Integral transforms and probability distributions involving a generalized hypergeometric function

Various extensions of the beta function together with their associated extended hypergeometric and confluent hypergeometric functions have been introduced and investigated. In this paper, using the very recently contrived extended beta function, we aim to introduce an extension F v p , q ; λ ; σ , τ u {{}_{u}F_{v}^{p,q;\lambda;\sigma,\tau}} of the generalized hypergeometric function F v u {{}_{u}F_{v}} and investigate certain classes of transforms and several identities of a generalized probability distribution involving this extension. In fact, we present some interesting formulas of Jacobi, Gegenbauer, pathway, Laplace, and Legendre transforms of this extension multiplied by a polynomial. We also introduce a generalized probability distribution to investigate its several related properties. Further, we consider some special cases of our main results with an argument about the derived process of a known result.

Bernard Lonergan e a Probabilidade Emergente

In Insight, an essay on human understanding, (1957, 1st edition) Lonergan presents a heuristic model of emerging probability in order to define, explain and extract norms from the dynamism common to all nature, including human nature, a dynamism that mirrors the reality of intellection. Continuity between different levels of nature discloses a directed, upward, but indeterminate dynamism of the emerging generalized probability. In addition to the ethical consequences that he elaborates, Lonergan remains in an open hermeneutic framework, beyond being proportionate to discursive reason; he opens the way for a surprising final manifestation of this universal dynamism through what he calls transcendent conjugated forms of generalized probability emerging – faith, hope and charity – that are proposed to human freedom.

Quantum mechanics as a generalized probability theory

When quantum mechanics is understood as a new generalized theory of probability - to be called the quantum probability theory - mysteries and controversies regarding quantum mechanics are dissolved. In the classical probability theory, that a measurement of some system requires an additional measurement apparatus is of insignificant importance - in the quantum probability theory, this comes to change. For one central single reason around a particular classical probability equation, the generalized probability view has not gained much traction, despite the fact that this essentially echoes (and provides logical underpinnings of) the conventional wisdom that `quantum mechanics just works as it is.' A classical probability axiom is just an initial intuition - there is no reason why we have to dogmatically cling onto axioms that can clearly be generalized. Issues with the principle of indifference in the classical probability theory are emphasized, along with the quantum reconstruction project of deriving quantum mechanics from epistemic requirements and potential quantum gravity consequences from the principle of maximum entropy.