Let (αn(a, k), βn(a, k)) be a WP-Bailey pair. Assuming the limits exist, let (α n(a), β ∗ n(a)) = lim k→1 ( αn(a, k), βn(a, k) 1− k ) be the derived WP-Bailey pair. By considering a particular limiting case of a transformation due to George Andrews, we derive some transformation and summation formulae for derived WP-Bailey pairs. We then use the formula to derive new identities for various thet...

A vector space is a group of objects that closed under finite addition and scalar multiplication. This paper discusses multiplication binomial coefficients defined in combinatorial geometric series. The series with derived from the multiple summations idea can enable scientific researchers to solve real world problems.

conclusions large amounts of preclinical studies have been made of stem cells and there is an increasing interest to perform these studies within the human population but preclinical studies are insufficient; therefore, much more and efficient studies should be conducted to evaluate the efficacy and safety of stem cells. context musculoskeletal injuries may be painful, troublesome, life limitin...

An exact series expansion for the field radiated by a monopole ring source with angular variation in source strength is derived from a previously developed expression for the field from a finite disk. The derived series can be used throughout the field, via the use of a reciprocity relation, and can be readily integrated to find the field radiated by arbitrary circular sources of finite extent,...

Citation: Seehaus TC, Marinsek S, Skvarca P, van Wessem JM, Reijmer CH, Seco JL and Braun MH (2016) Dynamic Response of Sjögren Inlet Glaciers, Antarctic Peninsula, to Ice Shelf Breakup Derived from Multi-Mission Remote Sensing Time Series. Front. Earth Sci. 4:66. doi: 10.3389/feart.2016.00066 Dynamic Response of Sjögren Inlet Glaciers, Antarctic Peninsula, to Ice Shelf Breakup Derived from Mul...

In this paper a useful formulation for the multivariate Rayleigh probability density function (pdf) with an exponential form for its correlation matrix is presented. Furthermore, the corresponding multivariate Rayleigh cumulative distribution function (cdf) is derived in a nested infinite series representation, which converges rapidly and can be efficiently used in many wireless applications wi...

Low-temperature series have been derived for the q-state Potts model on the Kagomé lattice (q = 3 and 4) and for the 3-state honeycomb lattice. The series are derived by the finitelattice method, with many additional terms obtained by noting the structure of the correction terms. Accurate estimates of the critical points are found, which confirm exact, but not rigorous results in the case of th...

tyrosinase is a multifunctional oxidase that is widely distributed in nature. it is a key enzyme in melanin biosynthesis and is involved in determining the color of mammalian skin and hair. in addition it is responsible for the undesirable enzymatic browning that occurs in plant-derived foods, limiting the shelf -life of fresh-cut products with the resultant economic loss.  in recent years ther...

This paper discusses an abelian group, also called a commutative under addition of the binomial coefficients combinatorial geometric series. The series is derived from multiple summations coefficient for each term in refers to coefficient. idea can enable scientific researchers solve real life problems.