Kummer’s Formula for Multiple Gamma Functions ^{∗}

Shin-ya Koyama (Keio University) Nobushige Kurokawa (Tokyo Institute of Technology)

Running title. Kummer’s Formula Abstract. We generalize Kummer’s formula on an expression of the gamma function to that of multiple gamma functions using the functional equation for multiple Hurwitz zeta functions. 2000 Mathematics Subject Classification: 11M06

1 Introduction

Kummer’s formula for the usual gamma function Γ(x) is the following identity:

log Γ(x) = \frac{1}{π} ∑_{n=1}^{∞} \frac{(log n) sin(2πnx)}{n} + \frac{log(2π) + γ}{π} ∑_{n=1}^{∞} \frac{sin(2πnx)}{n}

+ \frac{1}{2} ∑_{n=1}^{∞} \frac{cos(2πnx)}{n} + \frac{1}{2} log(2π) (1.1)

for 0 < x < 1, which was discovered by Kummer [9] (1847). A spectacular application is given by letting x be a rational number: we have formulas for special values of Dirichlet L-functions using the gamma function. For example, when x = 1/4 we get

L′(1, χ_{−4}) = −π log Γ (\frac{1}{4}) + \frac{π}{4} (γ + 3 log π + 2 log 2)

due to Malmstén [20] (1849), where

L(s, χ_{−4}) = ∑_{n:odd}(−1)^{\frac{n−1}{2}} n^{−s}

is the Dirichlet L-function for the non-trivial character χ_{−4} modulo 4. This result is equivalently written as

L′(0, χ_{−4}) = 2 log Γ (\frac{1}{4}) − \frac{3}{2} log 2 − log π

∗Presented at the conference on ”Zetas and Trace Formulas” in Okinawa, November 2002.

via the functional equation. We remark that L(0, χ_{−4}) = 1/2 and L(1, χ_{−4}) = π/4, so Malmstén’s result gives the (difficult) second term of the Taylor expansion for L(s, χ_{−4}) at s = 0 and s = 1.

Marmstén’s study motivated the following excellent formula

log Γ(x) = ζ′(0, x) + \frac{1}{2} log(2π)  (1.2)

of Lerch [18] (1894), where

ζ(s, x) = ∑_{n=0}^{∞}(n + x)^{−s}

is the Hurwitz zeta function ([8]) and the differentiation is in the first variable s. Combining this with Kronecker’s limit formula Lerch [19] (1897) obtained the formula

∑_{λ=1}^{Δ} (\frac{D}{λ}) log Γ (\frac{λ}{Δ}) = h log Δ − \frac{h}{3} log 2π − ∑_{(a,b,c)} log a + \frac{2}{3} ∑_{(a,b,c)} log(ϑ′_{1}(0|α)ϑ′_{1}(0|β)), (1.3)

where ϑ_{1} is the usual notation of theta function (so ϑ′_{1} is essentially η^{3}), ℚ(\sqrt{D}) is an imaginary quadratic field of discriminant D and class number h, (a, b, c) runs over h quadratic forms corresponding to the ideal classes, and Δ = |D|. (It would be convenient to look at the formula (163) of Landau [17].) In recent years this formula is called “Chowla-Selberg formula” after the 70 years late paper of Chowla-Selberg [2] (1967) with no mention to Lerch [19] unfortunately. Concerning the history around Kummer’s formula (1.1) and the above two Lerch’s formulae (1.2), (1.3) we refer to the excellent survey [17] written by Landau, which is the famous first paper on the prime ideal theorem generalizing the usual prime number theorem to algebraic number fields.

The purpose of this paper is to present a generalization of Kummer’s formula in the case of the multiple gamma function Γ_{r}(x). This multiple gamma function originates from Barnes [1] and it is defined as

Γ_{r}(x) = exp(ζ′_{r}(0, x))

where

ζ_{r}(s, x) = ∑_{n_{1},...,n_{r}=0}^{∞}(n_{1} + ⋯ + n_{r} + x)^{−s}

= ∑_{n=0}^{∞} \binom{n + r − 1}{r − 1}(n + x)^{−s}

is the multiple Hurwitz zeta function. We have many applications of Γ_{r}(x) in Sarnak [23], Voros [25], Shintani [24] and our previous papers [10] - [15]. From Γ_{1}(x) = Γ(x)/\sqrt{2π}, we can write Kummer’s formula as follows:

log Γ_{1}(x) = ζ′(0, x)
= \frac{1}{π} ∑_{n=1}^{∞} \frac{(log n) sin(2πnx)}{n}

+\frac{log(2π) + γ}{π} ∑_{n=1}^{∞} \frac{sin(2πnx)}{n} + \frac{1}{2} ∑_{n=1}^{∞} \frac{cos(2πnx)}{n}.  (1.4)

Our result is a generalization of this formula:

Theorem 1 Let 0 < x < 1. Then we have

(1)

log Γ_{2}(x) = −\frac{1}{2π^{2}} ∑_{n=1}^{∞} \frac{(log n) cos(2πnx)}{n^{2}} − \frac{log(2π) + γ − 1}{2π^{2}} ∑_{n=1}^{∞} \frac{cos(2πnx)}{n^{2}}

+ \frac{1}{4π} ∑_{n=1}^{∞} \frac{sin(2πnx)}{n^{2}} + (1 − x) log Γ_{1}(x).

(2)

log Γ_{3}(x) = −\frac{1}{4π^{3}} ∑_{n=1}^{∞} \frac{(log n) sin(2πnx)}{n^{3}} − \frac{2 log(2π) + 2γ − 3}{8π^{3}} ∑_{n=1}^{∞} \frac{sin(2πnx)}{n^{3}}

− \frac{1}{8π^{2}} ∑_{n=1}^{∞} \frac{cos(2πnx)}{n^{3}} + (\frac{3}{2} − x) log Γ_{2}(x) − \frac{(x − 1)^{2}}{2} log Γ_{1}(x).

