{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,29]],"date-time":"2025-10-29T13:18:44Z","timestamp":1761743924685,"version":"build-2065373602"},"reference-count":13,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2014,4,10]],"date-time":"2014-04-10T00:00:00Z","timestamp":1397088000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>From Kemp [1], we have a family of confluent q-Chu- Vandermonde distributions, consisted by three members I, II and III, interpreted as a family of q-steady-state distributions from Markov chains. In this article, we provide the moments of the distributions of this family and we establish a continuous limiting behavior for the members I and II, in the sense of pointwise convergence, by applying a q-analogue of the usual Stirling asymptotic formula for the factorial number of order n. Specifically, we initially give the q-factorial moments and the usual moments for the family of confluent q-Chu- Vandermonde distributions and then we designate as a main theorem the conditions under which the confluent q-Chu-Vandermonde distributions I and II converge to a continuous Stieltjes-Wigert distribution. For the member III we give a continuous analogue. Moreover, as applications of this study we present a modified q-Bessel distribution, a generalized q-negative Binomial distribution and a generalized over\/underdispersed (O\/U) distribution. Note that in this article we prove the convergence of a family of discrete distributions to a continuous distribution which is not of a Gaussian type.<\/jats:p>","DOI":"10.3390\/axioms3020140","type":"journal-article","created":{"date-parts":[[2014,4,10]],"date-time":"2014-04-10T11:41:08Z","timestamp":1397130068000},"page":"140-152","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Continuous Stieltjes-Wigert Limiting Behaviour of a Family of Confluent q-Chu-Vandermonde Distributions"],"prefix":"10.3390","volume":"3","author":[{"given":"Andreas","family":"Kyriakoussis","sequence":"first","affiliation":[{"name":"Department of Informatics and Telematics, Harokopio University, 70 El. Venizelou str., Athens 17671, Greece"}]},{"given":"Malvina","family":"Vamvakari","sequence":"additional","affiliation":[{"name":"Department of Informatics and Telematics, Harokopio University, 70 El. Venizelou str., Athens 17671, Greece"}]}],"member":"1968","published-online":{"date-parts":[[2014,4,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1016\/j.jspi.2005.02.009","article-title":"Steady-state Markov chain models for certain q-confluent hypergeometric distributions","volume":"2005","author":"Kemp","year":"2005","journal-title":"J. Stat. Plan. Infer."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1016\/0097-3165(73)90038-1","article-title":"Central and local limit theorem applied to asymptotic enumeration","volume":"15","author":"Bender","year":"1973","journal-title":"J. Combin. Theory A"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1016\/0097-3165(77)90019-X","article-title":"Central and local limit theorems for the coefficients of polynomials of binomial type","volume":"23","author":"Canfield","year":"1977","journal-title":"J. Combin. Theory A"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1016\/0097-3165(90)90056-3","article-title":"Gaussian limiting distributions for the number of components in combinatorial structures","volume":"53","author":"Flajolet","year":"1990","journal-title":"J. Comb. Theory Ser. A"},{"key":"ref_5","unstructured":"Graham, R.L., Gr\u00f6tschel, M., and Lov\u00e1sz, L. (1995). Asymptotic Enumeration Methods, Elsevier Science Publishers."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1007\/s11009-011-9231-1","article-title":"On a q-analogue of the stirling formula and a continuous limiting behaviour of the q-Binomial distribution-numerical calculations","volume":"15","author":"Kyriakoussis","year":"2013","journal-title":"Methodol. Comput. Appl. Probabil."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Koekoek, R., Lesky, P.A., and Swarttouw, R.F. (2010). Hypergeometric Orthogonal Polynomilas and Their q-Analogues, Springer Monographs in Mathematics, Springer Verlag.","DOI":"10.1007\/978-3-642-05014-5"},{"key":"ref_8","unstructured":"Koekoek, R., and Swarttouw, R.F. The Askey-Scheme of Hypergeometric Orthogonal Polynomials and Its q-analogue. Available online: http:\/\/aw.twi.tudelft.nl\/\"koekoek\/askey.html."},{"key":"ref_9","unstructured":"Christiansen, J.S. (2004). Indeterminate Moment Problems within the Askey-Schem. [Ph.D. Thesis, Institute of Mathematical Sciences, University of Copenhagen]."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Ismail, M.E.H. (2004). Classical and Quantum Orthogonal Polynomials, Cambridge University Press.","DOI":"10.1017\/CBO9781107325982"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1016\/j.jspi.2005.02.006","article-title":"Moments of a class of discrete q-distributions","volume":"135","author":"Charalambides","year":"2005","journal-title":"J. Stat. Plan. Infer."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"169","DOI":"10.4310\/MAA.1994.v1.n2.a3","article-title":"The Nevanlinna parametrization for some indeterminate Stieltjes moment problems associated with birth and death processes","volume":"1","author":"Berg","year":"1994","journal-title":"Methods Appl. Anal."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1016\/S0096-3003(96)00054-9","article-title":"Some combinatorial identities associated with the Vandermonde convolution","volume":"84","author":"Gould","year":"1997","journal-title":"Appl. Math. Comput."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/3\/2\/140\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:10:10Z","timestamp":1760217010000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/3\/2\/140"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,4,10]]},"references-count":13,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2014,6]]}},"alternative-id":["axioms3020140"],"URL":"https:\/\/doi.org\/10.3390\/axioms3020140","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2014,4,10]]}}}