By Ryuei Nishii, Shin-ichiro Ei, Miyuki Koiso, Hiroyuki Ochiai, Kanzo Okada, Shingo Saito, Tomoyuki Shirai
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Extra resources for A Mathematical Approach to Research Problems of Science and Technology: Theoretical Basis and Developments in Mathematical Modeling
1]) at negative integer points. 22 M. Wakayama We remark that the “residual part” of a differential Eisenstein series is in general given by a rational function in τ , whence it can not be handled in a framework of Eichler integrals. Moreover, one should note that only w4 (t) one can give its explicit expression by a sum of two such differential Eisenstein series . Furthermore, to understand the structure, especially the dimension of a space of residual modular forms, it is important to consider the Eichler cohomology groups  associated with several (2)-modules made by a set of certain functions on the Poincaré upper half plane, such as the space (field) of rational functions C(τ ), the space of holomorphic/meromorphic functions with some decay condition at the infinity (cusps), etc.
T. Ichinose, M. Wakayama, Special values of the spectral zeta function of the non-commutative harmonic oscillator and confluent Heun equations. Kyushu J. Math. 59, 39–100 (2005) Remarks on Quantum Interaction Models 33 13. T. Ichinose, M. Wakayama, On the spectral zeta function for the noncommutative harmonic oscillator. Rep. Math. Phys. 59, 421–432 (2007) 14. K. Kimoto, M. Wakayama, Elliptic curves arising from the spectral zeta function for noncommutative harmonic oscillators and 0 (4)-modular forms, in Proceedings of the Conference on L-functions, ed.
2 Heun Differential Operators In this section, we follow the results from . Recall the operator R ∈ U (sl2 ). Then, one observes ⎥ ⎤ 1 2 −2 − 2 coth 2κ) θz + a (R) = (z + z 2 2(εν)2 1 2ν ⎦ 1 (θz + − ν) + , + (a − )(z 2 − z −2 ) + 2 sinh 2κ 2 sinh 2κ where θz = zϕz . Hence, conjugating by z a−1 one obtains the following lemma. 28 M. Wakayama Lemma 5 For each integer a one has ⎤ 1 z −a+1 a (R)z a−1 = (z 2 + z −2 − 2 coth 2κ)(θz + a − ) 2 2ν ⎦ 1 2(εν)2 1 2 −2 (θz + a − − ν) + . + (a − )(z − z ) + 2 sinh 2κ 2 sinh 2κ Furthermore, notice that the operators a (H ), a (E) and a (F) are invariant under the symmetry z ≤ −z.
A Mathematical Approach to Research Problems of Science and Technology: Theoretical Basis and Developments in Mathematical Modeling by Ryuei Nishii, Shin-ichiro Ei, Miyuki Koiso, Hiroyuki Ochiai, Kanzo Okada, Shingo Saito, Tomoyuki Shirai