In the text we show some applications. We notice that our method is quite simple to obtain ζ′_{r}(0, x) using the functional equation ζ_{r}(s, x) = ξ_{r}(r −s, x) where ξ_{r}(s, x) is a Dirichlet series intimately related to ζ_{r}(s, x):

log Γ_{r}(x) = −ξ′_{r}(r, x).

This natural method was discovered by Hardy [4] to reprove the original Kummer’s formula. In [5, 6], Hardy studied the case of the double gamma function, but he did not obtain the generalization of Kummer’s formula.

Acknowledgement. This paper was presented at the symposium “Zetas and Trace Formulas” in Okinawa from November 27 to 29 of 2002. The authors would like to express their hearty thanks to all the participants of Okinawa symposium. We also thank Laguna Garden Hotel and Okinawa Convension Center for supplying the excellent atmosphere with blue sky and sea.

2 Multiple Gamma Functions

We prepare on the multiple gamma function Γ_{r}(x). We refer to Barnes[1] for the general theory. First we know that the function Γ_{r}(x) is a meromorphic function of order r (Γ_{r}(x)^{−1} being an entire function of order r), and it has a simple pole at x = 0:

Γ_{r}(x) ∼ \frac{1}{ρ_{r}x} (as x → 0).

We remark that Barnes [1] used Γ_{r}^{B}(x) normalized as

Γ_{r}^{B}(x) = ρ_{r}Γ_{r}(x) ∼ \frac{1}{x} (as x → 0).

For later use we calculate ρ_{r} for r = 1, 2, 3 here.

Theorem 2 (1) ρ_{1} = \sqrt{2π}.

(2)

ρ_{2} = \sqrt{2π}e^{−ζ′(−1)} = exp (−\frac{ζ′(2)}{2π^{2}} + \frac{7 log(2π) + γ − 1}{12}) .

(3)

ρ_{3} = \sqrt{2π}e^{− \frac{3}{2} ζ′(−1)− \frac{1}{2} ζ′(−2)} = exp (\frac{ζ(3)}{8π^{2}} − \frac{3ζ′(2)}{4π^{2}} + \frac{5 log(2π) + γ − 1}{8}) .

Proof. From

ζ_{r}(s, x + 1) = ζ_{r}(s, x) − ζ_{r−1}(s, x)

we have

Γ_{r}(x + 1) = Γ_{r}(x)Γ_{r−1}(x)^{−1}.

Hence letting x → 0 we get

Γ_{r}(1) = \frac{ρ_{r−1}}{ρ_{r}} .

(Here we understand that ζ_{0}(s, x) = x^{−s}, Γ_{0}(x) = x^{−1} and ρ_{0} = 1.) The first result ρ_{1} = \sqrt{2π} follows from Lerch’s formula

Γ_{1}(x) = \frac{Γ(x)}{\sqrt{2π}} ∼ \frac{1}{\sqrt{2π}x} (as x → 0).

Now we calculate

ρ_{2} = ρ_{1}Γ_{2}(1)^{−1} = \sqrt{2π}Γ_{2}(1)^{−1}

and

ρ_{3} = ρ_{2}Γ_{3}(1)^{−1} = \sqrt{2π}Γ_{2}(1)^{−1}Γ_{3}(1)^{−1}.

Since

ζ_{2}(s, x) = ∑_{n=0}^{∞}(n + 1)(n + x)^{−s}

= ∑_{n=0}^{∞}((n + x) + (1 − x))(n + x)^{−s}

= ∑_{n=0}^{∞}(n + x)^{−(s−1)} + (1 − x) ∑_{n=0}^{∞}(n + x)^{−s}
= ζ(s − 1, x) + (1 − x)ζ(s, x)

and

ζ_{3}(s, x) = ∑_{n=0}^{∞} \frac{(n + 2)(n + 1)}{2}(n + x)^{−s}

= \frac{1}{2} ∑_{n=0}^{∞}((n + x)^{2} + (3 − 2x)(n + x) + (x − 1)(x − 2))(n + x)^{−s}

= \frac{1}{2}ζ(s − 2, x) + \frac{3 − 2x}{2}ζ(s − 1, x) + \frac{(x − 1)(x − 2)}{2}ζ(s, x),

we obtain

ζ′_{2}(0, x) = ζ′(−1, x) + (1 − x)ζ′(0, x)

and

ζ′_{3}(0, x) = \frac{1}{2}ζ′(−2, x) + \frac{3 − 2x}{2}ζ′(−1, x) + \frac{(x − 1)(x − 2)}{2}ζ′(0, x)

Hence, letting x = 1 and remarking ζ(s, 1) = ζ(s), we have

Γ_{2}(1) = exp(ζ′_{2}(0, 1)) = exp(ζ′(−1))

and

Γ_{3}(1) = exp(ζ′_{3}(0, 1)) = exp (\frac{1}{2}ζ′(−2) + \frac{1}{2}ζ′(−1)) .

Thus we have

ρ_{2} = \sqrt{2π} exp(−ζ′(−1))

and

ρ_{3} = \sqrt{2π} exp (−\frac{3}{2}ζ′(−1) − \frac{1}{2}ζ′(−2)) .

Now we use the functional equation for ζ(s):

ζ(1 − s) = 2(2π)^{−s}Γ(s) cos (\frac{πs}{2}) ζ(s).

By differentiating both sides of this functional equation at s = 2, we have

−ζ′(−1) = 2(2π)^{−2}Γ(2)(cos π)ζ′(2) + 2(2π)^{−2}(− log(2π))Γ(2)(cos π)ζ(2)
+2(2π)^{−2}Γ′(2)(cos π)ζ(2)
= −\frac{ζ′(2)}{2π^{2}} + \frac{log(2π)}{12} − \frac{1 − γ}{12} ,

where we used ζ(2) = π^{2}/6 and Γ′(2) = 1 − γ. Hence

ρ_{2} = \sqrt{2π} exp (−\frac{ζ′(2)}{2π^{2}} + \frac{log(2π)}{12} − \frac{1 − γ}{12} )
= exp (−\frac{ζ′(2)}{2π^{2}} + \frac{7 log(2π) + γ − 1}{12}) .

