Complex Difference Equations of Malmquist Type

Transcription

1 Computational Methods and Function Theory Volume ), No. 1, Complex Difference Equations of Malmquist Type Janne Heittokangas, Risto Korhonen, Ilpo Laine, Jarkko Rieppo, and Kazuya Tohge Abstract. In a recent paper [1], Ablowitz, Halburd and Herbst applied Nevanlinna theory to prove some results on complex difference equations reminiscent of the classical Malmquist theorem in complex differential equations. A typical example of their results tells us that if a complex difference equation yz + 1) + yz 1) = Rz, y) with Rz, y) rational in both arguments admits a transcendental meromorphic solution of finite order, then deg y Rz, y) 2. Improvements and extensions of such results are presented in this paper. In addition to order considerations, a result see Theorem 13) is proved to indicate that solutions having Borel exceptional zeros and poles seem to appear in special situations only. Keywords. Complex difference equation, value distribution, Nevanlinna characteristic, Borel exceptional values MSC. Primary 39A10; Secondary 30D35, 39A Introduction The celebrated Malmquist theorem states that a complex differential equation of the form 1) y = Rz, y), where the right-hand side is rational in both arguments, and which admits a transcendental meromorphic solution y in the complex plane, reduces into a Riccati differential equation 2) y = az) + bz)y + cz)y 2 with rational coefficients. For more details concerning the equations 1) and 2), as well as for generalizations of the Malmquist theorem, see [8]. Received June 20, J.H., R.K., I.L., and J.R. have been partially supported by the INTAS project grant K.T. has been partially supported by the Academy of Finland and the JSPS the Grants-in-Aid for Scientific Research System, No ). ISSN /$ 2.50 c 2001 Heldermann Verlag

2 28 J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo, and K. Tohge CMFT In this paper we are studying growth and value distribution of meromorphic solutions of some complex difference equations. These difference equations arise from reasoning of Malmquist type, see [1] and a more detailed explanation below. In what follows, a meromorphic function means meromorphic in the complex plane C. For a meromorphic function y, let ρy) be the order of growth and µy) be the lower order of y. Further, let λy) resp. λ 1/y)) be the exponent of convergence of the zeros resp. poles) of y. We also assume that the reader is familiar with the fundamental results and standard notations of the Nevanlinna theory, see e.g. [6], [7] or [8]. Moreover, we use the notation d := deg y Rz, y) for the degree of Rz, y) with respect to y, where Rz, y) is rational in y with meromorphic coefficients. We always assume that Rz, y) is irreducible in y. We first recall some existence results for solutions meromorphic in the complex plane. The following two such results have been proved by S. Shimomura [9, Theorem 2.5] and N. Yanagihara [10, Corollary 6], respectively. Theorem 1 Shimomura). For any non-constant polynomial P y), the difference equation 3) yz + 1) = P yz)) has a non-trivial entire solution. Theorem 2 Yanagihara). For any non-constant rational function Ry), the difference equation 4) yz + 1) = Ryz)) has a non-trivial meromorphic solution in the complex plane. The equation 4) above is closely connected with the Schröder equation 5) fct) = Rft)), c C, c > 1. In fact, let f be a transcendental meromorphic solution of 5). Then, by the change of variable t = e z log c, the function yz) = ft) is a solution of 4). It was shown in [5, Theorem 3.2] that ρf) = log d)/log c ), where d is the degree of R and R is supposed to be irreducible in f). Hence ρy) =, provided that d 2. The Schröder equation is closely connected to complex dynamics, see [3]. Suppose now that y is a meromorphic solution of 4). Let Πz) be any periodic entire function of period 1. For example, Πz) = e 2πiz. Define w 1 z) := yπz) + z). Then w 1 z + 1) = yπz) + z + 1). Denoting t := Πz) + z, we see that w 1 z + 1) = yt + 1) = Ryt)) = Rw 1 z)), that is, w 1 z) satisfies 4) too. However, the growth of w 1 z) is much faster than the growth of yz), see e.g. [4]. Furthermore, by continuing the process above, we obtain a sequence of functions w n ) each solving 4). Moreover, w n+1 grows

