Hyperbolic beta integrals are analogues of Euler’s beta integral in which the role of Euler’s gamma function is taken over by Ruijsenaars’ hyperbolic gamma function. They may be viewed as (q, q̃ )-bibasic analogues of the beta integral in which the two bases q and q̃ are interrelated by modular inversion, and they entail q-analogues of the beta integral for |q| = 1. The integrals under considerat...

In Euclidean geometry, it is well-known that the k-order Voronoi diagram in R can be computed from the vertical projection of the k-level of an arrangement of hyperplanes tangent to a convex potential function in R: the paraboloid. Similarly, we report for the Klein ball model of hyperbolic geometry such a concave potential function: the northern hemisphere. Furthermore, we also show how to bui...

Motivated by the recent work on symmetric domains, this article investigates certain features of domain which are caused secant hyperbolic functions. Geometric characteristics analytic functions associated with discussed, include inclusion results, structural formula, sharp radii results such as radius starlikeness and convexity order α. It also finds a for ratios Euler numbers.

Hyperbolic beta integrals are analogues of Euler’s beta integral in which the role of Euler’s gamma function is taken over by Ruijsenaars’ hyperbolic gamma function. They may be viewed as (q, q̃ )-bibasic analogues of the beta integral in which the two bases q and q̃ are interrelated by modular inversion, and they entail q-analogues of the beta integral for |q| = 1. The integrals under considerat...

In the paper, authors find series expansions and identities for positive integer powers of inverse (hyperbolic) sine tangent, composite incomplete gamma function with hyperbolic sine, in terms first kind Stirling numbers, apply a newly established expansion to derive closed-form formula specific partial Bell polynomials representation generalized logsine function, deduce combinatorial involving...