Next, from the functional equation for ζ(s) we have

−ζ′(−2) = 2(2π)^{−3}Γ(3) (−\frac{π}{2} sin \frac{3π}{2} ) ζ(3).

Hence

ζ′(−2) = −\frac{ζ(3)}{4π^{2}}.

Thus

ρ_{3} = exp (\frac{log(2π)}{2} − \frac{3}{2} (\frac{ζ′(2)}{2π^{2}} − \frac{log(2π) + γ − 1}{12}) − \frac{1}{2} (−\frac{ζ(3)}{4π^{2}}))
= exp (\frac{ζ(3)}{8π^{2}} − \frac{3ζ′(2)}{4π^{2}} + \frac{5 log(2π) + γ − 1}{8}) .

3 Generalized Kummer’s Formula

To prove Theorem 1 we show Theorem 3 Let k ≥ 1 be an integer.

(1) When k is odd,

ζ′(−k, x) = \frac{2(−1)^{\frac{k+1}{2}} k!}{(2π)^{k+1}} \{∑_{n=1}^{∞} \frac{(log n) cos(2πnx)}{n^{k+1}}

+ (log(2π) + γ − (1 + \frac{1}{2} + ⋯ + \frac{1}{k})) ∑_{n=1}^{∞} \frac{cos(2πnx)}{n^{k+1}}

−\frac{π}{2} ∑_{n=1}^{∞} \frac{sin(2πnx)}{n^{k+1}}\} .

(2) When k is even,

ζ′(−k, x) = \frac{2(−1)^{\frac{k}{2}} k!}{(2π)^{k+1}} \{∑_{n=1}^{∞} \frac{(log n) sin(2πnx)}{n^{k+1}}

+ (log(2π) + γ − (1 + \frac{1}{2} + ⋯ + \frac{1}{k})) ∑_{n=1}^{∞} \frac{sin(2πnx)}{n^{k+1}}

+\frac{π}{2} ∑_{n=1}^{∞} \frac{cos(2πnx)}{n^{k+1}}\} .

Proof. We have the following functional equation for ζ(s, x) proved by Hurwitz [8]:

ζ(s, x) = 2(2π)^{s−1}Γ(1 − s) (sin (\frac{πs}{2}) ∑_{n=1}^{∞} \frac{cos(2πnx)}{n^{1−s}} + cos (\frac{πs}{2}) ∑_{n=1}^{∞} \frac{sin(2πnx)}{n^{1−s}}) .

Hence we obtain

ζ′(−k, x) = 2(2π)^{−k−1}(log(2π) · k! − Γ′(1 + k))

× (− sin (\frac{πk}{2} ) ∑_{n=1}^{∞} \frac{cos(2πnx)}{n^{k+1}} + cos (\frac{πk}{2} ) ∑_{n=1}^{∞} \frac{sin(2πnx)}{n^{k+1}})

+2(2π)^{−k−1}k! (− sin (\frac{πk}{2} ) ∑_{n=1}^{∞} \frac{(log n) cos(2πnx)}{n^{k+1}}

+ cos (\frac{πk}{2} ) ∑_{n=1}^{∞} \frac{(log n) sin(2πnx)}{n^{k+1}}

+\frac{π}{2} cos (\frac{πk}{2} ) ∑_{n=1}^{∞} \frac{cos(2πnx)}{n^{k+1}}

+ \frac{π}{2} sin (\frac{πk}{2} ) ∑_{n=1}^{∞} \frac{sin(2πnx)}{n^{k+1}}) .

Thus, using

\frac{Γ′}{Γ} (k + 1) = 1 + \frac{1}{2} + ⋯ + \frac{1}{k} − γ

we get (1) and (2).

Proof of Theorem 1. From the proof of Theorem 2 we see the relations

log Γ_{2}(x) = ζ′_{2}(0, x) = ζ′(−1, x) + (1 − x)ζ′(0, x) = ζ′(−1, x) + (1 − x) log Γ_{1}(x) and

log Γ_{3}(x) = ζ′_{3}(0, x) = \frac{1}{2}ζ′(−2, x) + \frac{3 − 2x}{2}ζ′(−1, x) + \frac{(x − 1)(x − 2)}{2}ζ′(0, x)
= \frac{1}{2}ζ′(−2, x) + (\frac{3}{2} − x) ζ′_{2}(0, x) − \frac{(x − 1)^{2}}{2} ζ′(0, x)
= \frac{1}{2}ζ′(−2, x) + (\frac{3}{2} − x) log Γ_{2}(x) − \frac{(x − 1)^{2}}{2} log Γ_{1}(x).

Hence we have Theorem 1 from Theorem 3.

4 Applications

As the first application of our generalization of Kummer’s formula we calculate some special values Γ_{r}(x) for r = 2 and 3. Before this we explain our method in the simple case r = 1: letting x = 1/2 in Kummer’s formula (1.4) for Γ_{1}(x) we have

Γ_{1} (\frac{1}{2}) = exp (\frac{1}{2} ∑_{n=1}^{∞} \frac{(−1)^{n}}{n} )

= exp (−\frac{1}{2} log 2)
= \frac{1}{\sqrt{2}}

and

Γ_{1}^{B} (\frac{1}{2}) = ρ_{1}Γ_{1} (\frac{1}{2})
= \sqrt{2π} · \frac{1}{\sqrt{2}}
= \sqrt{π}.

(Actually in this case r = 1 these results are also direct from Γ_{1}(x) = Γ(x)/\sqrt{2π} and Γ_{1}^{B}(x) = Γ(x) using the well-known Γ(1/2) = \sqrt{π}.)