3 1 2001), No. 1 Complex Difference Equations of Malmquist Type 29 faster than w n for each n. Hence, no concrete upper bound can be obtained for the characteristic function of the solutions of 4). This has been observed in a recent paper [1] by M. Ablowitz, R. Halburd and B. Herbst, the main results of which will be discussed below. An example of a complex difference equation combining existence and growth restriction has been offered by S. Bank and R. Kaufman in [2, Proposition 16]: Theorem 3 Bank-Kaufman). For any rational function Rz), the difference equation yz + 1) yz) = Rz) always has a meromorphic solution y such that T r, y) = Or). Ablowitz, Halburd and Herbst [1] studied complex difference equations related to 1) and 2), namely the equations 6) yz + 1) + yz 1) = a 0z) + a 1 z)y + + a p z)y p b 0 z) + b 1 z)y + + b q z)y q and 7) yz + 1) + yz 1) = az) + bz)y + cz)y 2, where the coefficients are meromorphic functions to be specified later on. Also, the equation 8) yz + 1)yz 1) = a 0z) + a 1 z)y + + a p z)y p b 0 z) + b 1 z)y + + b q z)y q, which is similar to 6), was studied in [1]. The difference equations 6) and 8) are closely related to Painlevé differential equations, as noted in [1]. The following three results, reminiscent of the classical Malmquist theorem, were proved in [1], see Theorems 3 5, respectively. Theorem 4 Ablowitz-Halburd-Herbst). If the difference equation 6) with polynomial coefficients a i z), b i z) admits a transcendental meromorphic solution of finite order, then d = max{p, q} 2. Theorem 5 Ablowitz-Halburd-Herbst). Suppose that the coefficients az), bz) in the difference equation 7) are polynomials and that cz) is a non-zero complex constant. Then any transcendental entire solution of 7) is of infinite order. Theorem 6 Ablowitz-Halburd-Herbst). If the difference equation 8) with polynomial coefficients a i z), b i z) admits a transcendental meromorphic solution of finite order, then d = max{p, q} 2. We now pose a list of questions related to these theorems. These questions will be considered in the remaining part of this paper. 1. What happens if some of the coefficients of the equations 6) 8) are meromorphic functions other than polynomials?

4 30 J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo, and K. Tohge CMFT 2. What if we replace 1 and 1 in the equations 6) 8) by arbitrary complex constants? 3. What happens if the left-hand sides of the equations 6) 8) are generalized to have n terms? 4. Is it possible to find any lower bound for the characteristic functions of transcendental meromorphic solutions of equations 6) 8)? 5. Is it possible to say something about the distribution of zeros, resp. poles, of meromorphic solutions of 6) 8)? As a previous result related to Question 4 above, we mention the following result due to N. Yanagihara [10, Theorem 1]. Theorem 7 Yanagihara). Any transcendental meromorphic solution y of 4) is of infinite order, unless d 1. More precisely, T r, y) Kd1 ε)) r, where ε > 0 is arbitrary and K > 0 is a constant not depending on ε. This paper has been organized as follows. As hinted above, the essential growth problem for meromorphic solutions of complex difference equations is to find a lower bound for their characteristic functions. Section 2 offers partial results in this direction concerning the difference equations 6) and 7), generalized as anticipated in questions 1, 2 and 3 above. Section 3 is devoted to considering a generalized form of the difference equation 8). More precisely, we show that in special cases only, it may happen that zeros and poles are Borel exceptional values of a meromorphic solution. 2. Growth of meromorphic solutions We begin by giving two remarks concerning Theorem 4. Remarks. 1) In the original statement of Theorem 4, see [1], the equality d = 2 was asserted. However, the case d = 1 may occur. In fact, yz) = z + e z solves yz + 1) + yz 1) = 2 e 1 ) z + e + 1 ) yz). e e A similar observation holds for Theorem 6. 2) Theorem 4 obviously extends to rational coefficients. It is easy to obtain the following two propositions. Proposition 8. Let c 1,..., c n C \ {0}. If the difference equation 9) yz + c i ) = a 0z) + a 1 z)y + + a p z)y p b 0 z) + b 1 z)y + + b q z)y q with rational coefficients a i z), b i z) admits a transcendental meromorphic solution of finite order, then d n.

