## Abstract

The Stieltjes constants have been of interest for over a century, yet their detailed behaviour remains under investigation. These constants appear in the Laurent expansion of the Hurwitz zeta function about . We obtain novel single and double summatory relations for , including single summation relations for and , where *a* and *b* are real and *p* and *q* are positive integers. In addition, we obtain new integration formulae for the Hurwitz zeta function and a new expression for the Stieltjes constants . Portions of the presentation show an intertwining of the theory of the hypergeometric function with that of the Hurwitz zeta function.

## 1. Introduction

The Stieltjes (or generalized Euler) constants appear as expansion coefficients in the Laurent series for the Hurwitz zeta function about its simple pole at *s*=1 (Stieltjes 1905; Hardy 1912; Kluyver 1927; Briggs 1955; Mitrović 1962; Israilov 1979). In an earlier paper (Coffey 2006), we derived new relations among the values , and we demonstrated one of the very recent conjectures put forward by Kreminski (2003) on the relationship between and as . The constants are very important in relation to the derivatives of the Riemann xi function at *s*=1, where is the Gamma function and is the Riemann zeta function (Riemann 1859–1860; Edwards 1974; Ivić 1985; Karatsuba & Voronin 1992). For the Stieltjes constants, there are relatively few formulae known for calculating them to high precision and little is known as to their arithmetical properties.

In this paper, we first present a summatory relation for . Our starting point is an old formula of Wilton (1927). Even special cases of the resulting expansions seem to provide interesting summations. We then present novel integration relations for the Hurwitz zeta function. All of these equations would also imply additional integral–summatory relations for the Stieltjes constants. Here, special cases include integrals of the Hurwitz zeta function with polynomials. Again, as for the first set of summatory results, the theory of the hypergeometric function is combined with that of the Hurwitz zeta function. We then proceed to obtain a new summatory relation for , where is rational and *b* is real. We present a new general expression for the Stieltjes constants . We mention that an advantage of this form is an improved rate of convergence, compared to some apparently simpler expressions. Therefore, it may find applicability in numerical computations. Before concluding the paper with a summary, we demonstrate a double summation relation for the Stieltjes constants with many useful special cases.

The Hurwitz zeta function, initially defined by for , has an analytic continuation to the whole complex plane (Berndt 1972; Titchmarsh 1986; Karatsuba & Voronin 1992; Laurinčikas & Garunkštis 2002). In the case of , reduces to the Riemann zeta function . In this instance, by convention, the Stieltjes constants are simply denoted as (Hardy 1912; Kluyver 1927; Briggs 1955; Mitrović 1962; Israilov 1979; Kreminski 2003).

## 2. Wilton formula

We first recall a formula of Wilton for the Hurwitz zeta function, and mention an alternative proof. In modified notation from Wilton (1927), we have(2.1)where is the Gamma function. Wilton wrote for , and equation (2.1) was obtained by binomial expansion and rearranging a double series. On the other hand, the integral representation,(2.2)can be used to derive equation (2.1) by forming and then expanding the exponential factor in power series and integrating term by term (Klusch 1992). Initially valid for , equation (2.1) converges absolutely for and then gives an analytic continuation of to the whole complex *s*-plane.

By using the relation (Hansen & Patrick 1962)(2.3)and taking various special cases of equation (2.1) and linear combinations thereof, one may obtain a variety of expressions for the Hurwitz zeta function when the parameter *c* is rational. Although a few of the resulting summatory forms of the Riemann zeta function may be new, they appear to be closely related to those given by Jensen (1887) and Ramaswami (1934) many years ago.

## 3. Summatory relation for **γ**_{k}(a−b)

**γ**

_{k}(a−b)From equation (2.1) it follows that(3.1)The purpose of this section is to uncover the summatory relation for the Stieltjes constants implied by this relation. The defining relation for these constants in terms of a Laurent expansion is(3.2)Applying this definition to equation (3.1), cancelling the polar terms, and using , gives(3.3)The first sum on the right-hand side of equation (3.3) can be performed with the aid of the following.