Theorem 4 (1)

Γ_{2} (\frac{1}{2}) = exp (−\frac{ζ′(2)}{4π^{2}} + \frac{log π + γ − 1}{24} − \frac{log 2}{4}) .

(2)

Γ_{2}^{B} (\frac{1}{2}) = exp (−\frac{3ζ′(2)}{4π^{2}} + \frac{5 log π + γ − 1}{8} + \frac{log 2}{3}) .

Γ_{3} (\frac{1}{2}) = exp (\frac{3ζ(3)}{32π^{2}} − \frac{ζ′(2)}{4π^{2}} + \frac{log π + γ − 1}{24} − \frac{3 log 2}{16}) .

(4)

Γ_{3}^{B} (\frac{1}{2}) = exp (\frac{7ζ(3)}{32π^{2}} − \frac{ζ′(2)}{π^{2}} + \frac{4 log π + γ − 1}{6} + \frac{7 log 2}{16}) .

Proof. (1). Let x = 1/2 in Theorem 1(1). Then we have

log Γ_{2} (\frac{1}{2}) = −\frac{1}{2π^{2}} ∑_{n=1}^{∞} \frac{(log n) cos(πn)}{n^{2}} − \frac{log(2π) + γ − 1}{2π^{2}} ∑_{n=1}^{∞} \frac{cos(πn)}{n^{2}}

+ \frac{1}{2} log Γ_{1} (\frac{1}{2}) .

We notice

∑_{n=1}^{∞} cos(πn)n^{−s} = − ∑_{n=1}^{∞}(−1)^{n−1}n^{−s}
= −(1 − 2 · 2^{−s})ζ(s)
= (2^{1−s} − 1)ζ(s)

and its differentiation

∑_{n=1}^{∞}(log n) cos(πn)n^{−s} = (log 2)2^{1−s}ζ(s) − (2^{1−s} − 1)ζ′(s).

Hence

∑_{n=1}^{∞} \frac{cos(πn)}{n^{2}} = −\frac{1}{2}ζ(2) = −\frac{π^{2}}{12}

and

∑_{n=1}^{∞} \frac{(log n) cos(πn)}{n^{2}} = \frac{log 2}{2}ζ(2) + \frac{1}{2}ζ′(2)
= \frac{π^{2} log 2}{12} + \frac{ζ′(2)}{2}.

Thus

log Γ_{2} (\frac{1}{2}) = −\frac{log 2}{24} − \frac{ζ′(2)}{4π^{2}} + \frac{log(2π) + γ − 1}{24} − \frac{log 2}{4}
= −\frac{ζ′(2)}{4π^{2}} + \frac{log π + γ − 1}{24} − \frac{log 2}{4}.

This gives (1).

(2). Since Γ_{2}^{B}(1/2) = ρ_{2}Γ_{2}(1/2), (2) follows from (1) and the formula for ρ_{2} in Theorem 2.

(3). Set x = 1/2 in Theorem 1(2). Then

log Γ_{3} (\frac{1}{2}) = −\frac{1}{8π^{2}} ∑_{n=1}^{∞} \frac{cos(πn)}{n^{3}} + log Γ_{2} (\frac{1}{2}) − \frac{1}{8} log Γ_{1} (\frac{1}{2})

= \frac{3ζ(3)}{32π^{2}} + log Γ_{2} (\frac{1}{2}) + \frac{log 2}{16},

where we used

∑_{n=1}^{∞} \frac{cos(πn)}{n^{3}} = (2^{1−3} − 1)ζ(3) = −\frac{3}{4}ζ(3)

and Γ_{1}(1/2) = 1/\sqrt{2}. Hence we obtain (3) from (1).

(4). Since Γ_{3}^{B}(1/2) = ρ_{3}Γ_{3}(1/2), (4) follows from (3) and the fomula for ρ_{3} in Theorem 2.

Values of Γ_{r}(x) at rational numbers are also expressed via special values of Dirichlet L-functions besides ζ(s) through the generalized Kummer’s formula. Here we report the following typical example.

Theorem 5

Γ_{2} (\frac{1}{4}) = exp (\frac{L(2, χ_{−4})}{4π} − \frac{3L′(1, χ_{−4})}{4π} − \frac{ζ′(2)}{16π^{2}} + \frac{19 log π + 19γ + log 2 − 1}{96}) ,

where L(s, χ_{−4}) is the Dirichlet L-function for the non-trivial character χ_{−4} modulo 4.

Proof. Let x = 1/4 in Theorem 1(1). Then we have

log Γ_{2} (\frac{1}{4}) = −\frac{1}{2π^{2}} ∑_{n=1}^{∞} \frac{(log n) cos(\frac{πn}{2})}{n^{2}} − \frac{log(2π) + γ − 1}{2π^{2}} ∑_{n=1}^{∞} \frac{cos(\frac{πn}{2})}{n^{2}}

+ \frac{1}{4π} ∑_{n=1}^{∞} \frac{sin(\frac{πn}{2})}{n^{2}} + \frac{3}{4} log Γ_{1} (\frac{1}{4}) .

We notice

∑_{n=1}^{∞} \frac{cos(\frac{πn}{2})}{n^{s}} = ∑_{n=1}^{∞} \frac{(−1)^{n}}{(2n)^{s}} = 2^{−s}(2^{1−s} − 1)ζ(s) = (2^{1−2s} − 2^{−s})ζ(s)

and its differentiation

∑_{n=1}^{∞} \frac{(log n) cos(\frac{πn}{2})}{n^{s}} = −(2^{1−2s} − 2^{−s})ζ′(s) + (2^{−s} − 2^{2−2s})(log 2)ζ(s).