5 1 2001), No. 1 Complex Difference Equations of Malmquist Type 31 Proposition 9. Let c 1,..., c n C \ {0}. If the difference equation n 10) yz + c i ) = a 0z) + a 1 z)y + + a p z)y p b 0 z) + b 1 z)y + + b q z)y q with rational coefficients a i z), b i z) admits a transcendental meromorphic solution of finite order, then d n. The proofs of Proposition 8 and Proposition 9 are straightforward adaptations of the proof of Theorem 4 in [1] and hence omitted. The following example shows that the case d = n may occur in both of the two results above. See also Examples 3 5 concerning the equation 10). Example 1. Let c C be a constant such that c π m, where m Z. Since 2 we see that yz) = tan z solves tanz + c) = 11) yz + c) = 1 C where C := tan c 0,. tan z + tan c 1 tan z tan c, yz) C yz) + 1/C, As a generalization of Theorems 5 and 7, we prove Theorem 10. Let c 1,..., c n C \ {0} and let m 2. Suppose y is a transcendental meromorphic solution of the difference equation m 12) a i z)yz + c i ) = b i z)yz) i with rational coefficients a i z), b i z). Denote C := max{ c 1,..., c n }. 1) If y is entire or has finitely many poles, then there exist constants K > 0 and r 0 > 0 such that log Mr, y) Km r/c holds for all r r 0. 2) If y has infinitely many poles, then there exist constants K > 0 and r 0 > 0 such that nr, y) Km r/c holds for all r r 0. Remark. Observe that the assertion is independent of the number n N of terms on the left-hand side of 12). i=0

6 32 J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo, and K. Tohge CMFT Proof. We multiply out the denominators of the coefficients a i z), b i z) in 12) to obtain m 13) P i z)yz + c i ) = Q i z)yz) i, where the coefficients P i z), Q i z) are polynomials. The proof is now divided into two parts. 1) Suppose that y, the solution of 13) and 12), is transcendental entire. Furthermore, denote p i = deg P i and q i = deg Q i. The maximum modulus principle yields M r + C, y z)) M r, y z + c i )) for all i = 1,..., n. By choosing s > max{p 1,..., p n }, it follows that ) ) m M r, Q i z) y z) i = M r, P i z) y z + c i ) i=0 when r is large enough. Furthermore Thus M M r, r, m 1 i=0 i=0 r s M r, ) y z + c i ) nr s M r + C, y z)), Q i z) y z) i ) = o M r, y z) m )). ) m Q i z) y z) i 1 2 M r, Q mz)y z)) m, i=0 when r is sufficiently large. Hence 14) log M r + C, y z)) m log M r, y z)) + g r), where g r) < K log r for some K > 0, when r is large enough. By iterating 14), we have 15) log M r + jc, y z)) m j log M r, y z)) + E j r), where E j r) = m j 1 g r) + m j 2 g r + C) + + g r + j 1) C) j 1 Km j 1 k=0 log r + kc) m k Km j 1 k=0 log r + kc) m k. Since log r + kc) log r) log kc) for r and k sufficiently large, we observe that the series above converges whenever m 2. Hence 16) E j r) K m j log r.

7 1 2001), No. 1 Complex Difference Equations of Malmquist Type 33 Since, by the hypothesis, y is transcendental entire, we have the inequality log Mr, y) 2K log r for r large enough. Thus 15) and 16) imply 17) log Mr + jc, yz)) K m j log r, which holds for r sufficiently large, say r r 0. By choosing r [r 0, r 0 + C) arbitrarily and letting j for each choice of r, we see that log Ms, yz)) K m s/c holds for all s s 0 := r 0 + C, where K := K m r0+c)/c log r 0. We have proved the assertion in the case of y being entire. Suppose then that y, the solution of 13) and 12), is meromorphic with finitely many poles. Then there exists a polynomial Sz) such that wz) = Sz)yz) is entire. Substituting yz) = wz)/sz) into 13) and again multiplying away the denominators, we will obtain a difference equation with polynomial coefficients similar to 13). Applying the reasoning above to w, we obtain, by the growth properties of polynomials, that log Mr, y) = log Mr, w) + log M r, 1 ) S K ε)m r/c = K m r/c, which holds for all r r 1 r 0. We have now proved Part 1). 2) Finally we suppose that y, the solution of 13) and 12), is meromorphic with infinitely many poles. Choose a pole z 0 of y having multiplicity τ 1 such that z 0 is not a zero of Q m z). Then the right-hand side of 13) has a pole of multiplicity mτ at z 0. Hence, there exists at least one index l 1 {1,..., n} such that z 0 + c l1 is a pole of y of multiplicity ν 1 mτ. Substitute z 0 + c l1 for z in 13) to obtain m 18) P i z 0 + c l1 )yz 0 + c l1 + c i ) = Q i z 0 + c l1 )yz 0 + c l1 ) i. We now have two possibilities: i) If z 0 + c l1 is a zero of Q m z), this process will be terminated and we have to choose another pole z 0 of y in the way we did above. ii) If z 0 + c l1 is not a zero of Q m z), then we see that the right-hand side of 18) has a pole of multiplicity mν 1 at z 0 + c l1. Hence, there exists at least one index l 2 {1,..., n} such that z 0 +c l1 +c l2 is a pole of y of multiplicity ν 2 mν 1 m 2 τ. At this point we note that, as a polynomial, the coefficient Q m z) has finitely many zeros only that can terminate our process above), all being inside of a finite disc z < R. We proceed to follow the steps i) and ii) above. Since there are infinitely many poles of y, we will find a pole z 0 of y such that i=0 z 0 + c l1 + + c lk =: ζ k

