q-Completely monotonic and q-Bernstein functions
نویسندگان
چکیده
منابع مشابه
COMPLETELY MONOTONIC FUNCTIONS INVOLVING THE GAMMA AND q-GAMMA FUNCTIONS
We give an infinite family of functions involving the gamma function whose logarithmic derivatives are completely monotonic. Each such function gives rise to an infinitely divisible probability distribution. Other similar results are also obtained for specific combinations of the gamma and q-gamma functions.
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In the paper, a function relating to the q -trigamma function is proved to be completely monotonic. As by-products, two functions relating to the logarithmic function are also proved to be completely monotonic. Mathematics subject classification (2010): Primary 33D05; Secondary 26A48.
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(1.1) f(aq + b) = f(aq) + f(b) or f(aq + b) = f(aq)f(b) for all r ≥ 0, a ≥ 0 and 0 ≤ b < q. Note that these equations force f(0) = 0 or f(0) = 1 respectively. These functions are called q-additive and q-multiplicative, respectively. It is easy to see that the functional equations imply that these functions are defined for all integers, when the values f(aq) are known for 1 ≤ a ≤ q − 1 and all r...
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If, in addition, f(x) is continuous at x = a, then it is called completely monotonic over [a, b), with similar definitions for (a, b] and [a, b]. A function is absolutely monotonic if all its derivatives are non-negative. A detailed study of these concepts can be found, for example, in [11, Chapter IV]. Our chief concern here is for the standard case in which a = 0, b = + 00. For this, S. N. Be...
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In this paper, we introduce the notion of q-quasiadditivity of arithmetic functions, as well as the related concept of q-quasimultiplicativity, which generalise strong q-additivity and -multiplicativity, respectively. We show that there are many natural examples for these concepts, which are characterised by functional equations of the form f(qk+ra+b) = f(a)+f(b) or f(qk+ra+b) = f(a)f(b) for al...
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ژورنال
عنوان ژورنال: Journal of Applied Mathematics, Statistics and Informatics
سال: 2014
ISSN: 1336-9180
DOI: 10.2478/jamsi-2014-0012