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Hypertranscendence of the multiple sine function for a complex period.

Let [[omega].sub.1], ..., [[omega].sub.r] [member of] C all lie on the same side of some straight line through the origin. We put [omega] := ([[omega].sub.1], ..., [[omega].sub.r]). We define the multiple Hurwitz zeta, gamma and sine functions by

[mathematical expression not reproducible].

The multiple gamma and sine functions were introduced by Barnes [2 4] and Kurokawa [9], respectively. When r = 1, the function Sinx is the usual sine function:

[Sin.sub.1](x, [omega])= 2 sin ([pi]x)/[omega]).

It is known that the multiple sine function has interesting applications: the Kronecker limit formula for real quadratic fields ([13]), expressions of special values of the Riemann zeta and Dirichlet L-functions ([9]), the calculation of the gamma factors of Selberg zeta functions ([9]), expression of solutions to the quantum Knizhnik-Zamolodchikov equation ([7]) and so on. Concerning basic properties of the multiple sine functions, we refer to [9].

The multiple sine function has similar properties to the usual sine function. Kurokawa and Wakayama [11] showed that, when the period [omega] is "rational", that is, there exists a positive number c satisfying [omega] [member of] c x [Q.sup.r], [Sin.sub.r](x, [omega]) satisfies the algebraic differential equation

F(x,y,y', ..., [y.sup.(n))] = 0

(n [member of] [Z.sub.[greater than or equal to]0],F(x,[Y.sub.0],[Y.sub.1], ..., [Y.sub.n]) [member of] C(x)[[Y.sub.0], [Y.sub.1], ..., [Y.sub.n]]).

In particular, when [omega] = (1, ..., 1), y = [Sin.sub.r](x, [omega]) satisfies the algebraic differential equation

y" + [([pi][Q.sub.r](x).sup.-1] - 1)[(y').sup.2][y.sup.-1] - [Q'.sub.r](x)[Q.sub.r][(x).sup.-1]y' + [pi][Q.sub.r](x)y = 0

with [mathematical expression not reproducible]. (See [10, Theorem 2.2(d)].) However, for a general period [omega], the differential algebraicity of the multiple sine function is still obscure.

On the other hand, in [8], we showed that the double cotangent function [Cot.sub.2](x, (1,[tau])) (the logarithmic derivative of the double sine function) degenerates to the digamma function (the logarithmic derivative of the gamma function) as [tau] tends to infinity. This proposition enables us to regard the double cotangent function as a generalization of the digamma function. Thus it is natural to ask whether properties of the digamma and gamma functions can be extended to the double cotangent and sine functions.

One of the important properties of the gamma function is its hypertranscendence: It does not satisfy any algebraic differential equation. This theorem was proved by Holder [6].

The purpose of this paper is, by generalizing this Holder's proof, to show the hypertranscendence of the multiple sine function for a "complex" period:

Theorem 0.1. Let r [greater than or equal to] 2. If there exists a non-real element in the set {[[omega].sub.j]/[[omega].sub.i]|1 [less than or equal to] i < j [less than or equal to] r}, then the r-ple sine function Sinr(x, [omega]) is hypertranscendental.

When all elements in the set {[[omega].sub.i]/[[omega].sub.j] | 1 [less than or equal to] i < j [less than or equal to] r} are positive real number and at least one element is irrational, it remains unclear whether or not the r-ple sine function Sinr(x, [omega]) is hypertranscendental. It may be possible that a totally different method from ours (for example, the Galois correspondence in differential Galois theory) provides a solution to this problem.

1. Hypertranscendence of a solution of a certain difference equation. In this section, by generalizing the argument of Holder [6], we establish the following general result, which will be used in the proof of Theorem 0.1.

Proposition 1.1. If a function f (x) satisfies the difference equation

(1.1) f (x + r)=f (x)[(2sin([pi]x)).sup.-1]

for a non-real constant [tau], then f (x) is hypertranscendental over C(x, [e.sup.[pi]ix]); that is, y = f (x) does not satisfy any algebraic differential equation over C(x,[e.sup.[pi]ix]), which is given by (1.2) F (x,y,y', ..., [y.sup.(n)]) = 0,

n [member of] [Z.sub.[greater than or equal to]0],

F (x, [Y.sub.0],[Y.sub.1], ..., [Y.sub.n]) [member of] C(x, [e.sup.[pi]ix])[[Y.sub.0],[Y.sub.1], ..., [Y.sub.n]].

To prove Proposition 1.1, we use the hypertranscendence criteria, established in the differential Galois theory. We will briefly recall this criteria and then, with the aid of it, prove Proposition 1.1. (For details on the hypertranscendence criteria, we refer to [5] and the references therein.)