8 34 J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo, and K. Tohge CMFT is a pole of y of multiplicity ν k for all k N. Since ν k m k τ, as k, and since y does not have essential singularities in the finite plane, we must have ζ k, as k. It is clear that, for k large enough, say k k 0, τm k τ1 + m + + m k ) n ζ k, y) n z 0 + kc, y) nt + kc, y), where t [ z 0, z 0 + C) can be chosen arbitrarily. Letting k for each choice of t, we see that nr, y) Km r/c holds for all r r 0 := k 0 + 1)C + z 0, where K := τm z 0 +C)/C. The fact that r 0 and K both depend on z 0 is not a problem, since z 0 is fixed. Remarks. 1) For any rational function yz) it is easy to find rational coefficients a i z), b i z) such that 12) holds. 2) It is easily seen that all transcendental meromorphic solutions of 12) have infinite lower order. 3) The assumption m 2 in Theorem 10 is essential. For Part 1), see Remark 1) in the beginning of this section. For Part 2), we note that the gamma function Γ solves yz + 1) + yz 1) = z + 1 ) yz) z 1 and ργ) = 1. 4) Suppose that n = 2 in 12). All non-constant periodic functions having two primitive periods c 1 and c 2 are elliptic and hence cannot be entire. This means that the reasoning used after Theorem 2 does not apply in this case. For n 3 we do not have a non-constant periodic function meromorphic or entire) π : C Ĉ having points c 1,..., c n as periods, where c 3,..., c n are not lattice points of any periodic parallelogram formed by c 1 and c 2. Example 2. Fix n = m N \ {1}. Let c i C be constants such that e c i = i for all i = 1,..., n. Then yz) = e ez /z solves z + c i )yz + c i ) = z i yz) i. This is an example of Theorem 10, Part 1). As a result similar to Theorem 10, corresponding to Theorems 4 and 7, we now assume that the coefficients are small functions, i.e. of growth ot r, y)). Theorem 11. Let c 1,..., c n C \ {0} and suppose that y is a non-rational meromorphic solution of 19) d i z)yz + c i ) = a 0z) + a 1 z)y + + a p z)y p b 0 z) + b 1 z)y + + b q z)y, q

