Discrete q-Distributions - download pdf or read online

By Charalambos A. Charalambides

ISBN-10: 1119119049

ISBN-13: 9781119119043

A self-contained learn of a few of the purposes and advancements of discrete distribution theory

Written by way of a widely known researcher within the box, Discrete q-Distributions features an prepared presentation of discrete q-distributions in accordance with the stochastic version of a chain of self sufficient Bernoulli trials. so as to preserve the e-book self-contained, the writer covers all the important easy q-sequences and q-functions.

The e-book starts with an creation of the notions of a q-power, a q-factorial, and a q-binomial coefficient and proceeds to debate the elemental q-combinatorics and q-hypergeometric sequence. subsequent, the booklet addresses discrete q-distributions with good fortune likelihood at a tribulation various geometrically, with fee q, both with the variety of earlier trials or with the variety of prior successes. additional, the ebook examines attention-grabbing stochastic versions with luck chance at any trial various geometrically either with the variety of trials and the variety of successes and provides neighborhood and international restrict theorems. Discrete q-Distributions also features:

  • Discussions of the definitions and theorems that spotlight key ideas and results
  • Several labored examples that illustrate the functions of the provided theory
  • Numerous workouts at various degrees of hassle that consolidate the strategies and effects in addition to supplement, expand, or generalize the results
  • Detailed tricks and solutions to all of the workouts in an appendix to assist less-experienced readers achieve a greater knowing of the content
  • An updated bibliography that incorporates the most recent developments and advances within the box and offers a collective resource for extra research
  • An Instructor’s suggestions handbook to be had on a spouse website

A distinct reference for researchers and practitioners in records, arithmetic, physics, engineering, and different technologies, Discrete q-Distributions is additionally a suitable textbook for graduate-level classes in discrete statistical distributions, distribution conception, and combinatorics.

 

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Extra info for Discrete q-Distributions

Example text

Proof. 26), and since ( ) − n2 −r n [t − r]n,q = q ([t]q − [r]q )([t]q − [r + 1]q ) · · · ([t]q − [r + n − 1]q ), we have ([t]q − [r]q )([t]q − [r + 1]q ) · · · ([t]q − [r + n − 1]q ) = n ∑ sq (n, k; r)[t]kq , k=1 for n = 2, 3, … The ith factor of the product of the left-hand side, fi (t; q) = [t]q − [r + i]q , i = 0, 1, … , n − 1, is a monomial with constant term −[r + i]q . Executing the multiplications, the kth-order power of [t]q is formed by multiplying the constant terms of any n − k factors {i1 , i2 , … , in−k }, out of the n factors {0, 1, … , n − 1}, together with the first-order terms [t]q of the remaining k factors.

4 A Gauss summation formula. Show that { [2m] ! ( )m n 1−q q [ ] ∑ , n = 2m, k n (−1) = [m]q2 ! 1+q k q 0, n = 2m + 1, k=0 for m a nonnegative integer. 5 Let x, y, and q be real numbers, with q ≠ 1, and let n be a positive integer. Using q-Vandermonde’s formula, show that n ∑ ( k (−1) q k+1 2 ) +k(y−n) k=0 [ ] [x]k,q [y]n,q n = , [x + y]n,q k q [x + y]k,q for x + y ≠ 0, 1, … , n − 1, and conclude that n ∑ ( (−1)k q k+1 2 )[ k=0 n k ] [x]k,q q [x + n]k,q = n ∑ k=0 ( (−1)k q [ = 1∕ k+1 2 )[ n k ] [ ] [ ] x x+n ∕ k q q k q ] x+n .

Kr ]q ! where kr = n − k1 − k2 − · · · − kr−1 , for ki = 0, 1, … , n, i = 1, 2, … , r and n = 0, 1, … , is called q-multinomial coefficient. The multivariate analogue of the Rogers-Szegö polynomial may be defined as [ ] ∑ n k k kr−1 Hn (t1 , t2 , … , tr−1 ; q) = t 1 t 2 · · · tr−1 , k1 , k2 , … , kr−1 q 1 2 where the summation is extended over all ki = 0, 1, … , n, i = 1, 2, … , r, such that k1 + k2 + · · · + kr−1 + kr = n. Derive its q-exponential generating function as ∞ ∑ un Hn (t1 , t2 , … , tr−1 ; q) = eq (u)eq (ut1 )eq (ut2 ) · · · eq (utr−1 ), [n]q !

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Discrete q-Distributions by Charalambos A. Charalambides


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