On an attempt to introduce a notion of bounded index for the Fueter regular functions of the quaternionic variable

Abstract

There is introduced a concept of index for the Fueter regular function of the quaternionic variables. There are considered three approaches (Fueter, Sudbery and Mariconda) constructing the Fueter regular function from a holomorphic function of complex variable. Using Mariconda's approach there are constucted some analogs of such elementary functions as the exponent, the sine and the cosine. For the Mariconda analogs we proved that they have bounded index and their indices equal 1, 2, 2, respectively. Using recent results on sum of entire functions whose derivatives are of bounded index it is established that the Fueter regular function constructed by Mariconda's approach is of bounded index, if the derivatives of its addends have bounded index. Also there was examined a function of the form $H(q)=f_1(x_0+ix_1)+jf_2(x_2+ix_3)$, where $f_1$ and $f_2$ are entire functions of complex variable. For the function $H$ it is proved its Fueter regularity and index boundedness if the first order derivatives of $f_1$ and $f_2$ have bounded index. Moreover, the index of the function $H$ does not exceed the maximum of indices of the functions $f'_1$ and $f'_2$ increased by $1$.