9 1 2001), No. 1 Complex Difference Equations of Malmquist Type 35 where all coefficients in 19) are of growth ot r, y)) without an exceptional set, as r, and d i s are non-vanishing. If d = max{p, q} > n, then for any ε, 0 < ε < d n)/d + n), there exists an r 0 > 0 such that )) r/c d 1 ε T r, y) K n 1 + ε for all r r 0, where C := max{ c 1,..., c n } and K > 0 is a constant. Proof. Let 0 < ε < d n)/d + n). Applying [1, Lemma 1] we see that T r, yz + c i )) 1 + ε)t r + C, y) + M for all r 1/ε and i = 1,..., n, where M = Mε) > 0 is some constant. Keeping this in mind, we then apply the Valiron-Mokhon ko theorem see [8, Theorem and Corollary 2.2.7]) on the right-hand side of 19) to obtain 20) T r + C, y) d n 1 ε 1 + ε ) T r, y), which holds for r r 0 1/ε. Observe that Valiron-Mokhon ko theorem may be applied without an exceptional set, see [5]. Iterating 20), we get )) j d 1 ε T r + jc, y) T r, y), n 1 + ε which holds for r r 0 and j N. [r 0, r 0 + C), we obtain d 1 ε T s, y) n 1 + ε )) s r0 C)/C Letting j for each choice of r T r 0, y) K d n which holds for all s r 0 + C and for some K = Kε) > 0. )) s/c 1 ε, 1 + ε Remarks. 1) The condition ε < d n)/d + n) in Theorem 11 is needed to obtain the inequality d1 ε))/n1 + ε))) > 1. 2) Under the assumptions of Theorem 11 it is easy to see that µy) =. 3) By an appropriate modification of the assumption on d > n, we may replace the left-hand side of 19) by a more general rational expression of yz), yz + c 1 ),..., yz + c n ) with meromorphic coefficients of the growth ot r, y)). Developing Theorem 11 further, we next suppose that the coefficients are of growth ot r, y)) outside an exceptional set of finite linear measure, that is, of growth Sr, y). Theorem 12. Suppose that all coefficients in 19) are of growth Sr, y) and that all the other assumptions of Theorem 11 hold. Then µy) =.

10 36 J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo, and K. Tohge CMFT Proof. Let 0 < ε < d n)/d+n). Following the proof of Theorem 11, we apply the Valiron-Mokhon ko theorem on the right-hand side of 19) and [1, Lemma 1] on the left-hand side of 19). In addition, we need to apply [8, Lemma 1.1] to deal with the exceptional sets. We conclude that, for every σ > 1 there exists an r 0 > 0 such that T σ 2 r + C, y) d ) 1 ε T r, y) =: γt r, y) n 1 + ε and T r, y) > 1 holds for all r r 0, where γ > 1. Inductively, for any j N, we have ) 21) T σ 2j r + σ2j 1 σ 2 1 C, y γ j T r, y) for all r r 0. Let r [r 0, σ 2 r 0 + C) and denote s = sj, r) = σ 2j r + σ2j 1 σ 2 1 C. We note that s = sj, r) runs through each value in [σ 2 r 0 +C, ), provided j, r) covers N [r 0, σ 2 r 0 + C). Now, ) 1 sσ 2 j = 2 log σ log 1) + C rσ 2 1) + C ) 1 2 log σ log sσ 2 1) + C σ 2 r 0 + C)σ 2 1) + C )) 1 log s log σ 2 r 0 + σ2 2 log σ σ 2 1 C. Hence, by 21), for all s σ 2 r 0 + C we get log s log log T s, y) log γ 2 log σ σ 2 r 0 + σ2 σ 2 1 C )) + log T r 0, y). Dividing this by log s and letting s in fact, j ), we obtain Finally, let σ 1+, and we are done. µy) log γ 2 log σ.

11 1 2001), No. 1 Complex Difference Equations of Malmquist Type Meromorphic solutions with few zeros and poles We now proceed to consider the distribution of zeros and poles of solutions of the equation 10). The following result tells us that solutions having Borel exceptional zeros and poles appear in special situations only. Theorem 13. Let c 1,..., c n C \ {0} and suppose that y is a non-rational meromorphic solution of n 22) yz + c i ) = a 0z) + a 1 z)y + + a p z)y p b 0 z) + b 1 z)y + + b q z)y q with meromorphic coefficients a i z), b i z) of growth Sr, y) such that a p z)b q z) 0. If 23) max λy), λ1/y)) < ρy), then 22) is of the form 24) n yz + c i ) = cz)yz) k, where cz) is meromorphic, T r, c) = Sr, y) and k Z. Proof. Denote hz) = n yz + c i). Fix constants β and γ such that max λy), λ1/y)) < β < γ < ρy). Using 23) and the lemma of the logarithmic derivative, we get Similarly, T r, y /y) = Nr, y) + Nr, 1/y) + Sr, y) = Or β ) + Sr, y). T r, h /h) = Nr, h /h) + mr, h /h) nnr + C, yz)) + nnr + C, 1/yz)) + Sr, h) = Or β ) + Sr, y), where C := max{ c 1,..., c n }. Here we have applied the Valiron-Mokhon ko theorem to the equation 10) to conclude that T r, h) = dt r, y) + Sr, y) and so Sr, h) = Sr, y). Since zeros and poles are Borel exceptional by 23), we may apply a result due to Whittaker, see [7, Satz 13.4], to deduce that y is of regular growth. Hence there exists r 0 > 0 such that T r, y) > r γ for r r 0. It follows that T r, y /y) = Sr, y) and T r, h /h) = Sr, y). Rewriting 22) in the form 25) b q z) P z, y) hz) = a p z) Qz, y) = uz, y),