Hence

∑_{n=1}^{∞} \frac{cos(\frac{πn}{2})}{n^{2}} = −\frac{ζ(2)}{8} = −\frac{π^{2}}{48}

and

∑_{n=1}^{∞} \frac{(log n) cos(\frac{πn}{2})}{n^{2}} = \frac{ζ′(2)}{8}.

Using

∑_{n=1}^{∞} \frac{sin(\frac{πn}{2})}{n^{s}} = ∑_{{n=1 \atop n:odd}}^{∞} \frac{(−1)^{\frac{n−1}{2}}}{n^{s}} = L(s, χ_{−4})

we have

Γ_{2} (\frac{1}{4}) = −\frac{ζ′(2)}{16π^{2}} + \frac{log(2π) + γ − 1}{96} + \frac{L(2, χ_{−4})}{4π} + \frac{3}{4} log Γ_{1} (\frac{1}{4}) .

Thus we obtain Theorem 5 from the result of Malmstén [20]

log Γ_{1} (\frac{1}{4}) = −\frac{L′(1, χ_{−4})}{π} + \frac{log π + γ}{4} ,

which is obtained by setting x = 1/4 in Kummer’s formula (1.4):

log Γ_{1} (\frac{1}{4}) = \frac{1}{π} ∑_{n=1}^{∞} \frac{(log n) sin(\frac{πn}{2})}{n}

+\frac{log(2π) + γ}{π} ∑_{n=1}^{∞} \frac{sin(\frac{πn}{2})}{n} + \frac{1}{2} ∑_{n=1}^{∞} \frac{cos(\frac{πn}{2})}{n}  (4.1)

= −\frac{1}{π} L′(1, χ_{−4}) + \frac{log(2π) + γ}{π} L(1, χ_{−4}) + \frac{1}{2} ∑_{n=1}^{∞} \frac{(−1)^{n}}{2n}  (4.2)
= −\frac{1}{π} L′(1, χ_{−4}) + \frac{log π + γ}{4} ,  (4.3)

where we used L(1, χ_{−4}) = π/4 and ∑_{n=1}^{∞} \frac{(−1)^{n−1}}{n} = log 2.

Now let S_{r}(x) = Γ_{r}(x)^{−1}Γ_{r}(r − x)^{(−1)^{r}} be the multiple sine function studied in [13]. As the second application of the generalized Kummer’s formula we prove basic properties of S_{r}(x) for r = 2 and 3. This generalizes the usual sine function:

S_{1}(x) = Γ_{1}(x)^{−1}Γ_{1}(1 − x)^{−1} = \frac{2π}{Γ(x)Γ(1 − x)} = 2 sin(πx).

We notice that

S_{1}(x) = exp (− ∑_{n=1}^{∞} \frac{cos(2πnx)}{n})

for 0 < x < 1 from Kummer’s formula for Γ_{1}(x).

Theorem 6 (1)
S_{2}(x) = exp (−\frac{1}{2π} ∑_{n=1}^{∞} \frac{sin(2πnx)}{n^{2}} + (x − 1) ∑_{n=1}^{∞} \frac{cos(2πnx)}{n})

= exp (−\frac{1}{2π} ∑_{n=1}^{∞} \frac{sin(2πnx)}{n^{2}}) · S_{1}(x)^{1−x}

for 0 < x < 1.

(2)

\frac{S′_{2}}{S_{2}} (x) = −π(x − 1) cot(πx).

(3)

S_{2} (\frac{1}{2}) = \sqrt{2}.

(4)

S_{2}(x) = 2πxe^{−x} ∏_{n=1}^{∞} ((1 + \frac{x}{n})^{1+n} (1 − \frac{x}{n})^{1−n} e^{−2x}) .

(5)

S_{2}(x) = exp (−π ∫_{0}^{x−1} t cot(πt)dt) .

Proof. Since

S_{2}(x) = Γ_{2}(x)^{−1}Γ_{2}(2 − x)

and

Γ_{2}(2 − x) = Γ_{2}(1 − x)Γ_{1}(1 − x)^{−1},

we have

S_{2}(x) = Γ_{2}(x)^{−1}Γ_{2}(1 − x)Γ_{1}(1 − x)^{−1}.

Hence, from (1) of Theorem 1 we have

log S_{2}(x) = −\frac{1}{2π} ∑_{n=1}^{∞} \frac{sin(2πnx)}{n^{2}} − (x − 1) log S_{1}(x)

and

\frac{S′_{2}}{S_{2}} (x) = −(x − 1)\frac{S′_{1}}{S_{1}} (x) − ∑_{n=1}^{∞} \frac{cos(2πnx)}{n} − log S_{1}(x)
= −(x − 1)\frac{S′_{1}}{S_{1}} (x)
= −(x − 1)π cot(πx),

where we used

log S_{1}(x) = − log Γ_{1}(x) − log Γ_{1}(1 − x)
= − ∑_{n=1}^{∞} \frac{cos(2πnx)}{n}.

Thus (1) and (2) are proved. We have (3) by setting x = 1/2 in (1). For the proof of (4) let

S_{2}(z) = e^{z} ∏_{n=1}^{∞} ((\frac{1 − \frac{z}{n}}{1 + \frac{z}{n}} )^{n} e^{2z})

following Hölder [7]. Then we easily obtain

\frac{S′_{2}}{S_{2}} (x) = πx cot(πx)

and S_{2}(1/2) = \sqrt{2} ([7, 13]). Hence it is sufficient to show that

S_{2}(x) = S_{2}(x)^{−1}S_{1}(x).

We know S_{2}(1/2) = \sqrt{2} and S_{1}(1/2) = 2. Thus the both sides are \sqrt{2} at x = 1/2, and the differentiations of the both sides turn out to be −(x − 1) cot(πx). Hence we get (4). Lastly

(5) follows from (2) since both sides are 1 at x = 1.

Remark 1 These results have further applications containing expressions of the difficult special values of Dirichlet L-functions ([13]). Especially (5) gives the calculation of the gamma factor of the Selberg zeta function for a Riemann surface (Sarnak [23] and Voros [25]). The situation is similar in the higher dimensional case, where we use the multiple sine function S_{r}(x) with r being the dimension ([13]).