To describe the criteria, we introduce some definitions. A ([phi],[delta])-ring (R, [phi], [delta]) is a ring R endowed with a ring automorphism [phi] and a derivation [delta] : R [right arrow] R (this means that [delta] is an additive map satisfying Leibniz rule [delta](ab) = [delta](a)b + a[delta](b) for all a,b [member of] R) such that [phi] commutes with [delta]. If R is a field, then (R, [phi], [delta]) is called a ([phi], [delta])-field.

Given a ([phi], [delta])-ring (R,[phi],[delta]), a [mathematical expression not reproducible] is a ([phi], [delta])-algebras if [??] is a ring extension of R, [mathematical expression not reproducible]; in this case, we will often denote [??] by [phi] and [delta] by [delta].

Let K be a ([phi], [delta])-field K. A [delta]-polynomial in the differential indeterminate y is a polynomial in the indeterminates {[[delta].sup.j]y | j [member of] [Z.sub.[greater than or equal to]0]} with coefficients in K. Let R be a K-([phi], [delta])-algebras and a [member of] R. If there exists a nonzero [delta]-polynomial P(y) in the differential indeterminate y such that P(a) = 0, then we say that a is hyperalgebraic over K.

The hypertranscendency criteria is as follows:

Proposition 1.2 ([5], Proposition 2.6). Let K be a ([phi], [delta])-field with k :={f [member of] K | [phi](f) = f} algebraically closed and let a [member of] [K.sup.x]. Let R be a K-([phi], [delta])-algebra and let v [member of] R \{0}. Assume that v is invertible in R. If [phi](v) = av and if [v] is hyperalgebraic over K, then there exists a nonzero linear homogeneous [delta]-polynomial L(y) and an element f [member of] K such that

L([delta](a)/a) = [phi](f) - f.

The converse is also true if [R.sup.[phi]] = k.

Remark 1.3. When

K = C(x), [phi](f (x)) = f (x + 1), [delta] = d/dx,

R = C(x,[GAMMA](x), [[GAMMA].sup.(1)](x), [[GAMMA].sup.(2)](x), ...),

the former part of Proposition 1.2 was proved by Holder [6].

We will prove Proposition 1.1. Suppose that y = f (x) satisfies the algebraic differential equation (1.2). Then, by applying Proposition 1.1 with

K = C(x, [e.sup.[pi]ix]), [phi](f (x)) = f (x + t), [delta] = d/dx,

R = C(x,[e.sup.[pi]ix], f, [f.sup.(1)], ...), v = f(x),

we find that there exist an integer n [greater than or equal to] 0, [A.sub.j] [member of] C not all zeros and R [member of] C(x, [e.sup.[pi]ix]) such that

(1.3) [n.summation over j=0] [A.sub.j] [[d.sup.j]/d[x.sup.j]] cot([pi]x) = R(x + [tau]) - R(x).

Since x = 0 is a pole of the left hand side of (1.3), at least one of R(x + [tau]) or R(x) also must have a pole at x = 0.

We cosider the case where R(x + [tau]) has a pole at x = 0. Then R(x) has a pole at x = [tau]. Since x = [tau] is not a pole of the left hand side of (1.3), R(x + [tau]) must have a pole at x = [tau]. Thus the function R(x) have a pole at x = 2[tau]. By repeating this process, it follows that the set of poles of R(x) contains {[tau], 2[tau], 3[tau], ...}. This contradicts the fact that imaginary parts of zeros and poles of an arbitrary elements of C(x,[e,sup,2[pi]ix]) are bounded.

Similarly, when the function R(x) has a pole at x = 0, we find that the set of poles of R(x) contains {0, -[tau], -2[tau], -3[tau], ...}, which also leads to a contradiction. Therefore we obtain the proposition.

2. Proof of Theorem 0.1. In this section, by applying Proposition 1.1, we prove Theorem 0.1. As a byproduct of Proposition 1.1, we also show that Appell's O-functions, introduced by Appell [1], is hypertranscendental.

We will use the following result, which is due to Ostrowski [12]:

Proposition 2.1 (Ostrowski [12]). Let [Mer.sup.DA] be the set of the meromorphic functions over C which satisfy algebraic differential equations. Then we have the following;

a) The set [Mer.sup.DA] is a field.

b) For elements f,g [member of] [Mer.sup.DA], the composition f [omicron] g belongs to [Mer.sup.DA].

We also recall the quasiperiodicity of the multiple sine function:

Proposition 2.2 ([9], Theorem 2.1 (a)). The multiple sine function satisfies the difference equation

Sinr(x + [[omega].sub.i], [omega]) = [Sin.sub.r](x,[omega])[Sin.sub.r- 1][(x,[omega](i)).sup.-1],

where we put [omega](i) = ([[omega].sub.1], ..., [[omega].sub.i-1], [[omega].sub.i+1], ..., [[omega].sub.r]) and [Sin.sub.0](x, *) = -1.

