Non-central generalized q-factorial coefficients and q-Stirling numbers

Discrete q-distributions on Bernoulli trials with a geometrically varying success probability

“…The probability function (4.15) is readily deduced from (4.4) by setting q j =1Àq r + j À 1 =(1À q)[r + j À 1] q , j = 1,2,y, and using (2.25), with y ¼ 1Àq. Introducing the following explicit expression of the noncentral generalized q-Stirling numbers of the second kind (see Charalambides, 2004)…”

“…obtained the probability function of the size (width) of the chain decomposition of an acyclic random graph G n,p of n notes in the form (4.15). The probability functions (4.13) and (4.15) and the factorial moments (4.14) were deduced as a corollary of the corresponding expressions under a more general model in Charalambides (2004Charalambides ( , 2005a.…”

“…This model was introduced and studied by in connection with a variety of probabilistic problems in the algorithmic graph theory. Charalambides (2004), inspired by the q-distributions obtained by Crippa and Simon, introduced and studied the noncentral generalized q-factorial coefficients and used them to express these distributions. The probability function and factorial moments of the number X n of successes up to the nth trial are derived in the following corollary of Theorem 5.1.…”

“…The probability function (5.6) is readily deduced from (5.1) by setting a i = À(1 À q À a(r + i À 1) ), i= 1,2,y, b j =1À q b (j À 1) , j = 1,2,y,i, and using (2.28), with q replaced by q a . Note that, introducing the following explicit expression of the noncentral generalized q-factorial coefficients (see Charalambides, 2004) jC q Àa ðn,x; Às,ÀrÞj ¼ q Àað…”

On Asymptotics for the Signless Noncentral q‐Stirling Numbers of the First Kind

“…of the number of trials until the occurrence of the nth success are both expressed in terms of the signless noncentral q-Stirling numbers of the first kind if λ m,n = q m+k . These q-distributions have been studied by Charalambides [1] and Crippa et al [2].…”