Theorem 7 (1)
S_{3}(x) = exp (\frac{1}{4π^{2}} ∑_{n=1}^{∞} \frac{cos(2πnx)}{n^{3}} − \frac{2x − 3}{4π} ∑_{n=1}^{∞} \frac{sin(2πnx)}{n^{2}} − \frac{(x − 1)(x − 2)}{2} ∑_{n=1}^{∞} \frac{cos(2πnx)}{n})

= exp (\frac{1}{4π^{2}} ∑_{n=1}^{∞} \frac{cos(2πnx)}{n^{3}}) S_{2}(x)^{\frac{3}{2} −x}S_{1}(x)^{− \frac{(x−1)^{2}}{2}}
for 0 < x < 1.

(2)

\frac{S′_{3}}{S_{3}} (x) = \frac{π}{2} (x − 1)(x − 2) cot(πx).

S_{3} (\frac{1}{2}) = 2^{\frac{3}{8}} exp (−\frac{3}{16π^{2}} ζ(3)) .

(4)

S_{3}(x) = 2πe^{−ζ′(−2)}xe^{\frac{x^{2}}{4} − \frac{3}{2} x} ∏_{n=1}^{∞} ((1 + \frac{x}{n})^{\frac{(n+1)(n+2)}{2}} (1 − \frac{x}{n})^{\frac{(n−1)(n−2)}{2}} e^{\frac{x^{2}}{2} −3x}) .

Proof. Using Γ_{r}(x) = Γ_{r}(x − 1)Γ_{r−1}(x − 1)^{−1} we have

S_{3}(x) = Γ_{3}(x)^{−1}Γ_{3}(3 − x)^{−1}
= Γ_{3}(x)^{−1}(Γ_{3}(2 − x)Γ_{2}(2 − x)^{−1})^{−1}
= Γ_{3}(x)^{−1}(Γ_{3}(1 − x)Γ_{2}(1 − x)^{−1}Γ_{2}(1 − x)^{−1}Γ_{1}(1 − x))^{−1}
= Γ_{3}(x)^{−1}Γ_{3}(1 − x)^{−1}Γ_{2}(1 − x)^{2}Γ_{1}(1 − x)^{−1}.

Hence, from Theorem 1(2), for 0 < x < 1

log S_{3}(x) = \frac{1}{4π^{2}} ∑_{n=1}^{∞} \frac{cos(2πnx)}{n^{3}} − (\frac{3}{2} − x) log Γ_{2}(x) + \frac{(x − 1)^{2}}{2} log Γ_{1}(x)

− (\frac{1}{2} + x) log Γ_{2}(1 − x) + \frac{x^{2}}{2} log Γ_{1}(1 − x) + 2 log Γ_{2}(1 − x) − log Γ_{1}(1 − x)

= \frac{1}{4π^{2}} ∑_{n=1}^{∞} \frac{cos(2πnx)}{n^{3}} + (\frac{3}{2} − x) log(Γ_{2}(1 − x)Γ_{2}(x)^{−1})

+\frac{(x − 1)^{2}}{2} log Γ_{1}(x) + (\frac{x^{2}}{2} − 1) log Γ_{1}(1 − x)

= \frac{1}{4π^{2}} ∑_{n=1}^{∞} \frac{cos(2πnx)}{n^{3}} + (\frac{3}{2} − x) log(S_{2}(x)Γ_{1}(1 − x))

+\frac{(x − 1)^{2}}{2} log Γ_{1}(x) + (\frac{x^{2}}{2} − 1) log Γ_{1}(1 − x)

= \frac{1}{4π^{2}} ∑_{n=1}^{∞} \frac{cos(2πnx)}{n^{3}} + (\frac{3}{2} − x) log S_{2}(x) − \frac{(x − 1)^{2}}{2} log S_{1}(x),

where we used that

Γ_{2}(1 − x)Γ_{2}(x)^{−1} = S_{2}(x)Γ_{1}(1 − x)

and

Γ_{1}(x)^{−1}Γ_{1}(1 − x)^{−1} = S_{1}(x).

(2). From (1) we have

\frac{S′_{3}}{S_{3}} (x) = −\frac{1}{2π} ∑_{n=1}^{∞} \frac{sin(2πnx)}{n^{2}} − (log S_{2}(x) + (x − 1) log S_{1}(x))

+ ((\frac{3}{2} − x) \frac{S′_{2}}{S_{2}} (x) − \frac{(x − 1)^{2}}{2} \frac{S′_{1}}{S_{1}} (x)) .

Hence, using the previous formulas for log S_{r}(x) and S′_{r}(x)/S_{r}(x) (r = 1, 2), we obtain

\frac{S′_{3}}{S_{3}} (x) = −\frac{1}{2π} ∑_{n=1}^{∞} \frac{sin(2πnx)}{n^{2}} + \frac{1}{2π} ∑_{n=1}^{∞} \frac{sin(2πnx)}{n^{2}}

+π ((x − 1) (x − \frac{3}{2}) − \frac{(x − 1)^{2}}{2} ) cot(πx)

= \frac{π}{2} (x − 1)(x − 2) cot(πx).

(3). From (1)

S_{3} (\frac{1}{2}) = exp (\frac{1}{4π^{2}} ∑_{n=1}^{∞} \frac{cos(πnx)}{n^{3}}) S_{2} (\frac{1}{2}) S_{1} (\frac{1}{2})^{− \frac{1}{8}}

= 2^{\frac{3}{8}} exp (\frac{1}{4π^{2}} ∑_{n=1}^{∞} \frac{(−1)^{n}}{n^{3}} ) .

