×

On the \(k\)-gamma \(q\)-distribution. (English) Zbl 1202.05012

The \(q\)-analogue of the Pochhammer \(k\)-symbol is given by \([t]_{n,k}=\prod_{j=0}^{n-1}[t+jk]_q\), where \([t]_q=\frac{1-q^t}{1-q}\). In this paper, the authors provide combinatorial and probabilistic interpretations for the the \(q\)-analogue of the Pochhammer \(k\)-symbol. Moreover, they study the \(q\)-analogue of the \(k\)-gamma distribution \(\Gamma_{q,k}(t)\) by introducing \(q\)-analogues of the Mellin transform. In particular, they show that \(\Gamma_{q,k}(t)\) is given by \[ \Gamma_{q,k}(t)=(1-q)^{1-t/k}\sum_{n\geq0}\frac{q^{kn(n+1)/2}}{(1-q^{kn+t})(q^k-1)^n[n]_{q^k}!}. \]

MSC:

05A30 \(q\)-calculus and related topics
60E05 Probability distributions: general theory
05C30 Enumeration in graph theory
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Andrews G., Askey R., Roy R., Special functions, Cambridge University Press, Cambridge, 1999; · Zbl 0920.33001
[2] Callan D., A combinatorial survey of identities for the double factorial, preprint available at http://arxiv.org/abs/0906.1317;
[3] Cheung P., Kac V, Quantum Calculus, Springer-Verlag, Berlin, 2002;
[4] De Sole A., Kac V., On integral representations of q-gamma and q-beta functions, Atti. Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei 9. Mat. Appl., 2005, 16, 11-29; · Zbl 1225.33017
[5] Díaz R., Pariguan E., On hypergeometric functions and Pochhammer k-symbol, Divulg. Mat., 2007, 15, 179-192; · Zbl 1163.33300
[6] Díaz R., Pariguan E., On the Gaussian q-distribution, J. Math. Anal. Appl., 2009, 358, 1-9 http://dx.doi.org/10.1016/j.jmaa.2009.04.046; · Zbl 1166.60007
[7] Diaz R., Pariguan E., Super, Quantum and Non-Commutative Species, Afr. Diaspora J. Math, 2009, 8, 90-130; · Zbl 1239.16001
[8] Díaz R., Teruel C, q, k-generalized gamma and beta functions, J. Nonlinear Math. Phys., 2005, 12, 118-134 http://dx.doi.org/10.2991/jnmp.2005.12.1.10; · Zbl 1075.33010
[9] George G., Mizan R., Basic Hypergeometric series, Cambridge Univ. Press, Cambridge, 1990; · Zbl 0695.33001
[10] Gessel I., Stanley R, Stirling polynomials, J. Combin. Theory Ser. A, 1978, 24, 24-33 http://dx.doi.org/10.1016/0097-3165(78)90042-0;
[11] Kokologiannaki C.G., Properties and Inequalities of Generalized k-Gamma, Beta and Zeta Functions, Int. J. Contemp. Math. Sciences, 2010, 5, 653-660; · Zbl 1202.33003
[12] Kuba M., On Path diagrams and Stirling permutations, preprint available at http://arxiv4.library.cornell.edu/abs/0906.1672;
[13] Mansour M., Determining the k-Generalized Gamma Function Γk(x) by Functional Equations, Int. J. Contemp. Math. Sciences, 2009, 4, 1037-1042; · Zbl 1186.33002
[14] Zeilberger D., Enumerative and Algebraic Combinatorics, In: Gowers T. (Ed.), The Princeton Companion to Mathematics, Princeton University Press, Princeton, 2008;
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.