Recently, Simsek-Acikgoz [17] and Kim-Jang-Yi [9] have studied the q-extension of Bernstein polynomials. In the present paper, we suggest q-extensions of Bernstein polynomials of degree n which differ from the q-Bernstein polynomials of Simsek-Acikgoz [17] and Kim-Jang-Yi [9]. Using these q-Bernstein polynomials, we derive fermionic p-adic integral representations of several q-Bernstein-type polynomials. Finally, we investigate identities between q-Bernstein polynomials and q-Euler numbers.
The nonextensive statistical mechanics proposed by Tsallis is today an intense and growing research field. Probability distributions which emerges from the nonextensive formalism(q-distributions) have been applied to an impressive variety of problems. In particular, the role of q-distributions in the interdisciplinary field of complex systems has been expanding. Here, we make a brief review of q-exponential, q-Gaussian and q-Weibull distributions focusing some of their basic properties and recent applications. The richness of systems analyzed may indicate future directions in this field.
Abstract We give bijections on restricted growth functions and rook placements on stairstep Ferrers boards to show that the q -Stirling numbers of the second kind, described by Gould, arise as generating functions for various statistics on set partitions. We also describe a two-variable, p, q -Stirling number which is the generating function for the joint distribution of pairs of statistics.
T. F. Hatchette, Robert Hudson, Walter F. Schlech
et al.
In the spring of 1999 in rural Newfoundland, abortions in goats were associated with illness in goat workers. An epidemiologic investigation and a serologic survey were conducted in April 1999 to determine the number of infections, nature of illness, and risk factors for infection. Thirty-seven percent of the outbreak cohort had antibody titers to phase II Coxiella burnetii antigen >1:64, suggesting recent infection. The predominant clinical manifestation of Q fever was an acute febrile illness. Independent risk factors for infection included contact with goat placenta, smoking tobacco, and eating cheese made from pasteurized goat milk. This outbreak raises questions about management of such outbreaks, interprovincial sale and movement of domestic ungulates, and the need for discussion between public health practitioners and the dairy industry on control of this highly infectious organism.
The epidemiology of Q fever in Germany was examined by reviewing relevant studies since 1947 and by analyzing available surveillance data since 1962. The average annual Q fever incidence nationwide from 1979 to 1989 was 0.8 per million and from 1990 to 1999, 1.4 per million. The mean annual incidence from 1979 to 1999 ranged from a minimum of 0.1 per million in several northern states to 3.1 per million in Baden-Württemberg, in the South. We identified 40 documented outbreaks since 1947; in 24 of these sheep were implicated as the source of transmission. The seasonality of community outbreaks has shifted from predominantly winter- spring to spring-summer, possibly because of changes in sheep husbandry. The location of recent outbreaks suggests that urbanization of rural areas may be contributing to the increase in Q fever. Prevention efforts should focus on reducing sheep-related exposures, particularly near urban areas.
Abstract We present a notation for q-calculus, which leads to a new method for computations and classifications of q-special functions. With this notation many formulas of q-calculus become very natural, and the q-analogues of many orthogonal polynomials and functions assume a very pleasant form reminding directly of their classical counterparts. The first main topic of the method is the tilde operator, which is an involution operating on the parameters in a q-hypergeometric series. The second topic is the q-addition, which consists of the Ward–AlSalam q-addition invented by Ward 1936 [102, p. 256] and Al-Salam 1959 [5, p. 240], and the Hahn q-addition. In contrast to the the Ward–AlSalam q-addition, the Hahn q-addition, compare [57, p. 362] is neither commutative nor associative, but on the other hand, it can be written as a finite product. We will use the generating function technique by Rainville [76] to prove recurrences for q-Laguerre polynomials, which are q-analogues of results in [76]. We will also find q-analogues of Carlitz’ [26] operator expression for Laguerre polynomials. The notation for Cigler’s [37] operational calculus will be used when needed. As an application, q-analogues of bilinear generating formulas for Laguerre polynomials of Chatterjea [33, p. 57], [32, p. 88] will be found.