Here

∑_{n=1}^{∞} \frac{(−1)^{n}}{n^{3}} = − ∑_{n=1}^{∞} \frac{(−1)^{n−1}}{n^{3}}
= −(1 − 2 · 2^{−3})ζ(3)
= −\frac{3}{4}ζ(3).

Hence we obtain (3).

(4). Let

S_{3}(x) = e^{\frac{x^{2}}{2}} ∏_{n=1}^{∞} ((1 − \frac{x^{2}}{n^{2}} )^{n^{2}} e^{x^{2}}) .

We prove that

S_{3}(x) = C · S_{3}(x)^{2}S_{2}(x)^{−3}S_{1}(x)

with C = e^{2ζ′(−2)}. Then we get (4) using the product expressions for S_{3}(x), S_{2}(x) and S_{1}(x). First, we see

\frac{S′_{3}}{S_{3}} (x) = πx^{2} cot(πx)

from [12, 13, 21] and

2\frac{S′_{3}}{S_{3}} (x) − 3\frac{S′_{2}}{S_{2}} (x) + \frac{S′_{1}}{S_{1}} (x) = π (2 · \frac{(x − 1)(x − 2)}{2} + 3(x − 1) + 1) cot(πx)
= πx^{2} cot(πx).

Hence, it is sufficient to show that

C = S_{3} (\frac{1}{2}) S_{3} (\frac{1}{2})^{−2} S_{2} (\frac{1}{2})^{3} S_{1} (\frac{1}{2})^{−1} .

We calculated the value S_{3} (\frac{1}{2}) in [16] ([13]) as

S_{3} (\frac{1}{2}) = 2^{\frac{1}{4}} exp (−\frac{7}{8π^{2}} ζ(3)) ,

which is equivalent to the following result of Euler [3]:

ζ(3) = \frac{2π^{2}}{7} log 2 + \frac{16}{7} ∫0^{\frac{π}{2}} x log(sin x)dx.

We know

S_{3} (\frac{1}{2}) = 2^{\frac{3}{8}} exp (−\frac{3}{16π^{2}} ζ(3))

as above. Thus we have

S_{3} (\frac{1}{2}) S_{3} (\frac{1}{2})^{−2} S_{2} (\frac{1}{2})^{3} S_{1} (\frac{1}{2})^{−1}

= 2^{\frac{1}{4}} exp (−\frac{7}{8π^{2}} ζ(3)) (2^{\frac{3}{8}} exp (−\frac{3}{16π^{2}} ζ(3)))^{−2} (\sqrt{2})^{3} · 2^{−1}
= exp (−\frac{ζ(3)}{2π^{2}})
= exp(2ζ′(−2))
= C,

where we used ζ(3) = −4π^{2}ζ′(−2) coming from the functional equation for ζ(s).

Remark 2 For another approach to Theorems 6 and 7 we refer to [13], where we use multiplication formulas for multiple sine functions and the argument is more elaborate. The above proofs are quite direct because of our Kummer’s formula for multiple gamma functions.

5 Generalizations and Problems

Our investigation naturally leads to the case of Γ_{r}(x) for r ≧ 4. This is treated similarly as r = 2 and 3 by using

ζ_{r}(s, x) = \frac{1}{(r − 1)!}ζ(s − r + 1, x) + ∑_{k=1}^{r−1} b_{r,k}(x)ζ_{k}(s, x)

where b_{r,k}(x) is a polynomial in x determined by

\binom{n + r − 1}{r − 1} = \frac{(n + x)^{r−1}}{(r − 1)!} + ∑_{k=1}^{r−1} b_{r,k}(x)\binom{n + k − 1}{k − 1}

for all integers n ≧ 0. For example

b_{r,1}(x) = −\frac{(x − 1)^{r−1}}{(r − 1)!} .

Then we have

Γ_{r}(x) = exp (ζ′_{r}(0, x))
= exp (\frac{1}{(r − 1)!}ζ′(1 − r, x)) ∏_{k=1}^{r−1} Γ_{k}(x)^{b_{r,k}(x)}.

Hence our Theorem 3 gives the Kummer’s formula for Γ_{r}(x) inductively.

Beyond this we have a more general problem for Γ_{r}(x; (ω_{1}, ⋯, ω_{r})) with general periods (ω_{1}, ⋯, ω_{r}) defined by

Γ_{r}(x; (ω_{1}, ⋯, ω_{r})) = exp (ζ′_{r}(0, x, (ω_{1}, ⋯, ω_{r}))) ,

where

ζ_{r}(s, x, (ω_{1}, ⋯, ω_{r})) = ∑_{n_{1},...,n_{r}=0}^{∞}(n_{1}ω_{1} + ⋯ + n_{r}ω_{r} + x)^{−s}

is the multiple Hurwitz zeta function for general parameters. We have seen the case (ω_{1}, ⋯, ω_{r}) = (1, ..., 1) above. First, when (ω_{1}, ⋯, ω_{r}) is reduced to the rational parameters the situation is quite similar to the case (1, ..., 1). For example, if (ω_{1}, ω_{2}) = (1, 2), then

ζ_{2}(s, x, (1, 2)) = 2^{−s} (ζ_{2} (s, \frac{x}{2}) + ζ_{2} (s, \frac{x + 1}{2}))

and

ζ′_{2}(0, x, (1, 2)) = ζ′_{2} (0, \frac{x}{2}) + ζ′_{2} (0, \frac{x + 1}{2}) − (log 2) (ζ_{2} (0, \frac{x}{2}) + ζ_{2} (0, \frac{x + 1}{2}))

(see [15]). Since

ζ_{2} (0, \frac{x}{2}) + ζ_{2} (0, \frac{x + 1}{2}) = ζ_{2}, (0, x, (1, 2))
= \frac{1}{4} (x^{2} − 3x + \frac{11}{6}) ,

we have

log Γ_{2}(x, (1, 2)) = log Γ_{2} (\frac{x}{2}) + log Γ_{2} (\frac{x + 1}{2}) − \frac{log 2}{4} (x^{2} − 3x + \frac{11}{6}) .

