| 1. |
Masanobu Kaneko, Yoshinori Mizuno, Genus character L-functions of quadratic orders and class numbers, Journal of the London Mathematical Society, 10.1112/jlms.12313, 102, 1, 69-98, 2020.08, An explicit form of genus character (Formula presented.) -functions of quadratic orders is presented in full generality. As an application, we generalize a formula due to Hirzebruch and Zagier on the class number of imaginary quadratic fields expressed in term of the continued fraction expansion.. |
| 2. |
Masanobu Kaneko, Hirofumi Tsumura, MULTI-POLY-BERNOULLI NUMBERS AND RELATED ZETA FUNCTIONS, Nagoya Mathematical Journal, 10.1017/nmj.2017.16, 232, 19-54, 2018.12, We construct and study a certain zeta function which interpolates multi-poly-Bernoulli numbers at nonpositive integers and whose values at positive integers are linear combinations of multiple zeta values. This function can be regarded as the one to be paired up with the ξ-function defined by Arakawa and Kaneko. We show that both are closely related to the multiple zeta functions. Further we define multi-indexed poly-Bernoulli numbers, and generalize the duality formulas for poly-Bernoulli numbers by introducing more general zeta functions.. |
| 3. |
金子昌信, 山本修司, A new integral–series identity of multiple zeta values and regularizations, Selecta Mathematica New Series, 10.1007/s00029-018-0400-8, 24, 3, 2499-2521, 2018.07. |
| 4. |
M. Kaneko, K. Nagatomo, and Y. Sakai, The third order modular linear differential equations, Journal of Algebra, 485, 1, 332-352, 2017.02. |
| 5. |
Masanobu Kaneko, Mika Sakata, ON MULTIPLE ZETA VALUES OF EXTREMAL HEIGHT, BULLETIN OF THE AUSTRALIAN MATHEMATICAL SOCIETY, 10.1017/S0004972715001227, 93, 2, 186-193, 2016.04, We give three identities involving multiple zeta values of height one and of maximal height: an explicit formula for the height-one multiple zeta values, a regularised sum formula and a sum formula for the multiple zeta values of maximal height.. |
| 6. |
Masanobu Kaneko, Koji Tasaka, Double zeta values, double Eisenstein series, and modular forms of level 2, MATHEMATISCHE ANNALEN, 10.1007/s00208-013-0930-5, 357, 3, 1091-1118, 2013.11, We study the double shuffle relations satisfied by the double zeta values of level 2, and introduce the double Eisenstein series of level 2 which satisfy the double shuffle relations. We connect the double Eisenstein series to modular forms of level 2.. |
| 7. |
Kaneko Masanobu, Sakai Yuichi, THE RAMANUJAN-SERRE DIFFERENTIAL OPERATORS AND CERTAIN ELLIPTIC CURVES, PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 10.1090/S0002-9939-2013-11917-9, 141, 10, 3421-3429, 2013.10, For several congruence subgroups of low levels and their conjugates, we derive differential equations satisfied by the Eisenstein series of weight 4 and relate them to elliptic curves, whose associated new forms of weight 2 constitute the list of Martin and Ono of new forms given by eta-products/quotients.. |
| 8. |
Kaneko Masanobu, Nagatomo Kiyokazu, Sakai Yuichi, Modular Forms and Second Order Ordinary Differential Equations: Applications to Vertex Operator Algebras, LETTERS IN MATHEMATICAL PHYSICS, 10.1007/s11005-012-0602-5, 103, 4, 439-453, 2013.04, We study the relation between the Kaneko–Zagier equation and the Mathur–Mukhi–Sen classification, and extend it to the case of solutions with logarithmic terms, which correspond to pseudo-characters of non-rational vertex operator algebras. As an application, we prove a non-existence theorem of rational vertex operator algebras.. |
| 9. |
Yutaro Honda, Masanobu Kaneko, ON FOURIER COEFFICIENTS OF SOME MEROMORPHIC MODULAR FORMS, BULLETIN OF THE KOREAN MATHEMATICAL SOCIETY, 10.4134/BKMS.2012.49.6.1349, 49, 6, 1349-1357, 2012.11, We prove a congruence modulo a prime of Fourier coefficients of several meromorphic modular forms of low weights. We prove the result by establishing a generalization of a theorem of Garthwaite.. |
| 10. |
Hiroshi Yoshida, Yoshihiro Miwa, Masanobu Kaneko, Elliptic curves and Fibonacci numbers arising from Lindenmayer system with symbolic computation, APPLICABLE ALGEBRA IN ENGINEERING COMMUNICATION AND COMPUTING, 10.1007/s00200-011-0143-7, 22, 2, 147-164, 2011.03, Starting from an egg, the multicell becomes a set of cells comprising a variety of types to serve functions. This phenomenon brings us a bio-motivated Lindenmayer system. To investigate conditions for a variety of cell types, we have constructed a stochastic model over Lindenmayer systems. This model considers interactive behaviors among cells, yielding complicated polynomials. Using symbolic computation, we have derived explicit relations between cell-type diversity and cell-type ratio constraint. These relations exhibit elliptic curve-and Fibonacci number-related patterns. This is the first example of elliptic curves to appear in the Lindenmayer context. A survey of the rational points and the quadratic irrational numbers on the derived curves has revealed Fibonacci-related periodic and quasiperiodic patterns. Further we have found that in some region, there are only two elliptic curve-related periodic patterns.. |