Proof of Theorem 0.1. We prove the theorem by induction on r. By Proposition 2.2 and Proposition 1.1, the theorem is obviously true for r = 2.

Suppose that the theorem is true for r and that [Sin.sub.r+1](x, ([[omega].sub.1], ..., [[omega].sub.r+1])) satisfies an algebraic differential equation. For simplicity, we put [omega] := ([[omega].sub.1], ..., [[omega].sub.r+1]). By the condition of the theorem, without loss of generality, we can assume that [[omega].sub.2]/[[omega].sub.1] is a non-real complex number. Proposition 2.2 gives

[Sin.sub.r+1] (x + [[omega].sub.r+1], [omega]) = [Sin.sub.r+1](x, [omega])[Sin.sub.r][(x, [omega](r + 1)).sup.-1],

which means that, by applying Proposition 2.1, Sinr(x, ([[omega].sub.1], ..., [[omega].sub.r])) also satisfies an algebraic differential equation. This contradicts the induction hypothesis. Thus we obtain the theorem.

Proposition 1.1 is also applicable to the proof of the hypertranscendence for Appell's O-function (also called q-Pochhammer symbol or q-shifted factorial). Appell's O-functions are defined as follows; Let r [greater than or equal to] 1 and let [omega] = ([[omega].sub.1], ..., [[omega].sub.r]) be a r-tuple consisting of complex numbers with positive imaginary part. We put

[mathematical expression not reproducible]

and define the functions [O.sub.r](x,[omega]) by

[mathematical expression not reproducible].

Theorem 2.3. The function [O.sub.r](x, [omega]) is hypertranscendental.

Proof. We put

f(x) = [O.sub.1](x, [[omega].sub.1]) exp([[pi]i/2] ([x.sup.2]/[[omega].sub.1] - (1 + 1/[[omega].sub.1])x)).

Then it is easy to see that f(x) satisfies the difference equation (1.1). Thus, by Proposition 2.1, the theorem is true for r = 1.

The remaining part of the proof is similar to that of Theorem 0.1, by observing that Appell's O-functions satisfy the following difference equations:

[mathematical expression not reproducible].

doi: 10.3792/pjaa.95.16

References

[1] P. Appell, Sur une classe de fonctions analogues aux fonctions Euleriennes, Math. Ann. 19 (1881), no. 1, 84 102. [2] E. W. Barnes, The genesis of the double gamma functions, Proc. Lond. Math. Soc. 31 (1899), 358 381.

[3] E. W. Barnes, The theory of the double gamma function, Philos. Trans. Roy. Soc. (A) 196 (1901), 265 388.

[4] E. W. Barnes, On the theory of the multiple gamma function, Trans. Cambridge Phil. Soc. 19 (1904), 374 425.

[5] T. Dreyfus, C. Hardouin, and J. Roques, Hypertranscendence of solutions of mahler equations, J. Eur. Math. Soc. (JEMS) 20 (2018), 2209-2238.

[6] O. Holder, Ueber die Eigenschaft der Gamma-function keiner algebraischen Differentialgleichung zu genugen, Math. Ann. 28 (1887), 1 13.

[7] M. Jimbo and T. Miwa, Quantum KZ equation with [absolute value of q] 1 and correlation functions of the XXZ model in the gapless regime, J. Phys. A 29 (1996), no. 12, 2923 2958.

[8] M. Kato, An addition type formula for the double cotangent function, Ph.D. Thesis, Kobe University (2017).

[9] N. Kurokawa and S. Koyama, Multiple sine functions, Forum Math. 15 (2003), no. 6, 839-876.

[10] S.-Y. Koyama and N. Kurokawa, Zeta functions and normalized multiple sine functions, Kodai Math. J. 28 (2005), no. 3, 534 550.

[11] N. Kurokawa and M. Wakayama, Differential algebraicity of multiple sine functions, Lett. Math. Phys. 71 (2005), no. 1, 75 82.

[12] A. Ostrowski, Uber Dirichletsche Reihen und algebraische Differentialgleichungen, Math. Z. 8 (1920), no. 3 4, 241 298.

[13] T. Shintani, On a Kronecker limit formula for real quadratic fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 24 (1977), no. 1, 167 199.

By Masaki KATO

Department of Mathematics, Graduate School of Science, Kobe University, 1-1 Rokkodai, Nada-ku, Kobe 657-8501, Japan

(Communicated by Masaki KASHIWARA, M.J.A., Jan. 15, 2019)

2010 Mathematics Subject Classification. Primary 33E30, 11J81, 11J91.
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