Discrete probability distributions generated by the generalized STER summation (Q1881442)

Given a discrete probability distribution \(\{P_j\,:\,j=0,1,\dots\}\), the STER distribution is given by \[ Q_i=A\,\sum_{j=i+1}^{\infty} P_j/j,\qquad i=0,1,\dots, \] where \(A\) is a normalizing constant. Here the author studies the following generalization: \[ Q_i(k,l)=A(k,l)\,\sum_{j=i}^{\infty} P_{j+k}(j+l)^{-1},\quad i,k=0,1,\dots;\;l=1,2,\dots. \] To each parent distribution \(\{P_j\}\) there corresponds a unique descendant \(\{Q_i(k,l)\}\); the converse is not true. Lemma 2.1 gives a recursive formula for the computation of \(Q_i(k,l)\). In the final section it is proved that the terms of the Yule distribution (and only those) are invariant with respect to the given summation.