determinants of the Laplacians, Hurwitz (or generalized) Zeta function, Glaisher-Kinkelin constant, polygamma functions MSC (2000): 33B99, llM06, l l M 9 9

1.

INTRODUCTION AND PRELIMINARIES

The multiple Gamma functions were defined and studied by Barnes (cf. [2], [3], [4], and [ 5 ] ) and others in about 1900. Although these functions did not appear in the tables of the most well-known special functions, yet the double Gamma function was cited in the exercises by Whittaker and Watson [21, p. 2641 and recorded also by Gradshteyn and Ryzhik [14, p. 661, Entry 6.441(4); p. 937, Entry 8.3331. Recently, these functions were revived in the study of the determinants of the Laplacians on the n-dimensional unit sphere Sn (see [6], [9], [lo], [16], [19], and [20]). More recently, Choi et al. ([7],[8], [lo], and [ll]) used these functions to evaluate the sums of several classes of series involving the Riemann Zeta function,

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J. CHOI A N D H.M.SRIVASTAVA

238

the subject of which can be traced back to an over two-century old theorem of Christian Goldbach (1690-1764) noted in the work of Srivastava [18, p. 11 who investigated this subject in a rather systematic and unified manner. Before their investigation by Barnes, the multiple Gamma functions had been introduced and used in a different form by several authors. In this note we aim at presenting closed-form evaluations of a significantly large family of series involving the Zeta function. Each of these evaluations is made here a t markedly different arguments from those considered in earlier works referred to above. We begin by recalling the Barnes G-function (1/G = rz is the so-called double Gamma function) which has several equivalent forms including the one given below: 1 1r.f~ 1))-1 = G(Z 1) := ( 2 r ) z / 2exp [-1 [(I 7) z2 z]]

+

+

+

+

where y denotes the Euler-Mascheroni constant given by y := lim 11-00

(25

- log n )

0.577 215 664 901 532 860 606 512.. . .

(2)

k=l

For sufficiently large real x and a E C , we have the following Stirling formula for the G-function: 1 3x2 ax a log(27r) - log A - - - log G(x a 1) = 2 12 4

+ +

+

+

logx+O(x-')

(

x

)

(3)

where A is the Glaisher-Kinkelin constant defined by

The C-function satisfies the following fundamental functional relationships: G(l) = 1

and

G(z

+ 1) = r ( z ) G(z),

(5)

where denotes the familiar Gamma function which also satisfies the basic relationships: and r ( z 1) = z r ( z ) . (6) r(1) = 1 The Riemann Zeta function [(s) is defined by

+

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SERIES ASSOCIATED WITH THE ZETA FUNCTION

239

which can, except for a simple pole at s = 1 with its residue 1, be continued analytically to the whole complex s-plane by means of a familiar contour integral representation (cf. Whittaker and Watson [21, p. 2661) or many other known integral representations (cf. Erdblyi et al. [12, p. 331). The Hurwitz (or generalized) Zeta function ( ( 8 , a) is defined by

( ( ) > l

aEC\Zc;

Z c : = Z - ~ ( 0 ) ; Z-:={-1,-2,

-3 ,...}),

which can, just as ((s), be continued analytically to the whole complex s-plane except for a simple pole at s = 1 (with its residue 1). Clearly, we have

The Catalan constant G (which is denoted also by A) is defined by

.

The Polygamma functions $ ~ ( ~ ) ((zn)E N := (1, 2, 3, . .}) are defined by d"+l dn $(n)(z) := -iogr(z) = -$(z) E N O := N u {o)), (11) dzn dzn+l where $(')(z) := $(z) denotes the Psi (or Digamma) function defined by

In terms of the Hurwitz Zeta function ((s, a) defined by (8), we can write $'"'(z) = (-1)"" 2.

n! [(n

+ 1,z)

( n E M).

(13)

RELATIONSHIPS AMONG MATHEMATICAL CONSTANTS

The following definite integral was evaluated by Gosper [13, p. 711:

Subsequently (but independently), in a markedly different way, Choi and Srivastava [8, p. 100, Equation (2.29)] showed that

By comparing the integral formulas (14) and (15), we obtain

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J. CHOI A N D H.M. SRIVASTAVA

('(2) = n 2

Y + - log ( 2 ~ -) 2 log

(6

If we apply the relationship (cf. Voros [20, p. 462, Equation (A.11)]): 1 l o g A = - -(I(-1) 12 in (16), we find that 1 C'P) = - 11 - 7 log (2i~)] 12 2iT2 ' which can also be obtained by appealing appropriately to Riemann's functional equation for ((s) (cf. Whittaker and Watson [21, p. 2751). The relationship (18) is recorded erroneously in the aforecited work of Voros [20, p. 453, Equation (6.25)]. Gosper [13, p. 711 also evaluated another definite integral which is rewritten here, by using (13) and (16), in the form:

+-

-

('(-1)

Now we recall Alexeiewsky's theorem (cf. Choi and Srivastava [7, p. 107, Equation (2.7)]):

0

which, upon setting z = 113, and using (5) and (6), yields

I t readily follows from (19) and (21) that G (113) can be expressed in terms of more recognizable and elementary constants:

If we use the following well-known reflection and duplication formulas for the Gamma function: r(i

- Z) = iT

sin ( i ~ z )

(Z $2

z := {o, &I, 5 2 , . . .I)

and J;;r(2~) we obtain

= 2"-'

r ( ~r)

(23)

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SERIES ASSOCIATED WITH THE ZETA FUNCTION

241

Gosper [13, p. 711 evaluated yet another definite integral which is rewritten here, by using (13) and (16), in the form:

The special case of (20) when z = 116 yields

where we have made use of (5) and (6). Combining (26) and (27) with (25), we obtain

Next, by setting z = 116 and z = 113 in the following duplication formula for the G-function (cf., e.g., Choi and Srivastava [8, p. 94, Equation (1.16)]):

and applying (25), we have

and

It follows from (22), (28), and (30) that

If we substitute from (22) and (32) into (31), we get

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J. CHOI A N D H.M.SRIVASTAVA

It is interesting to remark in passing that (cf. Choi and Srivastava [8, p. 941; Gosper [13, pp. 71-72])

are evaluated in terms of several mathematical constants including the Catalan constant G defined by (10). 3.

SERIES INVOLVING THE ZETA FUNCTION

Choi and Srivastava [8] presented many integral identities and closed-form evaluations of series involving the Hurwitz and Riemann Zeta functions by making use of the theory of the double Gamma function r2 = 1/G. In view of the relationships exhibited by (9), by setting a = 1 and a = 2 in some of their closed-form evaluations [8, pp. 106-107, Equations (4.8) to (4.11)], we obtain w

(-I)*

(0zkrl = [I - log(2n)l; k+l

z2 + (1 + 7)T + log G ( z + 1)

(34) (14 < I), which can also be deduced directly from (1) (cf. Choi and Srivastava [7, p. 106, Equation (2.1)]); k=2

which follows also from (34) in view of the elementary identity: 00

z2 Co *zk+l = log ( I + z) - z + k+l 2

(121 < 1);

k=2

03

z zktl = (1 - log(2n)l2

z2 - (1 + 7)- log G ( l - z)

2 k=2 which follows also upon replacing z by -z in (34);

which follows also upon replacing z by -z in (35);

( z