Open Access
May 2004 Derivatives of multiple sine functions
Nobushige Kurokawa
Proc. Japan Acad. Ser. A Math. Sci. 80(5): 65-69 (May 2004). DOI: 10.3792/pjaa.80.65
Abstract

We calculate derivatives of multiple sine functions to investigate coefficients appearing in the addition type formula. We present explicit expressions and we obtain an interesting modular form.

References

1.

Barnes, E. W.: On the theory of the multiple gamma function. Trans. Cambridge Philos. Soc., 19, 374–425 (1904). Barnes, E. W.: On the theory of the multiple gamma function. Trans. Cambridge Philos. Soc., 19, 374–425 (1904).

2.

Deninger, C.: Local $L$-factors of motives and regularized determinants. Invent. Math., 107, 135–150 (1992).  MR1135468 10.1007/BF01231885 Deninger, C.: Local $L$-factors of motives and regularized determinants. Invent. Math., 107, 135–150 (1992).  MR1135468 10.1007/BF01231885

3.

Jimbo, M., and Miwa, T.: Quantum $KZ$ equation with $|q|=1$ and correlation functions of the $XXZ$ model in the gapless regime. J. Phys. A, 29, 2923–2958 (1996).  MR1398600 10.1088/0305-4470/29/12/005 Jimbo, M., and Miwa, T.: Quantum $KZ$ equation with $|q|=1$ and correlation functions of the $XXZ$ model in the gapless regime. J. Phys. A, 29, 2923–2958 (1996).  MR1398600 10.1088/0305-4470/29/12/005

4.

Kurokawa, N.: Multiple sine functions and Selberg zeta functions. Proc. Japan Acad., 67A, 61–64 (1991).  MR1105522 10.3792/pjaa.67.61 euclid.pja/1195512182 Kurokawa, N.: Multiple sine functions and Selberg zeta functions. Proc. Japan Acad., 67A, 61–64 (1991).  MR1105522 10.3792/pjaa.67.61 euclid.pja/1195512182

5.

Kurokawa, N.: Multiple zeta functions: an example. Adv. Stud. Pure Math., 21, 219–226 (1992).  MR1210791 Kurokawa, N.: Multiple zeta functions: an example. Adv. Stud. Pure Math., 21, 219–226 (1992).  MR1210791

6.

Kurokawa, N.: Gamma factors and Plancherel measures. Proc. Japan Acad., 68A, 256–260 (1992).  MR1202627 10.3792/pjaa.68.256 euclid.pja/1195511631 Kurokawa, N.: Gamma factors and Plancherel measures. Proc. Japan Acad., 68A, 256–260 (1992).  MR1202627 10.3792/pjaa.68.256 euclid.pja/1195511631

7.

Kurokawa, N.: On the generalization of the sine function. Technical Rep. Tsuda Univ., 4, 1–25, (1992). (In Japanese). Kurokawa, N.: On the generalization of the sine function. Technical Rep. Tsuda Univ., 4, 1–25, (1992). (In Japanese).

8.

Kurokawa, N., and Koyama, S.: Multiple sine functions. Forum Math., 15, 839–876 (2003).  MR2010282 10.1515/form.2003.042 Kurokawa, N., and Koyama, S.: Multiple sine functions. Forum Math., 15, 839–876 (2003).  MR2010282 10.1515/form.2003.042

9.

Koyama, S., and Kurokawa, N.: Kummer's formula for multiple gamma functions. J. Ramanujan Math. Soc., 18, 87–107 (2003).  MR1966530 Koyama, S., and Kurokawa, N.: Kummer's formula for multiple gamma functions. J. Ramanujan Math. Soc., 18, 87–107 (2003).  MR1966530

10.

Kurokawa, N., Ochiai, H., and Wakayama, M.: Multiple trigonometry and zeta functions. J. Ramanujan Math. Soc., 17, 101–113 (2002).  MR1913896 Kurokawa, N., Ochiai, H., and Wakayama, M.: Multiple trigonometry and zeta functions. J. Ramanujan Math. Soc., 17, 101–113 (2002).  MR1913896

11.

Kurokawa, N., and Wakayama, M.: On $\zeta(3)$. J. Ramanujan Math. Soc., 16, 205–214 (2001).  MR1863604 Kurokawa, N., and Wakayama, M.: On $\zeta(3)$. J. Ramanujan Math. Soc., 16, 205–214 (2001).  MR1863604

12.

Kurokawa, N., and Wakayama, M.: Absolute tensor products. Int. Math. Res. Not., 2004-5, 249–260 (2004).  MR2038715 10.1155/S1073792804132327 Kurokawa, N., and Wakayama, M.: Absolute tensor products. Int. Math. Res. Not., 2004-5, 249–260 (2004).  MR2038715 10.1155/S1073792804132327

13.

Manin, Y.: Lectures on zeta functions and motives (according to Deninger and Kurokawa). Astérisque, 228, 121–163 (1995).  MR1330931 Manin, Y.: Lectures on zeta functions and motives (according to Deninger and Kurokawa). Astérisque, 228, 121–163 (1995).  MR1330931

14.

Shintani, T.: On a Kronecker limit formula for real quadrtic fields. J. Fac. Sci. Univ. Tokyo Sect. IA Math., 24, 167–199 (1977).  MR460283 Shintani, T.: On a Kronecker limit formula for real quadrtic fields. J. Fac. Sci. Univ. Tokyo Sect. IA Math., 24, 167–199 (1977).  MR460283
Copyright © 2004 The Japan Academy
Nobushige Kurokawa "Derivatives of multiple sine functions," Proceedings of the Japan Academy, Series A, Mathematical Sciences 80(5), 65-69, (May 2004). https://doi.org/10.3792/pjaa.80.65
Published: May 2004
Vol.80 • No. 5 • May 2004
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