12 38 J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo, and K. Tohge CMFT we may suppose that P and Q are monic polynomials in y with coefficients of growth Sr, y). Denote Y := y /y, U := u /u and observe that T r, U) = Sr, y) by 25). Since P Q P Q = u = Uu = UP Q 2 Q, we get 26) P Q P Q = UP Q. Writing y = Y y in 26), regarding then 26) as an algebraic equation in y with coefficients of growth Sr, y) and comparing the leading coefficients, we obtain p q)y = U. Therefore, uz) = αyz) p q for some α C, and so proving the assertion. Example 3. We observe that hz) = α a pz) b q z) yz)p q n tanz + c i ) is a rational function in tan z not being of the form 24). Since λtan z) = λ1/ tan z) = ρtan z) = 1, the condition 23) in Theorem 13 is necessary. Example 4. Condition 23) in Theorem 13 cannot be replaced by since yz) = sin z satisfies min{λy), λ1/y)} < ρy), yz 1)yz + 1) = yz) 2 sin 2 1. Example 5. This example shows that there really exist equations of the form 24) having meromorphic solutions. Let A C \ {0} and k Z. Fix constants α, β C satisfying Then the difference equation α k+2 = A and β + 1 β = k. yz + 1)yz 1) = A yz) k, which is clearly of the form 24), has an entire solution yz) = α exp Πz)e z log β). Here Πz) is any periodic entire function of period 1.

13 1 2001), No. 1 Complex Difference Equations of Malmquist Type 39 Acknowledgement. We would like to thank the referee for proposing the idea of the present proof of Theorem 13. In addition, the last author would like to thank the Department of Mathematics for its warm hospitality during his research visit at the University of Joensuu. References 1. M. J. Ablowitz, R. Halburd, and B. Herbst, On the extension of the Painlevé property to difference equations, Nonlinearity ), S. Bank and R. Kaufman, An extension of Hölder s theorem concerning the gamma function, Funkcialaj Ekvacioj ), L. Carleson and T. Gamelin, Complex Dynamics, Springer-Verlag, New York, J. Clunie, The composition of entire and meromorphic functions, 1970 Mathematical Essays Dedicated to A. J. Macintyre, Ohio University Press, Athens, Ohio, G. Gundersen, J. Heittokangas, I. Laine, J. Rieppo and D. Yang, Meromorphic solutions of generalized Schröder equations, Aequationes Math ), W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, G. Jank and L. Volkmann, Einführung in die Theorie der ganzen und meromorphen Funktionen mit Anwendungen auf Differentialgleichungen, Birkhäuser Verlag, Basel-Boston, I. Laine, Nevanlinna Theory and Complex Differential Equations, Walter de Gruyter, Berlin, S. Shimomura, Entire solutions of a polynomial difference equation, J. Fac. Sci. Univ. Tokyo Sect. IA Math ), N. Yanagihara, Meromorphic solutions of some difference equations, Funkcialaj Ekvacioj ), Janne Heittokangas heittoka@cc.joensuu.fi Address: University of Joensuu, Department of Mathematics, P. O. Box 111, FIN Joensuu, Finland Risto Korhonen rkorhone@cc.joensuu.fi Address: University of Joensuu, Department of Mathematics, P. O. Box 111, FIN Joensuu, Finland Ilpo Laine Ilpo.Laine@joensuu.fi Address: University of Joensuu, Department of Mathematics, P. O. Box 111, FIN Joensuu, Finland Jarkko Rieppo jrieppo@pelu.jns.fi Address: University of Joensuu, Department of Mathematics, P. O. Box 111, FIN Joensuu, Finland Kazuya Tohge tohge@t.kanazawa-u.ac.jp Address: Kanazawa University, Faculty of Technology, Kodatsuno, Kanazawa , Japan