*For* *and* *, we have*(3.4)

Method 1. We have (Gradshteyn & Ryzhik 1980)(3.5)where is the Pochhammer symbol and _{1}*F*_{0} is a hypergeometric function with a single numerator parameter. By integrating equation (3.5) from 0 to *z*, we obtain(3.6a)This relation together with equation (3.5) implies that(3.6b)proving the lemma. For a second method of proof, we may again start with the generalized binomial expansion of equation (3.5). We then put and use on the left-hand side. ▪

By expanding the result of lemma 3.1 in powers of and inserting in equation (3.3), we have(3.7)

The double series on the right-hand side of equation (3.7) can be written in several forms. We note in passing the following.

*(i) The* *term of the double sum on the right-hand side of equation* *(3.7)* *may be written as*(3.8)*where* *is the digamma function*. *(ii) The sum*(3.9)*where* _{p}*F*_{q} *is the generalized hypergeometric function.*

Part (i) follows from equation (3.5) and the fact that (Wilton 1922–1923; Zhang & Williams 1994). Part (ii) follows by writing and using the series definition of _{p}*F*_{q}.

In the _{k+1}*F*_{k} function on the right-hand side of equation (3.9), the numerator parameter exceeds the denominator parameter by exactly 1. Therefore, massive reduction of this function is possible. For instance, all sums of the form , where *p* is a positive integer, appearing on the left-hand side of equation (3.9) can be evaluated by repeated differentiation with respect to *b* and manipulation of the result of equation (3.5).

Putting *s*=1 in equation (3.7) gives(3.10a)(3.10b)The equality of the right-hand sides of equations (3.10a) and (3.10b) holds only if the interchange of the order of summations is justified. Otherwise, we must consider the relation between the left-hand side of equation (3.10*a*) and the right-hand side of equation (3.10*b*) as an approximation by the partial sums of the latter that is generally divergent. The sum on *j* in equation (3.10*b*) can be reduced to a finite summation with the use of the expansion (Gould 1978)(3.11)where are Stirling numbers of the second kind and . The use of equation (3.11) in equation (3.10*b*) results in(3.12)where is the Kronecker symbol. When evaluated at and , this equation gives(3.13)

Having just derived the lowest-order result (3.10*a*) coming from equation (3.7), we now indicate how the general case may be obtained. Obviously, equation (3.7) states that(3.14)where represents the coefficient of on the right-hand side of that equation. With a binomial expansion and the interchange of two sums, the right-hand side of equation (3.7) can be written as(3.15)The ratio of Gamma functions in (3.15) could be expanded by Taylor series about *s*=1 by evaluating the derivatives at this point in terms of the polygamma function. For instance, we have(3.16)However, an alternative is to appeal to the expansion(3.17)where are Stirling numbers of the first kind (Gould 1978), so that another binomial expansion gives(3.18)The insertion of equation (3.18) into expression (3.15) and the reordering of four sums gives the following.

*With* *as in equation* *(3.14)**, we have*(3.19)

## 4. Integral relations for the Hurwitz zeta function

In this section we demonstrate how many integral relations for the Hurwitz zeta function may be obtained from equation (3.1). In turn, all the resulting summation relations imply corresponding relations for the Stieltjes constants. We may demonstrate the following.

*For* , , , *and* *, we have*(4.1)*where* *is the Beta function and* *is the Gauss hypergeometric function. By additionally imposing* *, convergence on the unit circle* *may be obtained.*

We let in equation (3.1), multiply by and integrate on *t* from 0 to 1. The interchange of integration and summation on the right-hand side is justified by the absolute convergence of the sum. Using a standard integral representation for *F* (Gradshteyn & Ryzhik 1980) then gives the proposition. ▪

There are obviously many special cases of this proposition. When , and the summand on the right-hand side of equation (4.1) contains only Gamma function factors. Such a situation occurs again for , when(4.2)Of course, this case corresponds to and in equation (4.1). For , a further subcase occurs when and are integers and the beta factor in equation (4.1) can be expanded as factorials and in terms of binomial coefficients. As a very special case, we have(4.3)Neither do we have to restrict ourselves to definite integrals. In the same fashion as equation (4.1) is derived, we have for instance(4.4)We have found that proposition 4.1 subsumes the main theorem of Kanemitsu & Kumagai (2001), corresponding to the case .

The idea behind equation (4.1) can be extended to other integrals, including the use of an integral representation of the generalized hypergeometric function. With the introduction of a second summation, equations (4.3) and (4.4) permit the definite and indefinite integration of various polynomials of interest against the Hurwitz zeta function. In this regard on the interval [0,1], applications to the kindred Bernoulli polynomials follow.

