Some generalizations of the Euler-Knopp transformation

Abstract

Summary. The purpose of this paper is to construct a generalization of the Euler-Knopp transformation. Using this, one may recover previously known transformations, derive new transformations useful for numerical calculations and derive generating functions and other formulas of theoretical interest involving well-known functions.

The numerical performance of Tricomi's formula for inverting the Laplace transform

“…Theorem 3.3 and 3.11 are examples for the application of the transformed series (2.1). Concerning the generality of this approach we observe that the proof of the theorems consists in finding regions of the complex s-plane where the value Popt for series (2.1) is fully determined by certain "key singularities" of f(s) = S(F) and in showing that nothing changes if in these regions there are more singularities of f. Then we note that the cases of Theorems 3.3 and 3.11 are based on two specific expressions for Popt given in [4,Sect. 4], where more general situations are described.…”

“…4 [4,8,12], where it is shown that the value Popt may be determined when the locations of the singularities of the function represented by the series to be transformed are known.…”

Some Polynomial Systems Associated with a Certain Family of Differential Operators

“…Recently, several authors (see, for example, Gabutti and Lyness [3], Mathis and Sismondi [8], and Srivastava [19]) investigated many different families of generating functions associated with the Stirling numbers S n k defined by (3.1). We choose to recall here one of Srivastava's general results on these families of generating functions as Then, in terms of the Stirling numbers S n k defined by (3.1), the following family of generating functions,…”

The entrainment theorem for wall driven boundary layer flows

“…The first condition (17) has been obtained from f (∞) = 0, and the second one from f (0) = f w and f (0) = 1. The wall shear stress S is obtained in this approach as…”

“…(26) and (27) can be accelerated with the aid of the classical Euler-Knopp type series transformation [16]. In the present calculations, following the work of [17], an improved form of the Euler-Knopp transformation has been used. Applied, e.g., to the series (21), the transformation of Gabutti and Lyness gives…”