  The hyperbolic secant distribution has several generalizations with
applications in finance. In this study, we explore the dual geometric structure
of one such generalization, namely the beta-logistic distribution. Recent
findings also interpret Bernoulli and Euler polynomials as moments of specific
random variables, treating them as special cases within the framework of the
beta-logistic distribution. The current study also uncovers that the
beta-logistic distribution admits an $\alpha$-parallel prior for any real
number $\alpha$, that has the potential for application in geometric
statistical inference.
