When r/a≊0, Löwdin’s α function (1/r)αℓ(fLM‖a,r) is expressed as (1/r)αℓ ( fLM‖a,r) =2γ(LM‖ℓ )aL−1∑∞n=ℓ [∑imaxi=ℓ+M[hn,2n−i (LM‖ℓ )/(2n−i)!]a2n−ig(2n−i) (a)](r/a)2n−ℓ, where a is a separation between a new center placed on the origin and the old center located at a given point on the z axis that is the origin of the coordinate system defining a function f (R)YML(Θ,Φ) to be expanded around the new center [ f(R) is an arbitrary radial function, and YML(Θ,Φ), a complex spherical harmonics], and r is a distance from the new center. Here, imax=min{2n,2(L+ℓ )}, and the function g(j)(a) is expressed as g(j)(a)=[(d/dR)j ( f(iR)/RL−1)]R=a. The closed form of the coefficients hn,2n−i(LM‖ℓ ) is given by Eq. (27) in J. Math. Phys. 26, 3193 (1985) (referred to as Part II). Since they are expressed in terms of the different coefficients bKk(LM‖ℓ ) independent of the form of f(R), hn,2n−i(LM‖ℓ ) constitute a set of universal constants as do bKk(LM‖ℓ ). The explicit expression for the factor γ(LM‖ℓ ) in front of the summation symbol is given by Eq. (2.9) in J. Math. Phys. 25, 1133 (1984). Hereafter the coefficients expressed by hn,2n−i(LM‖ℓ )/(2n−i)! are symbolized by In,2n−i(LM‖ℓ ). Although they are expressed in a complex form with a triple sum, that expression can be reduced to two considerably simple forms by changing the summation indices to others by using the addition theorem for binomial coefficients and the condition for the sum in this theorem to vanish. Moreover, in the two special cases that i=ℓ+M and i=2(L+ℓ ) or 2(L+ℓ )−1, those two expressions are reduced to two single-term forms, respectively. Introducing into the expression for In,2n−i(LM‖ℓ ) in terms of bKk(LM‖ℓ ) the recursion formula for bKk(LM‖ℓ ) in only M given by Eq. (24) in Part II, leads to the recursion formula for In,2n−i(LM‖ℓ ) with respect to M. On the other side, the recursion formulas for In,2n−i(LM‖ℓ ) with M=L and ℓ as to i are obtained through a skillful manipulation. By connecting those recursion formulas for In,2n−i(LM‖ℓ ) to each other, we can obtain a procedure for calculating In,2n−i(LM‖ℓ ) successively. As the result of actual performance of the procedure, all the formulas expressing In,2n−i(LM‖ℓ ) with restriction 0≤M≤min{L,ℓ }≤max{L,ℓ }≤2 are presented as functions of the parameter n in a table.