Radial matrix elements with Dirac's functions in the theory of fine and hyperfine structures of hydrogen-like systems

Investigating the effects of fine and hyperfine structures of hydrogen spectrum there arises practical necessity to evaluate the different matrix elements with Dirac's radial functions f and g; usually such matrix elements being evaluated approximately. The article deals with the study of relativistic matrix elements by a new method, i.e. with the help of Appell's hypergeometrical functions F2(x,y) and their new properties. Such approach allows: (1) to get exact analytical expressions for these matrix elements by means of Appell's F2 (and Clausen's 3F2) functions; (2) to reveal latent symmetry of diagonal matrix elements the above symmetry is connected with the property of Appell's function F2(1,1) mirror-like symmetry; (3) to prove that the diagonal and off-diagonal matrix elements in a pole-domain of Euler's gamma-function are given exactly analytically by means of nonorientable series of type F2(x,y). The examples for the evaluation of diagonal matrix elements for the K- and L-shells and also of dipole matrix elements for Lyman's series are obtained