As a further example similar to equation (4.1), we have(4.5)

## 5. Summatory relation for *γ*_{k}(b+p/q)

_{k}(b+p/q)

For positive integers *q* and real *b*, we show the following

*(i)*(5.1)*(ii)*(5.2)

Part (ii) follows from part (i) by putting , separating the term of the sum on the right-hand side of equation (5.1), and using the functional equation of the digamma function. For part (i), we start from the relation (Hansen & Patrick 1962)(5.3)We substitute the Laurent expansion (3.2), expand the exponential in powers of *s*−1, and cancel polar terms on both sides of the resulting equation. By interchanging the two sums of the remaining double sum, proposition 5.1(i) follows. Equation (5.1) provides a extension of proposition 3 of Coffey (2006). ▪

The special case of in equation (5.1) or (5.2) reads(5.4)Since , this equation is equivalent to the multiplication formula for the digamma function (e.g. Gradshteyn & Ryzhik 1980); it can be written alternatively as(5.5a)or(5.5b)

## 6. New expression for the Stieltjes constants

We record and prove here the following new expression for the Stieltjes constants.

*We have for integers* (6.1)*where* *are the Bernoulli numbers.*

In place of equation (12*b*) of Coffey (2006), we write(6.2)and instead of equation (12*c*) of that reference, we write(6.3)based upon the generating function for the Bernoulli numbers. Inserting these expressions into equation (11) of Coffey (2006), rearranging sums and separating the polar term using equation (13) of Coffey (2006) gives the expression (6.1). ▪

We have been informed that R. Smith (2005, personal communication) has independently obtained a result very similar to equation (6.1), based upon transformation of an old identity of Kluyver's (1927). In fact, the equivalence is easy to see in the following way. One has (Kluyver 1927), where is the Bernoulli polynomial. Then a result equivalent to equation (6.1) is obtained by inserting the explicit expression for in terms of and applying Euler transformation of series. Although the Euler transformation applied to an alternating series introduces a summation, it drastically increases the rate of convergence of series such as (6.1).

Since the Bernoulli numbers vanish when is odd, the last term on the right-hand side of equation (6.1) vanishes whenever is even. The expression (6.1) yields double precision values with just the first 50 or so terms over *n* for small values of *m*.

## 7. Double summation of the Stieltjes constants

Let be an integer and . We demonstrate the following.

(7.1)*where* Li_{n} *is the polylogarithm function. Corresponding to* *, we have the special case*(7.2)*In addition, equation* (7.1) and (7.2) *continue to hold at the point* . *In particular, we have for* (7.3)*The case of* *in equation* *(7.2)* *gives*(7.4)

We write equation (3.2) as(7.5)and differentiate *j* times with respect to *z*, so that(7.6)We then set , multiply both sides of equation (7.6) by , sum over *m* from 1 to ∞, and use the series definition of Li_{n}. The proposition follows. ▪

We may note that the special case (7.2) recovers equation (3.10*a*).

## 8. Summary

We have used a variety of special functions and special numbers to derive single and double summation relations of the Stieltjes constants. These include new summatory relations for the Stieltjes constants in the forms of and , where *a* and *b* are real and *p* and *q* are positive integers. Even special cases of these results are of interest, showing the analytic and number theoretic information encapsulated in the Hurwitz and Riemann zeta functions. In addition, we obtained new integration formulae for the Hurwitz zeta function, including integrands with algebraic or polynomial factors. A theme of the paper is the use of a much earlier formula of Wilton (1927) that may be looked upon as a sort of functional equation of the Hurwitz zeta function in its second argument. Parts of the analysis show an intertwining of the theory of the hypergeometric function with that of the Hurwitz zeta function.

The special functions that we have encountered include the polygamma functions. This is to be expected since the Pochhammer symbol is a building block of the hypergeometric function and, in turn, its derivative introduces polygamma functions. In fact, it has been shown (Coffey 2006) that the Stieltjes constants are the same as the negative of a generalized digamma function. The re-expression of Pochhammer symbols as power series of the argument introduces the Stirling numbers of the first kind. We additionally made use of the Stirling numbers of the second kind and Bernoulli numbers. With the latter numbers and binomial coefficients, we presented a novel form of the Stieltjes constants appropriate for the Riemann zeta function.

## Footnotes

- Received December 29, 2005.
- Accepted February 10, 2006.

- © 2006 The Royal Society