A Note on the New Extended Beta Function with Its Application

The purpose of this research is to provide a systematic review of a new type of extended beta function and hypergeometric function using a confluent hypergeometric function, as well as to examine various belongings and formulas of the new type of extended beta function, such as integral representations, derivative formulas, transformation formulas, and summation formulas. In addition, we also investigate extended Riemann–Liouville (R-L) fractional integral operator with associated properties. Furthermore, we develop new beta distribution and present graphically the relation between moment generating function and ℓ .

Some Results of Extended Beta Function and Hypergeometric Functions by Using Wiman’s Function

The main aim of this research paper is to introduce a new extension of the Gauss hypergeometric function and confluent hypergeometric function by using an extended beta function. Some functional relations, summation relations, integral representations, linear transformation formulas, and derivative formulas for these extended functions are derived. We also introduce the logarithmic convexity and some important inequalities for extended beta function.

Curvature-dimension conditions under time change

We derive precise transformation formulas for synthetic lower Ricci bounds under time change. More precisely, for local Dirichlet forms we study how the curvature-dimension condition in the sense of Bakry–Émery will transform under time change. Similarly, for metric measure spaces we study how the curvature-dimension condition in the sense of Lott–Sturm–Villani will transform under time change.

Equivalency in joint signatures for binary/multi-state systems of different sizes

The joint signatures of binary-state and multi-state (semi-coherent or mixed) systems with i.i.d. (independent and identically distributed) binary-state components are considered in this work. For the comparison of pairs of binary-state systems of different sizes, transformation formulas of their joint signatures are derived by using the concept of equivalent systems and a generalized triangle rule for order statistics. Similarly, for facilitating the comparison of pairs of multi-state systems of different sizes, transformation formulas of their multi-state joint signatures are also derived. Some examples are finally presented to illustrate and to verify the theoretical results established here.

Corollaries and multiple extensions of Gessel and Stanton hypergeometric summation formulas

We find some new simple hypergeometric formulas in the footsteps of the important article by Gessel and Stanton. These are multiple reduction formulas, multiple summation formulas, as well as multiple transformation formulas for special Kampé de Fériet functions and Appell functions. The hypergeometric summation formulas have special function arguments in Q and parameter values in N or C. The proofs use Pfaff-Kummer transformation, Euler transformation, or an improved form of Slater reversion.

On Decomposition Formulas Related to the Gaussian Hypergeometric Functions in Three Variables

In this paper, by using certain inverse pairs of symbolic operators introduced by Choi and Hasanov in 2011, we establish several decomposition formulas associated with the Gaussian triple hypergeometric functions. Some transformation formulas for these functions have also been obtained.

Transformation formulas of incomplete hypergeometric functions via fractional calculus operators

The desire for present article is to derive from the application of fractional calculus operators a transformation that expresses a potentially useful incomplete hypergeometric function in various forms of a countable sum of lesser-order functions. Often listed are numerous (known or new) specific cases and implications of the findings described herein

On the Direct Limit from Pseudo Jacobi Polynomials to Hermite Polynomials

In this short communication, we present a new limit relation that reduces pseudo-Jacobi polynomials directly to Hermite polynomials. The proof of this limit relation is based upon 2F1-type hypergeometric transformation formulas, which are applicable to even and odd polynomials separately. This limit opens the way to studying new exactly solvable harmonic oscillator models in quantum mechanics in terms of pseudo-Jacobi polynomials.