| 11. |
Masanobu Kaneko, Poly-Bernoulli numbers and related zeta functions, MSJ Memoir, 21, 73--85, 2010.02. |
| 12. |
KANEKO Masanobu, OBSERVATIONS ON THE 'VALUES' OF THE ELLIPTIC MODULAR FUNCTION j(τ) AT REAL QUADRATICS, Kyushu Journal of Mathematics, 10.2206/kyushujm.63.353, 63, 2, 353-364, 2009.10, We define 'values' of the elliptic modular j-function at real quadratic irrationalities by using Hecke's hyperbolic Fourier expansions, and present some observations based on numerical experiments.. |
| 13. |
M. Kaneko, On an extension of the derivation relation for multiple zeta values, The Conference on $L$-functions, (L. Weng and M. Kaneko eds.), 89--94, 2007.01. |
| 14. |
Masanobu Kaneko, Masao Koike, On extremal quasimodular forms, KYUSHU JOURNAL OF MATHEMATICS, 60, 2, 457-470, vol. 60-2, 457--470, 2006.09, We define and study 'extremal' quasimodular forms. Some explicit descriptions of such forms are given. Connections with certain differential equations and Atkin's orthogonal polynomials, and the positivity of the Fourier coefficients, are also discussed..  |
| 15. |
H. Gangl, M. Kaneko and D. Zagier, Double zeta values and modular forms, Proceedings of the conference in memory of Tsuneo Arakawa, 71--106, 2006.07. |
| 16. |
K. Ihara, M. Kaneko and D. Zagier, Derivation and double shuffle relations for multiple zeta values, Compositio Math., vol. 142-02, 307--338, 2006.04. |
| 17. |
M. Kaneko, On the local factor of the zeta function of quadratic orders, Zeta functions, Topology, and Quantum Physics, Developments in Mathematics, 14, 75-79, Vol.14, 75--79, 2005.04. |
| 18. |
Arakawa Tsuneo, Kaneko Masanobu, On multiple L-values, Journal of the Mathematical Society of Japan, 10.2969/jmsj/1190905444, 56, 4, 967-991, vol. 56-4, 967--991, 2004.04, We formulate and prove reguralized double shuffle and derivation relations for multiple L-values. A description of principal part of a multiple L-function is also given.. |
| 19. |
Kaneko Masanobu, Koike Masao, On Modular Forms Arising from a Differential Equation of Hypergeometric Type, The Ramanujan Journal, 10.1023/A:1026291027692, 7, 1-3, 145-164, vol. 7, 145--164., 2003.09, Modular and quasimodular solutions of a specific second order differential equation in the upper-half plane, which originates from a study of supersingular j-invariants in the first author's work with Don Zagier, are given explicitly. Positivity of Fourier coefficients of some of the solutions as well as a characterization of the differential equation are also discussed.. |
| 20. |
Masanobu Kaneko, Nobushige Kurokawa, Masato Wakayama, A variation of Euler's approach to values of the Riemann zeta function, Kyushu J. Math., vol. 57-1, 175--192, 2003.03. |
| 21. |
KANEKO Masanobu, KUROKAWA Nobushige, WAKAYAMA Masato, A VARIATION OF EULER′S APPROACH TO VALUES OF THE RIEMANN ZETA FUNCTION, Kyushu Journal of Mathematics, 10.2206/kyushujm.57.175, 57, 1, 175-192, 2003.02. |
| 22. |
Masanobu Kaneko, On the zeros of certain modular forms, Number Theory and its Applications, 2, 193-197, 193--197., 1999.04. |
| 23. |
Masanobu Kaneko, Traces of singular moduli and the Fourier
coefficients of the elliptic modular function $j(\tau)$, CRM Proceedings and Lecture Notes, vol. 19, 173--176., 1999.04. |
| 24. |
T Arakawa, M Kaneko, Multiple zeta values, poly-Bernoulli numbers, and related zeta functions, NAGOYA MATHEMATICAL JOURNAL, 153, 189-209, 1999.03, We study the function
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and show that the poly-Bernoulli numbers introduced in our previous paper are expressed as special values at negative arguments of certain combinations of these functions. As a consequence of our study, we obtain a series of relations among multiple zeta values.. |
| 25. |
Masanobu Kaneko, Don Zagier, Supersingular j-invariants, hypergeometric series, and Atkin's
orthogonal polynomials, AMS/IP Studies in Advanced Mathematics, vol. 7, 97--126., 1998.04. |
| 26. |
M. Kaneko, Poly-Bernoulli numbers, J. de Theorie des Nombres de Bordeaux, 9, 199-206, 1997.04. |
| 27. |
T. Asai, M. Kaneko, H. Ninomiya, Zeros of certain modular functions and an application, Comment. Math. Univ. St. Pauli, 46, 1, 93-101, 1997.04. |
| 28. |
M. Kaneko, D. Zagier, A generalized Jacobi theta function and quasimodular forms, Progress in Math., 129, 165-172, 1995.06. |
| 29. |
M. Kaneko, A Recurrence Formula for the Bernoulli Numbers, Proc. of Japan Acad., 71A, 8, 192-193, 1995.06. |
| 30. |
M KANEKO, A GENERALIZATION OF THE CHOWLA-SELBERG FORMULA AND THE ZETA-FUNCTIONS OF QUADRATIC ORDERS, PROCEEDINGS OF THE JAPAN ACADEMY SERIES A-MATHEMATICAL SCIENCES, 66, 7, 201-203, 1990.09. |
| 31. |
M KANEKO, SUPERSINGULAR J-INVARIANTS AS SINGULAR MODULI MOD P, OSAKA JOURNAL OF MATHEMATICS, 26, 4, 849-855, 1989.12. |
| 32. |
Y. Ihara, M. Kaneko, A. Yukinari, On some properties of the universal power series for Jacobi sums, Advanced Studies in Pure Math., 12, 65-86, 1987.06. |