Hence we obtain Kummer’s formula for Γ_{2}(x, (1, 2)) from our Theorem 1(1).

The non-rational (or “non-commensurable”) parameter case, we face the problem to have the functional equation for ζ_{r}(s, x, (ω_{1}, ⋯, ω_{r})) of the form

ζ_{r}(s, x, (ω_{1}, ⋯, ω_{r})) = ξ_{r}(r − s, x, (ω_{1}, ⋯, ω_{r}))

by using the residue calculation as in Riemann [22] and Hurwitz [8]. This problem is in general highly non-trivial (delicate convergence) as investigated by Hardy [5, 6]. After that we would have the desired Kummer’s type formula:

log Γ_{r}(x; (ω_{1}, ⋯, ω_{r})) = −ξ′_{r}(r, x, (ω_{1}, ⋯, ω_{r})).

We postpone the detailed investigation to the next opportunity.

References

[1] E.W. Barnes: “On the theory of the multiple gamma function. Trans. Cambridge Philos. Soc., 19 (1904) 374-425

[2] S. Chowla and A. Selberg “On Epstein’s zeta-function” J. reine angew. Math. (Crelle’s J.) 227 (1967) 86-110.

[3] L. Euler: “Exercitationes analyticae” Novi Commentarii Academiae Scientiarum Petropolitanae, 17 (1772) 173-204. (Opera Omnia I-15, pp. 131-167)

[4] G.H. Hardy “On Kummer’s series for log Γ(a)” Quarterly J. Math. 37 (1906) 49-53. (Collected Papers: Vol IV, pp. 428-432)

[5] G.H. Hardy “The expression of the double zeta-function and double gamma-function in terms of elliptic functions” Trans. Cambridge Phil. Soc. 20 (1905) 1-35. (Collected Papers: Vol. IV, pp. 393-427)

[6] G.H. Hardy “On double Fourier series, and especially those which represent the double zeta-function with real and incommensurable parameters” Quarterly J. Math. 37 (1906) 53-79. (Collected Papers: Vol. IV, pp. 433-459)

[7] O. Hölder: “Ueber eine transcendente Function” Göttingen Nachrichten 1886, Nr. 16. pp. 514-522.

[8] A. Hurwitz “Einige Eigenschaften der Dirichlet’schen Funktionen F (s) = ∑(\frac{D}{n}) · \frac{1}{n^{s}}, die bei der Bestimmung der Klassenanzahlen binärer quadratischer Formen auftreten” Zeitschrift für Mathematik und Physik 27 (1882) 86-101. (Werke Vol. I, pp. 72-88)

[9] E. Kummer “Beitrag zur Theorie der Function Γ(x) = ∫_{0}^{∞} e^{−v}v^{x−1}dv” J. reine angew. Math. (Crelle’s J.) 35 (1847) 1-4. (Collected Papers Vol. II, pp.325-328)

[10] N. Kurokawa: “Multiple sine functions and Selberg zeta functions” Proc. Japan Acad. 67A (1991) 61-64.

[11] N. Kurokawa: “Gamma factors and Plancherel measures” Proc. Japan Acad. 68A (1992) 256-260.

[12] N. Kurokawa: “Multiple zeta functions: an example” In \underline{Zeta Functions in Geometry}, volume 21 of Advanced Studies in Pure Math., pages 219-226, Kinokuniya, Tokyo 1992.

[13] N. Kurokawa and S. Koyama: “Multiple sine functions” Forum Math. (in press)

[14] S. Koyama and N. Kurokawa: “Multiple zeta functions I” Composit. Math. (in press)

[15] S. Koyama and N. Kurokawa: “Normalized double sine functions” Proc. Japan Acad. 79 (2003) 14-18.

[16] N. Kurokawa and M. Wakayama: “On ζ(3)” J. Ramanujan Math. Soc. 16 (2001) 205-214.

[17] E. Landau “Über die zu einem algebraischen Zahlkörper gehörige Zetafunction und die Ausdehnung der Tschebyschefschen Primzahltheorie auf das Problem der Verteilung der Primideale” J. reine angew. Math. (Crelle’s J.) 125 (1902) 64-188. (Collected Works Vol. I, pp. 201-326)

[18] M. Lerch “Dalši studie v oboru Malmsténovských řad” Rozpravy České Akad. 3 (1894) no. 28, pp. 1-61.

[19] M. Lerch “Sur quelques formules relatives du nombre des classes” Bull. Sci. Math. 21 (1897) 290-304.

[20] C.J. Malmstén “De integralibus quibusdam definitis, seriebusque infinitis” J. reine angew. Math. (Crelle’s J.) 38 (1849) 1-39.

[21] Yu. I. Manin: “Lectures on zeta functions and motives (according to Deninger and Kurokawa)” Asterisque 228 (1995) 121-163.

[22] B. Riemann: “Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse” Monatsberichte der Berliner Akademie, November 1859, pp. 671-680. (Gesammelte Mathematische Werke, pp. 145-155)

[23] P. Sarnak: “Determinants of Laplacians” Comm. Math. Phys. 110 (1987) 113–120.

[24] T. Shintani: “On a Kronecker limit formula for real quadratic fields” J. Fac. Sci. Univ. Tokyo, 24 (1977) 167-199.

[25] A. Voros: “Spectral functions, special functions and Selberg trace formula” Comm. Math. Phys. 110 (1987) 439-465.

Shin-ya Koyama: Department of Mathemartics, Keio University, 3-14-1, Hiyoshi, Kohokuku, Kanagawa 223-8522, Japan. e-mail: koyama@math.keio.ac.jp Nobushige Kurokawa: Department of Mathematics, Tokyo Institute of Technology, 2-12-1, Oh-okayama, Meguro-ku, Tokyo 152-8551, Japan. e-mail: kurokawa@math.titech.ac.jp