  We construct a family of triangle-ladder diagrams which may be calculated by
making use of Belokurov-Usyukina loop reduction technique in d = 4 -2e
dimensions. The main idea of the approach proposed in the present paper
consists in generalization of this loop reduction technique existing in d = 4
dimensions. The recursive formula relating the result for L-loop triangle
ladder diagram of this family and the result for (L-1)-loop triangle ladder
diagram of the same family is derived. Since the method proposed in the present
paper combines analytic and dimensional regularizations, at the end of the
calculation we have to remove the analytic regularization by taking the double
uniform limit in which the parameters of the analytic regularization are
vanishing. In this limit on the left hand side of the recursive relations we
obtain in the position space the diagram in which the indices of the rungs are
1 and all the other indices are 1-e. Fourier transformation of the diagrams of
this type gives the momentum space diagrams which have indices of the rungs
equal to 1-e and all the other indices 1. Via conformal transformation of the
dual space image of this momentum space representation we relate such a family
of the triangle ladder momentum diagrams to a family of the box ladder momentum
diagrams in which the indices of the rungs are equal to 1-e and all the other
indices are 1. Since any diagram from this family can be reduced to one-loop
diagram, the proposed generalization of the Belokurov-Usyukina loop reduction
technique to non-integer number of dimensions allows us to calculate this
family of box-ladder diagrams in the momentum space explicitly in terms of
Appell's hypergeometric function F_4 without expanding in powers of parameter e
in an arbitrary kinematic region in the momentum space.
