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In this paper, simple analytic expressions for the radial functions r ll1l2( f (r)) and Glnl1l2( f (r)), which appear in the expressions for 𝒴m1l1(∇) fl2(r) Ym2l2(r̂) and ∇2n𝒴m1l1(∇) fl2(r) Ym2l2(r̂) when expressed as linear combinations of Yml(r̂), where 𝒴m1l1(∇) is obtained from the solid harmonic 𝒴m1l1(r) =r l1Ym1l1(r̂), are derived by replacing x, y, and z in its representation as a polynomial of degree l1 by ∂/∂x, ∂/∂y, and ∂/∂z, respectively. Writing r ll1l2( f (r)) =r ll1l20(r) f (r) and Glnl1l2( f (r)) =Glnl1l20(r) f (r), the expression for r ll1l20(r) (Glnl1l20(r)) is found to be (2/r)l1((2/r)l1+2n) times a product of commuting factors containing the single operator 1/2 r(d/dr) and not two noncommuting operators r and (1/r)(d/dr) in the many equivalent previously obtained results which are thus synthesized in this approach. Also, expressions for these operators manifest the symmetries in the problem. In addition, a simple connection Glnl1l20(r) =r ll1+2nl20(r) between these two operators and the corresponding radial functions Glnl1l2( f (r)) and r ll1l2( f (r)) are found.

It is important to classify covering subgroups of the fundamental group of a topological space using their topological properties in the topologized fundamental group. In this paper, we introduce and study some topologies on the fundamental group and use them to classify coverings, semicoverings, and generalized coverings of a topological space. To do this, we use the concept of subgroup topology on a group and discuss their properties. In particular, we explore which of these topologies make the fundamental group a topological group. Moreover, we provide some examples of topological spaces to compare topologies of fundamental groups.

Starting from an exact recursion relation of the form A(q,N)=Σi,jA(q−i,N−j)Cij, we present a recursion method for calculating the various moments μN(m)=Σq=0NqmA(q,N). These results are applied to obtain the occupation statistics for occupation of a 2 × N array by parallel and not necessarily parallel dumbbells.