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Highly accurate calculation of the auxiliary functions of the fourth order and five-body integrals

General theory and analytical formulae suitable for numerical calculations of the fourth order auxiliary functions A4(k, ℓ, m, n, a, b, c, d) are presented. The auxiliary functions of the fourth order A4(k, ℓ, m, n, a, b, c, d) are of great interest in various applications, e.g., they can be used to solve a number of five- and many-body problems in atomic, molecular and nuclear physics. The obtained analytical formulae have been extensively tested in actual calculations of the A4(k, ℓ, m, n, a, b, c, d) functions and one-centre five-body integrals I5. Numerical computations of the ground bound state in the Be atom with the use of Hylleraas variational expansion are briefly discussed. The one-centre, five-body integrals I5 can also be useful to determine the general five-body integrals .