Eisenstein series in Clifford analysis (Q2756068)

Finite dimensional generalizations of the complex function theory is lined out in two essential directions: Theory of several complex variables and the theory of a hypercomplex variable. The famous Swiss mathematician R. Fueter found in the 40s of the last century a function theory for the special case of one quaternionic variable. The Gent research group of R. Delanghe developed a corresponding function theory in real Clifford algebras and founded the so-called Clifford analysis. Clifford analysis have in recent years become increasingly important tools also in the analysis of partial differential equations and their applications in mathematical physics. For a successful work it is necessary to have classes of monogenic elementary functions. During the last years Sören Krausshar studied periodic Clifford monogenic functions. Roughly speaking monogenic functions belong to the kernel of a generalized Cauchy-Riemann system. This theory contains a higher-dimensional analogue of a Cauchy integral formula. His considerations cover the construction of such functions, their properties and the applications to analytic number theory. One of the main results of this theory is the generalization of trigonometric functions as tangens and cotangens in Clifford algebras. In order to define it he makes use of Eisenstein series NEWLINE\[NEWLINEG_n(z)= \sum_{(c,d)\in Z\times Z\{0, 0\}} {1\over (cz+d)^n} (n\geq 4;\text{Im} z>0).NEWLINE\]NEWLINE This procedure permits to avoid the use of products and quotients of monogenic functions. Note that it is well-known that a product of two monogenic functions never can be monogenic. The study of Eisenstein series is the common work of many famous mathematicians. Among them should be called C. L. Siegel who generalized Eisenstein series to several complex variables and R. Fueter who introduced non-analytical Eisenstein series and its connection to the Laplace-Beltrami operator attached to the upper half plane. Furthermore, he proves that any odd monogenic function which fulfils an associated double angle formula has to be the hypercomplex cotangens function. The cotangens function can be determined by NEWLINE\[NEWLINE{\mathcal C}{\mathcal O}{\mathcal T}^{(p)}: =\sum_{\omega \in\Omega_p} q_0(z+\omega) (1\leq p\leq k-1).NEWLINE\]NEWLINE Here \(\Omega_p\) defines a suited \(p\)-dimensional lattice. Similarely generalizations of the tangens, secans and cosecans function as periodic monogenic meromorphic function with a finite number of isolated zeros could be obtained. Completely analog to the classical complex case a generalized Riemann zeta function is deduced on the lattice \(\Omega_p\). In particular one has to accentuate that the dense connection between the generalized Riemann zeta function and Epstein's zeta function. Its construction makes use of the so-called \(q_n\)-functions which are defined as NEWLINE\[NEWLINEq_n(z-a)= {\partial^{|n|} \over\partial x^n} {\overline {z-a}\over |z-a|^{k+1}}NEWLINE\]NEWLINE with \(z_i=x_i- e_ix_0\). This representation permits to obtain a new recursion formula of derivatives of the Cauchy kernel. Stronger as up to now known estimates of such kernels follow. In generalization of results of Dixon, Fueter and Ryan S. Krausshar succeed in the introduction of generalized elliptic functions, which can be deduced from the generalized Weierstrass \(\zeta\)-function by differentiation. The latter one is by Sören Krausshar defined by: NEWLINE\[NEWLINE{\mathcal P}(z):= q_0(z-a)- q_0(z-b)+ \sum_{ \omega \in\Omega_{k+1} \setminus\{0\}} \biggl [\bigl( q_0(z-a+ \omega)-q_0(z-b+\omega) \bigr)- \bigl(q_0(\omega -a)-q_0 (\omega-b) \bigl)\biggr]NEWLINE\]NEWLINE where \(a,b \in {\mathcal A}_{k+1}\) and \(a-b\notin \Omega_{k+1}\). Results which are obtained by R. Fueter in the quaternionic case could also be proved in the case of a real Clifford algebra. It should be noticed that the method is completely different to Fueter's approach. The results show a lot of similarity between the two-dimensional case and the case of Clifford valued functions. S. Krausshar lines out possible applications of his results in the treatment of the modul problem, of the surface representation in the differential geometry and in the analytic number theory.